Tianbao Yang

AAAI Conference 2026 Conference Paper

CyPortQA: Benchmarking Multimodal Large Language Models for Cyclone Preparedness in Port Operation

• Chenchen Kuai
• Chenhao Wu
• Yang Zhou
• Bruce Wang
• Tianbao Yang
• Zhengzhong Tu
• Zihao Li
• Yunlong Zhang

As tropical cyclones intensify and track forecasts become increasingly uncertain, U.S. ports face heightened supply-chain risk under extreme weather conditions. Port operators need to rapidly synthesize diverse multimodal forecast products, such as probabilistic wind maps, track cones, and official advisories, into clear, actionable guidance as cyclones approach. Multimodal large language models (MLLMs) offer a powerful means to integrate these heterogeneous data sources alongside broader contextual knowledge, yet their accuracy and reliability in the specific context of port cyclone preparedness have not been rigorously evaluated. To fill this gap, we introduce CyPortQA, the first multimodal benchmark tailored to port operations under cyclone threat. CyPortQA assembles 2,917 real-world disruption scenarios from 2015 through 2023, spanning 145 U.S. principal ports and 90 named storms. Each scenario fuses multi-source data (i.e., tropical cyclone products, port operational impact records, and port condition bulletins) and is expanded through an automated pipeline into 117,178 structured question–answer pairs. Using this benchmark, we conduct extensive experiments on diverse MLLMs, including both open-source and proprietary model. MLLMs demonstrate great potential in situation understanding but still face considerable challenges in reasoning tasks, including potential impact estimation and decision reasoning.

TMLR Journal 2026 Journal Article

Single-loop Algorithms for Stochastic Non-Convex Optimization with Weakly-Convex Constraints

• Ming Yang
• Gang Li
• Quanqi Hu
• Qihang Lin
• Tianbao Yang

Constrained optimization with multiple functional inequality constraints has significant applications in machine learning. This paper examines a crucial subset of such problems where both the objective and constraint functions are weakly convex. Existing methods often face limitations, including slow convergence rates or reliance on double-loop algorithmic designs. To overcome these challenges, we introduce a novel single-loop penalty-based stochastic algorithm. Following the classical exact penalty method, our approach employs a hinge-based penalty, which permits the use of a constant penalty parameter, enabling us to achieve a state-of-the-art complexity for finding an approximate Karush-Kuhn-Tucker (KKT) solution. We further extend our algorithm to address finite-sum coupled compositional objectives, which are prevalent in artificial intelligence applications, establishing improved complexity over existing approaches. Finally, we validate our method through experiments on fair learning with receiver operating characteristic (ROC) fairness constraints and continual learning with non-forgetting constraints.

ICML Conference 2025 Conference Paper

A Near-Optimal Single-Loop Stochastic Algorithm for Convex Finite-Sum Coupled Compositional Optimization

• Bokun Wang
• Tianbao Yang

This paper studies a class of convex Finite-sum Coupled Compositional Optimization (cFCCO) problems with applications including group distributionally robust optimization (GDRO) and learning with imbalanced data. To better address these problems, we introduce an efficient single-loop primal-dual block-coordinate stochastic algorithm called ALEXR. The algorithm employs block-coordinate stochastic mirror ascent with extrapolation for the dual variable and stochastic proximal gradient descent updates for the primal variable. We establish the convergence rates of ALEXR in both convex and strongly convex cases under smoothness and non-smoothness conditions of involved functions, which not only improve the best rates in previous works on smooth cFCCO problems but also expand the realm of cFCCO for solving more challenging non-smooth problems such as the dual form of GDRO. Finally, we derive lower complexity bounds, demonstrating the (near-)optimality of ALEXR within a broad class of stochastic algorithms for cFCCO. Experimental results on GDRO and partial Area Under the ROC Curve (pAUC) maximization demonstrate the promising performance of our algorithm.

NeurIPS Conference 2025 Conference Paper

Advancing Interpretability of CLIP Representations with Concept Surrogate Model

• Nhat Hoang-Xuan
• Xiyuan Wei
• Wanli Xing
• Tianbao Yang
• My T. Thai

Contrastive Language-Image Pre-training (CLIP) generates versatile multimodal embeddings for diverse applications, yet the specific information captured within these representations is not fully understood. Current explainability techniques often target specific tasks, overlooking the rich, general semantics inherent in the representations. Our objective is to reveal the concepts encoded in CLIP embeddings by learning a surrogate representation, which is expressed as a linear combination of human-understandable concepts evident in the image. Our method, which we term EXPLAIN-R, introduces a novel approach that leverages CLIP's learned instance-instance similarity to train a surrogate model that faithfully mimics CLIP's behavior. From the trained surrogate, we derive concept scores for each input image; these scores quantify the contribution of each concept and act as the explanation for the representation. Quantitative evaluations on multiple datasets demonstrate our method's superior faithfulness over the baseline. Moreover, a user study confirms that our explanations are perceived as more relevant, complete, and useful. Our work provides a novel approach for interpreting CLIP image representations, enhancing the user interpretability of representations and fostering more trustworthy AI systems.

TMLR Journal 2025 Journal Article

AutoTrust: Benchmarking Trustworthiness in Large Vision Language Models for Autonomous Driving

• Shuo Xing
• Hongyuan Hua
• Xiangbo Gao
• Shenzhe Zhu
• Renjie Li
• Kexin Tian
• Xiaopeng Li
• Heng Huang

Recent advancements in large vision language models (VLMs) tailored for autonomous driving (AD) have shown strong scene understanding and reasoning capabilities, making them undeniable candidates for end-to-end driving systems. However, limited work exists on studying the trustworthiness of DriveVLMs—a critical factor that directly impacts public transportation safety. In this paper, we introduce AutoTrust, a comprehensive trustworthiness benchmark for large vision-language models in autonomous driving (DriveVLMs), considering diverse perspectives---including trustfulness, safety, robustness, privacy, and fairness. We constructed the largest visual question-answering dataset for investigating trustworthiness issues in driving scenarios, comprising over 10k unique scenes and 18k queries. We evaluated six publicly available VLMs, spanning from generalist to specialist, from open-source to commercial models. Our exhaustive evaluations have unveiled previously undiscovered vulnerabilities of DriveVLMs to trustworthiness threats. Specifically, we found that the general VLMs like LLaVA-v1.6 and GPT-4o-mini surprisingly outperform specialized models fine-tuned for driving in terms of overall trustworthiness. DriveVLMs like DriveLM-Agent are particularly vulnerable to disclosing sensitive information. Additionally, both generalist and specialist VLMs remain susceptible to adversarial attacks and struggle to ensure unbiased decision-making across diverse environments and populations. Our findings call for immediate and decisive action to address the trustworthiness of DriveVLMs--an issue of critical importance to public safety and the welfare of all citizens relying on autonomous transportation systems. We release all the codes and datasets in https://github.com/taco-group/AutoTrust.

NeurIPS Conference 2025 Conference Paper

DisCO: Reinforcing Large Reasoning Models with Discriminative Constrained Optimization

• Gang Li
• Ming Lin
• Tomer Galanti
• Zhengzhong Tu
• Tianbao Yang

The recent success and openness of DeepSeek-R1 have brought widespread attention to Group Relative Policy Optimization (GRPO) as a reinforcement learning method for large reasoning models (LRMs). In this work, we analyze the GRPO objective under a binary reward setting and reveal an inherent limitation of question-level difficulty bias arising from its group relative advantage function. We also identify a connection between GRPO and traditional discriminative methods in supervised learning. Motivated by these insights, we introduce a new Discriminative Constrained Optimization (DisCO) framework for reinforcing LRMs, grounded in the principle of discriminative learning: increasing the scores of positive answers while decreasing those of negative ones. The main differences between DisCO and GRPO and its recent variants are: (1) it replaces the group relative objective with a discriminative objective defined by a scoring function; (2) it abandons clipping-based surrogates in favor of non-clipping RL surrogate objectives used as scoring functions; (3) it employs a simple yet effective constrained optimization approach to enforce the KL divergence constraint. As a result, DisCO offers notable advantages over GRPO and its variants: (i) it completely eliminates difficulty bias by adopting discriminative objectives; (ii) it addresses the entropy instability in GRPO and its variants through the use of non-clipping scoring functions and a constrained optimization approach, yielding long and stable training dynamics; (iii) it allows the incorporation of advanced discriminative learning techniques to address data imbalance, where a significant number of questions have more negative than positive generated answers during training. Our experiments on enhancing the mathematical reasoning capabilities of SFT-finetuned models show that DisCO significantly outperforms GRPO and its improved variants such as DAPO, achieving average gains of 7\% over GRPO and 6\% over DAPO across six benchmark tasks for a 1. 5B model.

ICML Conference 2025 Conference Paper

Discovering Global False Negatives On the Fly for Self-supervised Contrastive Learning

• Vicente Balmaseda
• Bokun Wang
• Ching-Long Lin
• Tianbao Yang

In self-supervised contrastive learning, negative pairs are typically constructed using an anchor image and a sample drawn from the entire dataset, excluding the anchor. However, this approach can result in the creation of negative pairs with similar semantics, referred to as "false negatives", leading to their embeddings being falsely pushed apart. To address this issue, we introduce GloFND, an optimization-based approach that automatically learns on the fly the threshold for each anchor data to identify its false negatives during training. In contrast to previous methods for false negative discovery, our approach globally detects false negatives across the entire dataset rather than locally within the mini-batch. Moreover, its per-iteration computation cost remains independent of the dataset size. Experimental results on image and image-text data demonstrate the effectiveness of the proposed method. Our implementation is available at https: //github. com/vibalcam/GloFND.

ICML Conference 2025 Conference Paper

Discriminative Finetuning of Generative Large Language Models without Reward Models and Human Preference Data

• Siqi Guo 0003
• Ilgee Hong
• Vicente Balmaseda
• Changlong Yu
• Liang Qiu
• Xin Liu
• Haoming Jiang
• Tuo Zhao

Supervised fine-tuning (SFT) has become a crucial step for aligning pretrained large language models (LLMs) using supervised datasets of input-output pairs. However, despite being supervised, SFT is inherently limited by its generative training objective. To address its limitations, the existing common strategy is to follow SFT with a separate phase of preference optimization (PO), which relies on either human-labeled preference data or a strong reward model to guide the learning process. In this paper, we address the limitations of SFT by exploring one of the most successful techniques in conventional supervised learning: discriminative learning. We introduce Discriminative Fine-Tuning (DFT), an improved variant of SFT, which mitigates the burden of collecting human-labeled preference data or training strong reward models. Unlike SFT that employs a generative approach and overlooks negative data, DFT adopts a discriminative paradigm that increases the probability of positive answers while suppressing potentially negative ones, aiming for data prediction instead of token prediction. Our contributions include: (i) a discriminative probabilistic framework for fine-tuning LLMs by explicitly modeling the discriminative likelihood of an answer among all possible outputs given an input; (ii) efficient algorithms to optimize this discriminative likelihood; and (iii) extensive experiments demonstrating DFT’s effectiveness, achieving performance better than SFT and comparable to if not better than SFT$\rightarrow$PO. The code can be found at https: //github. com/Optimization-AI/DFT.

ICML Conference 2025 Conference Paper

Gradient Aligned Regression via Pairwise Losses

• Dixian Zhu
• Tianbao Yang
• Livnat Jerby

Regression is a fundamental task in machine learning that has garnered extensive attention over the past decades. The conventional approach for regression involves employing loss functions that primarily concentrate on aligning model prediction with the ground truth for each individual data sample. Recent research endeavors have introduced novel perspectives by incorporating label similarity into regression through the imposition of additional pairwise regularization or contrastive learning on the latent feature space, demonstrating their effectiveness. However, there are two drawbacks to these approaches: (i) their pairwise operations in the latent feature space are computationally more expensive than conventional regression losses; (ii) they lack theoretical insights behind these methods. In this work, we propose GAR (Gradient Aligned Regression) as a competitive alternative method in label space, which is constituted by a conventional regression loss and two pairwise label difference losses for gradient alignment including magnitude and direction. GAR enjoys: i) the same level efficiency as conventional regression loss because the quadratic complexity for the proposed pairwise losses can be reduced to linear complexity; ii) theoretical insights from learning the pairwise label difference to learning the gradient of the ground truth function. We limit our current scope as regression on the clean data setting without noises, outliers or distributional shifts, etc. We demonstrate the effectiveness of the proposed method practically on two synthetic datasets and on eight extensive real-world tasks from six benchmark datasets with other eight competitive baselines. Running time experiments demonstrate the superior efficiency of the proposed GAR compared to existing methods with pairwise regularization or contrastive learning in the latent feature space. Additionally, ablation studies confirm the effectiveness of each component of GAR. The code is open sourced at https: //github. com/DixianZhu/GAR.

TMLR Journal 2025 Journal Article

Learning to Rank with Top-$K$ Fairness

• Boyang Zhang
• Quanqi Hu
• Mingxuan Sun
• Qihang Lin
• Tianbao Yang

Fairness in ranking models is crucial, as disparities in exposure can disproportionately affect protected groups. Most fairness-aware ranking systems focus on ensuring comparable average exposure for groups across the entire ranked list, which may not fully address real-world concerns. For example, when a ranking model is used for allocating resources among candidates or disaster hotspots, decision-makers often prioritize only the top-$K$ ranked items, while the ranking beyond top-$K$ becomes less relevant. In this paper, we propose a list-wise learning-to-rank framework that addresses the issues of inequalities in top-$K$ rankings at training time. Specifically, we propose a top-$K$ exposure disparity measure that extends the classic exposure disparity metric in a ranked list. We then learn a ranker to balance relevance and fairness in top-$K$ rankings. Since direct top-$K$ selection is computationally expensive for a large number of items, we transform the non-differentiable selection process into a differentiable objective function and develop efficient stochastic optimization algorithms to achieve both high accuracy and sufficient fairness. Extensive experiments demonstrate that our method outperforms existing methods.

ICML Conference 2025 Conference Paper

Model Steering: Learning with a Reference Model Improves Generalization Bounds and Scaling Laws

• Xiyuan Wei
• Ming Lin 0002
• Fanjiang Ye
• Fengguang Song
• Liangliang Cao
• My T. Thai
• Tianbao Yang

This paper formalizes an emerging learning paradigm that uses a trained model as a reference to guide and enhance the training of a target model through strategic data selection or weighting, named model steering. While ad-hoc methods have been used in various contexts, including the training of large foundation models, its underlying principles remain insufficiently understood, leading to sub-optimal performance. In this work, we propose a theory-driven framework for model steering called DRRho risk minimization, which is rooted in Distributionally Robust Optimization (DRO). Through a generalization analysis, we provide theoretical insights into why this approach improves generalization and data efficiency compared to training without a reference model. To the best of our knowledge, this is the first time such theoretical insights are provided for the new learning paradigm, which significantly enhance our understanding and practice of model steering. Building on these insights and the connection between contrastive learning and DRO, we introduce a novel method for Contrastive Language-Image Pretraining (CLIP) with a reference model, termed DRRho-CLIP. Extensive experiments validate the theoretical insights, reveal a superior scaling law compared to CLIP without a reference model, and demonstrate its strength over existing heuristic approaches. Code is released at github. com/Optimization-AI/DRRho-CLIP

TMLR Journal 2025 Journal Article

Multi-Output Distributional Fairness via Post-Processing

• Gang Li
• Qihang Lin
• Ayush Ghosh
• Tianbao Yang

The post-processing approaches are becoming prominent techniques to enhance machine learning models' fairness because of their intuitiveness, low computational cost, and excellent scalability. However, most existing post-processing methods are designed for task-specific fairness measures and are limited to single-output models. In this paper, we introduce a post-processing method for multi-output models, such as the ones used for multi-task/multi-class classification and representation learning, to enhance a model's distributional parity, a task-agnostic fairness measure. Existing methods for achieving distributional parity rely on the (inverse) cumulative density function of a model’s output, restricting their applicability to single-output models. Extending previous works, we propose to employ optimal transport mappings to move a model's outputs across different groups towards their empirical Wasserstein barycenter. An approximation technique is applied to reduce the complexity of computing the exact barycenter and a kernel regression method is proposed to extend this process to out-of-sample data. Our empirical studies evaluate the proposed approach against various baselines on multi-task/multi-class classification and representation learning tasks, demonstrating the effectiveness of the proposed approach.

ICLR Conference 2025 Conference Paper

On Discriminative Probabilistic Modeling for Self-Supervised Representation Learning

• Bokun Wang
• Yunwen Lei
• Yiming Ying
• Tianbao Yang

We study the discriminative probabilistic modeling on a continuous domain for the data prediction task of (multimodal) self-supervised representation learning. To address the challenge of computing the integral in the partition function for each anchor data, we leverage the multiple importance sampling (MIS) technique for robust Monte Carlo integration, which can recover InfoNCE-based contrastive loss as a special case. Within this probabilistic modeling framework, we conduct generalization error analysis to reveal the limitation of current InfoNCE-based contrastive loss for self-supervised representation learning and derive insights for developing better approaches by reducing the error of Monte Carlo integration. To this end, we propose a novel non-parametric method for approximating the sum of conditional probability densities required by MIS through convex optimization, yielding a new contrastive objective for self-supervised representation learning. Moreover, we design an efficient algorithm for solving the proposed objective. We empirically compare our algorithm to representative baselines on the contrastive image-language pretraining task. Experimental results on the CC3M and CC12M datasets demonstrate the superior overall performance of our algorithm. Our code is available at https://github.com/bokun-wang/NUCLR.

NeurIPS Conference 2025 Conference Paper

Self-Supervised Contrastive Learning is Approximately Supervised Contrastive Learning

• Achleshwar Luthra
• Tianbao Yang
• Tomer Galanti

Despite its empirical success, the theoretical foundations of self-supervised contrastive learning (CL) are not yet fully established. In this work, we address this gap by showing that standard CL objectives implicitly approximate a supervised variant we call the negatives-only supervised contrastive loss (NSCL), which excludes same-class contrasts. We prove that the gap between the CL and NSCL losses vanishes as the number of semantic classes increases, under a bound that is both label-agnostic and architecture-independent. We characterize the geometric structure of the global minimizers of the NSCL loss: the learned representations exhibit augmentation collapse, within-class collapse, and class centers that form a simplex equiangular tight frame. We further introduce a new bound on the few-shot error of linear-probing. This bound depends on two measures of feature variability—within-class dispersion and variation along the line between class centers. We show that directional variation dominates the bound and that the within-class dispersion's effect diminishes as the number of labeled samples increases. These properties enable CL and NSCL-trained representations to support accurate few-shot label recovery using simple linear probes. Finally, we empirically validate our theoretical findings: the gap between CL and NSCL losses decays at a rate of $\mathcal{O}(\frac{1}{\#\text{classes}})$; the two losses are highly correlated; minimizing the CL loss implicitly brings the NSCL loss close to the value achieved by direct minimization; and the proposed few-shot error bound provides a tight estimate of probing performance in practice.

NeurIPS Conference 2025 Conference Paper

Stochastic Momentum Methods for Non-smooth Non-Convex Finite-Sum Coupled Compositional Optimization

• Xingyu Chen
• Bokun Wang
• Ming Yang
• Qihang Lin
• Tianbao Yang

Finite-sum Coupled Compositional Optimization (FCCO), characterized by its coupled compositional objective structure, emerges as an important optimization paradigm for addressing a wide range of machine learning problems. In this paper, we focus on a challenging class of non-convex non-smooth FCCO, where the outer functions are non-smooth weakly convex or convex and the inner functions are smooth or weakly convex. Existing state-of-the-art result face two key limitations: (1) a high iteration complexity of $O(1/\epsilon^6)$ under the assumption that the stochastic inner functions are Lipschitz continuous in expectation; (2) reliance on vanilla SGD-type updates, which are not suitable for deep learning applications. Our main contributions are two fold: (i) We propose stochastic momentum methods tailored for non-smooth FCCO that come with provable convergence guarantees; (ii) We establish a **new state-of-the-art** iteration complexity of $O(1/\epsilon^5)$. Moreover, we apply our algorithms to multiple inequality constrained non-convex optimization problems involving smooth or weakly convex functional inequality constraints. By optimizing a smoothed hinge penalty based formulation, we achieve a **new state-of-the-art** complexity of $O(1/\epsilon^5)$ for finding an (nearly) $\epsilon$-level KKT solution. Experiments on three tasks demonstrate the effectiveness of the proposed algorithms.

TMLR Journal 2025 Journal Article

Stochastic Primal-Dual Double Block-Coordinate for Two- way Partial AUC Maximization

• Linli Zhou
• Bokun Wang
• My T. Thai
• Tianbao Yang

Two-way partial AUC (TPAUC) is a critical performance metric for binary classification with imbalanced data, as it focuses on specific ranges of the true positive rate (TPR) and false positive rate (FPR). However, stochastic algorithms for TPAUC optimization remain under-explored, with existing methods either limited to approximated TPAUC loss functions or burdened by sub-optimal complexities. To overcome these limitations, we introduce two innovative stochastic primal-dual double block-coordinate algorithms for TPAUC maximization. These algorithms utilize stochastic block-coordinate updates for both the primal and dual variables, catering to both convex and non-convex settings. We provide theoretical convergence rate analyses, demonstrating significant improvements over prior approaches. Our experimental results, based on multiple benchmark datasets, validate the superior performance of our algorithms, showcasing faster convergence and better generalization. This work advances the state of the art in TPAUC optimization and offers practical tools for real-world machine learning applications.

JMLR Journal 2025 Journal Article

Universal Online Convex Optimization Meets Second-order Bounds

• Lijun Zhang
• Yibo Wang
• Guanghui Wang
• Jinfeng Yi
• Tianbao Yang

Recently, several universal methods have been proposed for online convex optimization, and attain minimax rates for multiple types of convex functions simultaneously. However, they need to design and optimize one surrogate loss for each type of functions, making it difficult to exploit the structure of the problem and utilize existing algorithms. In this paper, we propose a simple strategy for universal online convex optimization, which avoids these limitations. The key idea is to construct a set of experts to process the original online functions, and deploy a meta-algorithm over the linearized losses to aggregate predictions from experts. Specifically, the meta-algorithm is required to yield a second-order bound with excess losses, so that it can leverage strong convexity and exponential concavity to control the meta-regret. In this way, our strategy inherits the theoretical guarantee of any expert designed for strongly convex functions and exponentially concave functions, up to a double logarithmic factor. As a result, we can plug in off-the-shelf online solvers as black-box experts to deliver problem-dependent regret bounds. For general convex functions, it maintains the minimax optimality and also achieves a small-loss bound. Furthermore, we extend our universal strategy to online composite optimization, where the loss function comprises a time-varying function and a fixed regularizer. To deal with the composite loss functions, we employ a meta-algorithm based on the optimistic online learning framework, which not only enjoys a second-order bound, but also can utilize estimations for upcoming loss functions. With suitable configurations, we show that the additional regularizer does not contribute to the meta-regret, thus ensuring the universality in the composite setting. [abs] [ pdf ][ bib ] &copy JMLR 2025. ( edit, beta )

NeurIPS Conference 2024 Conference Paper

Adaptive Preference Scaling for Reinforcement Learning with Human Feedback

• Ilgee Hong
• Zichong Li
• Alexander Bukharin
• Yixiao Li
• Haoming Jiang
• Tianbao Yang
• Tuo Zhao

Reinforcement learning from human feedback (RLHF) is a prevalent approach to align AI systems with human values by learning rewards from human preference data. Due to various reasons, however, such data typically takes the form of rankings over pairs of trajectory segments, which fails to capture the varying strengths of preferences across different pairs. In this paper, we propose a novel adaptive preference loss, underpinned by distributionally robust optimization (DRO), designed to address this uncertainty in preference strength. By incorporating an adaptive scaling parameter into the loss for each pair, our method increases the flexibility of the reward function. Specifically, it assigns small scaling parameters to pairs with ambiguous preferences, leading to more comparable rewards, and large scaling parameters to those with clear preferences for more distinct rewards. Computationally, our proposed loss function is strictly convex and univariate with respect to each scaling parameter, enabling its efficient optimization through a simple second-order algorithm. Our method is versatile and can be readily adapted to various preference optimization frameworks, including direct preference optimization (DPO). Our experiments with robotic control and natural language generation with large language models (LLMs) show that our method not only improves policy performance but also aligns reward function selection more closely with policy optimization, simplifying the hyperparameter tuning process.

NeurIPS Conference 2024 Conference Paper

Communication-Efficient Federated Group Distributionally Robust Optimization

• Zhishuai Guo
• Tianbao Yang

Federated learning faces challenges due to the heterogeneity in data volumes and distributions at different clients, which can compromise model generalization ability to various distributions. Existing approaches to address this issue based on group distributionally robust optimization (GDRO) often lead to high communication and sample complexity. To this end, this work introduces algorithms tailored for communication-efficient Federated Group Distributionally Robust Optimization (FGDRO). Our contributions are threefold: Firstly, we introduce the FGDRO-CVaR algorithm, which optimizes the average top-K losses while reducing communication complexity to $O(1/\epsilon^4)$, where $\epsilon$ denotes the desired precision level. Secondly, our FGDRO-KL algorithm is crafted to optimize KL regularized FGDRO, cutting communication complexity to $O(1/\epsilon^3)$. Lastly, we propose FGDRO-KL-Adam to utilize Adam-type local updates in FGDRO-KL, which not only maintains a communication cost of $O(1/\epsilon^3)$ but also shows potential to surpass SGD-type local steps in practical applications. The effectiveness of our algorithms has been demonstrated on a variety of real-world tasks, including natural language processing and computer vision.

NeurIPS Conference 2024 Conference Paper

Single-Loop Stochastic Algorithms for Difference of Max-Structured Weakly Convex Functions

• Quanqi Hu
• Qi Qi
• Zhaosong Lu
• Tianbao Yang

In this paper, we study a class of non-smooth non-convex problems in the form of $\min_{x}[\max_{y\in\mathcal Y}\phi(x, y) - \max_{z\in\mathcal Z}\psi(x, z)]$, where both $\Phi(x) = \max_{y\in\mathcal Y}\phi(x, y)$ and $\Psi(x)=\max_{z\in\mathcal Z}\psi(x, z)$ are weakly convex functions, and $\phi(x, y), \psi(x, z)$ are strongly concave functions in terms of $y$ and $z$, respectively. It covers two families of problems that have been studied but are missing single-loop stochastic algorithms, i. e. , difference of weakly convex functions and weakly convex strongly-concave min-max problems. We propose a stochastic Moreau envelope approximate gradient method dubbed SMAG, the first single-loop algorithm for solving these problems, and provide a state-of-the-art non-asymptotic convergence rate. The key idea of the design is to compute an approximate gradient of the Moreau envelopes of $\Phi, \Psi$ using only one step of stochastic gradient update of the primal and dual variables. Empirically, we conduct experiments on positive-unlabeled (PU) learning and partial area under ROC curve (pAUC) optimization with an adversarial fairness regularizer to validate the effectiveness of our proposed algorithms.

ICML Conference 2024 Conference Paper

Stability and Generalization of Stochastic Compositional Gradient Descent Algorithms

• Ming Yang
• Xiyuan Wei
• Tianbao Yang
• Yiming Ying

Many machine learning tasks can be formulated as a stochastic compositional optimization (SCO) problem such as reinforcement learning, AUC maximization and meta-learning, where the objective function involves a nested composition associated with an expectation. Although many studies have been devoted to studying the convergence behavior of SCO algorithms, there is little work on understanding their generalization, that is, how these learning algorithms built from training data would behave on future test examples. In this paper, we provide the stability and generalization analysis of stochastic compositional gradient descent algorithms in the framework of statistical learning theory. Firstly, we introduce a stability concept called compositional uniform stability and establish its quantitative relation with generalization for SCO problems. Then, we establish the compositional uniform stability results for two notable stochastic compositional gradient descent algorithms, namely SCGD and SCSC. Finally, we derive dimension-independent excess risk bounds for SCGD and SCSC by balancing stability results and optimization errors. To the best of our knowledge, these are the first-ever known results on stability and generalization analysis of stochastic compositional gradient descent algorithms.

ICML Conference 2024 Conference Paper

To Cool or not to Cool? Temperature Network Meets Large Foundation Models via DRO

• Zi-Hao Qiu
• Siqi Guo 0003
• Mao Xu
• Tuo Zhao
• Lijun Zhang 0005
• Tianbao Yang

The temperature parameter plays a profound role during training and/or inference with large foundation models (LFMs) such as large language models (LLMs) and CLIP models. Particularly, it adjusts the logits in the softmax function in LLMs, which is crucial for next token generation, and it scales the similarities in the contrastive loss for training CLIP models. A significant question remains: “ Is it viable to learn a neural network to predict a personalized temperature of any input data for enhancing LFMs? " In this paper, we present a principled framework for learning a small yet generalizable temperature prediction network (TempNet) to improve LFMs. Our solution is composed of a novel learning framework with robust losses underpinned by constrained distributionally robust optimization (DRO), and a properly designed TempNet with theoretical inspiration. TempNet can be trained together with a large foundation model from scratch or learned separately given a pretrained foundation model. It is not only useful for predicting personalized temperature to promote the training of LFMs but also generalizable and transferable to new tasks. Our experiments on LLMs and CLIP models demonstrate that TempNet greatly improves the performance of existing solutions or models.

TMLR Journal 2023 Journal Article

Attentional-Biased Stochastic Gradient Descent

• Qi Qi
• Yi Xu
• Wotao Yin
• Rong Jin
• Tianbao Yang

In this paper, we present a simple yet effective provable method (named ABSGD) for addressing the data imbalance or label noise problem in deep learning. Our method is a simple modification to momentum SGD where we assign an individual importance weight to each sample in the mini-batch. The individual-level weight of a sampled data is systematically proportional to the exponential of a scaled loss value of the data, where the scaling factor is interpreted as the regularization parameter in the framework of distributionally robust optimization (DRO). Depending on whether the scaling factor is positive or negative, ABSGD is guaranteed to converge to a stationary point of an information-regularized min-max or min-min DRO problem, respectively. Compared with existing class-level weighting schemes, our method can capture the diversity between individual examples within each class. Compared with existing individual-level weighting methods using meta-learning that require three backward propagations for computing mini-batch stochastic gradients, our method is more efficient with only one backward propagation at each iteration as in standard deep learning methods. ABSGD is flexible enough to combine with other robust losses without any additional cost. Our empirical studies on several benchmark datasets demonstrate the effectiveness of the proposed method.

ICML Conference 2023 Conference Paper

Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization

• Quanqi Hu
• Zi-Hao Qiu
• Zhishuai Guo
• Lijun Zhang 0005
• Tianbao Yang

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve $m\gg 1$ lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling $I$ blocks and sampling $B$ samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of $O(\frac{m\epsilon^{-3}\mathbb{I}(I \textless m)}{I\sqrt{I}}+\frac{m\epsilon^{-3}}{I\sqrt{B}})$ for finding an $\epsilon$-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms.

JMLR Journal 2023 Journal Article

Fast Objective & Duality Gap Convergence for Non-Convex Strongly-Concave Min-Max Problems with PL Condition

• Zhishuai Guo
• Yan Yan
• Zhuoning Yuan
• Tianbao Yang

This paper focuses on stochastic methods for solving smooth non-convex strongly-concave min-max problems, which have received increasing attention due to their potential applications in deep learning (e.g., deep AUC maximization, distributionally robust optimization). However, most of the existing algorithms are slow in practice, and their analysis revolves around the convergence to a nearly stationary point. We consider leveraging the Polyak-Lojasiewicz (PL) condition to design faster stochastic algorithms with stronger convergence guarantee. Although PL condition has been utilized for designing many stochastic minimization algorithms, their applications for non-convex min-max optimization remain rare. In this paper, we propose and analyze a generic framework of proximal stage-based method with many well-known stochastic updates embeddable. Fast convergence is established in terms of both the primal objective gap and the duality gap. Compared with existing studies, (i) our analysis is based on a novel Lyapunov function consisting of the primal objective gap and the duality gap of a regularized function, and (ii) the results are more comprehensive with improved rates that have better dependence on the condition number under different assumptions. We also conduct deep and non-deep learning experiments to verify the effectiveness of our methods. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

NeurIPS Conference 2023 Conference Paper

Federated Compositional Deep AUC Maximization

• Xinwen Zhang
• Yihan Zhang
• Tianbao Yang
• Richard Souvenir
• Hongchang Gao

Federated learning has attracted increasing attention due to the promise of balancing privacy and large-scale learning; numerous approaches have been proposed. However, most existing approaches focus on problems with balanced data, and prediction performance is far from satisfactory for many real-world applications where the number of samples in different classes is highly imbalanced. To address this challenging problem, we developed a novel federated learning method for imbalanced data by directly optimizing the area under curve (AUC) score. In particular, we formulate the AUC maximization problem as a federated compositional minimax optimization problem, develop a local stochastic compositional gradient descent ascent with momentum algorithm, and provide bounds on the computational and communication complexities of our algorithm. To the best of our knowledge, this is the first work to achieve such favorable theoretical results. Finally, extensive experimental results confirm the efficacy of our method.

ICML Conference 2023 Conference Paper

FeDXL: Provable Federated Learning for Deep X-Risk Optimization

• Zhishuai Guo
• Rong Jin 0001
• Jiebo Luo 0001
• Tianbao Yang

In this paper, we tackle a novel federated learning (FL) problem for optimizing a family of X-risks, to which no existing FL algorithms are applicable. In particular, the objective has the form of $\mathbb{E}_{\mathbf{z}\sim \mathcal{S}_1} f(\mathbb{E}_{\mathbf{z}’\sim\mathcal{S}_2} \ell(\mathbf{w}; \mathbf{z}, \mathbf{z}’))$, where two sets of data $\mathcal S_1, \mathcal S_2$ are distributed over multiple machines, $\ell(\cdot; \cdot, \cdot)$ is a pairwise loss that only depends on the prediction outputs of the input data pairs $(\mathbf{z}, \mathbf{z}’)$. This problem has important applications in machine learning, e. g. , AUROC maximization with a pairwise loss, and partial AUROC maximization with a compositional loss. The challenges for designing an FL algorithm for X-risks lie in the non-decomposability of the objective over multiple machines and the interdependency between different machines. To this end, we propose an active-passive decomposition framework that decouples the gradient’s components with two types, namely active parts and passive parts, where the active parts depend on local data that are computed with the local model and the passive parts depend on other machines that are communicated/computed based on historical models and samples. Under this framework, we design two FL algorithms (FeDXL) for handling linear and nonlinear $f$, respectively, based on federated averaging and merging and develop a novel theoretical analysis to combat the latency of the passive parts and the interdependency between the local model parameters and the involved data for computing local gradient estimators. We establish both iteration and communication complexities and show that using the historical samples and models for computing the passive parts do not degrade the complexities. We conduct empirical studies of FeDXL for deep AUROC and partial AUROC maximization, and demonstrate their performance compared with several baselines.

ICML Conference 2023 Conference Paper

Generalization Analysis for Contrastive Representation Learning

• Yunwen Lei
• Tianbao Yang
• Yiming Ying
• Ding-Xuan Zhou

Recently, contrastive learning has found impressive success in advancing the state of the art in solving various machine learning tasks. However, the existing generalization analysis is very limited or even not meaningful. In particular, the existing generalization error bounds depend linearly on the number $k$ of negative examples while it was widely shown in practice that choosing a large $k$ is necessary to guarantee good generalization of contrastive learning in downstream tasks. In this paper, we establish novel generalization bounds for contrastive learning which do not depend on $k$, up to logarithmic terms. Our analysis uses structural results on empirical covering numbers and Rademacher complexities to exploit the Lipschitz continuity of loss functions. For self-bounding Lipschitz loss functions, we further improve our results by developing optimistic bounds which imply fast rates in a low noise condition. We apply our results to learning with both linear representation and nonlinear representation by deep neural networks, for both of which we derive Rademacher complexity bounds to get improved generalization bounds.

ICML Conference 2023 Conference Paper

Label Distributionally Robust Losses for Multi-class Classification: Consistency, Robustness and Adaptivity

• Dixian Zhu
• Yiming Ying
• Tianbao Yang

We study a family of loss functions named label-distributionally robust (LDR) losses for multi-class classification that are formulated from distributionally robust optimization (DRO) perspective, where the uncertainty in the given label information are modeled and captured by taking the worse case of distributional weights. The benefits of this perspective are several fold: (i) it provides a unified framework to explain the classical cross-entropy (CE) loss and SVM loss and their variants, (ii) it includes a special family corresponding to the temperature-scaled CE loss, which is widely adopted but poorly understood; (iii) it allows us to achieve adaptivity to the uncertainty degree of label information at an instance level. Our contributions include: (1) we study both consistency and robustness by establishing top-$k$ ($\forall k\geq 1$) consistency of LDR losses for multi-class classification, and a negative result that a top-$1$ consistent and symmetric robust loss cannot achieve top-$k$ consistency simultaneously for all $k\geq 2$; (2) we propose a new adaptive LDR loss that automatically adapts the individualized temperature parameter to the noise degree of class label of each instance; (3) we demonstrate stable and competitive performance for the proposed adaptive LDR loss on 7 benchmark datasets under 6 noisy label and 1 clean settings against 13 loss functions, and on one real-world noisy dataset. The method is open-sourced at https: //github. com/Optimization-AI/ICML2023_LDR.

ICML Conference 2023 Conference Paper

Learning Unnormalized Statistical Models via Compositional Optimization

• Wei Jiang 0029
• Jiayu Qin 0001
• Lingyu Wu
• Changyou Chen
• Tianbao Yang
• Lijun Zhang 0005

Learning unnormalized statistical models (e. g. , energy-based models) is computationally challenging due to the complexity of handling the partition function. To eschew this complexity, noise-contrastive estimation (NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. However, as found in previous works, NCE may perform poorly in many tasks due to its flat loss landscape and slow convergence. In this paper, we study a direct approach for optimizing the negative log-likelihood of unnormalized models from the perspective of compositional optimization. To tackle the partition function, a noise distribution is introduced such that the log partition function can be written as a compositional function whose inner function can be estimated with stochastic samples. Hence, the objective can be optimized by stochastic compositional optimization algorithms. Despite being a simple method, we demonstrate that it is more favorable than NCE by (1) establishing a fast convergence rate and quantifying its dependence on the noise distribution through the variance of stochastic estimators; (2) developing better results for one-dimensional Gaussian mean estimation by showing our objective has a much favorable loss landscape and hence our method enjoys faster convergence; (3) demonstrating better performance on multiple applications, including density estimation, out-of-distribution detection, and real image generation.

NeurIPS Conference 2023 Conference Paper

Maximization of Average Precision for Deep Learning with Adversarial Ranking Robustness

• Gang Li
• Wei Tong
• Tianbao Yang

This paper seeks to address a gap in optimizing Average Precision (AP) while ensuring adversarial robustness, an area that has not been extensively explored to the best of our knowledge. AP maximization for deep learning has widespread applications, particularly when there is a significant imbalance between positive and negative examples. Although numerous studies have been conducted on adversarial training, they primarily focus on robustness concerning accuracy, ensuring that the average accuracy on adversarially perturbed examples is well maintained. However, this type of adversarial robustness is insufficient for many applications, as minor perturbations on a single example can significantly impact AP while not greatly influencing the accuracy of the prediction system. To tackle this issue, we introduce a novel formulation that combines an AP surrogate loss with a regularization term representing adversarial ranking robustness, which maintains the consistency between ranking of clean data and that of perturbed data. We then devise an efficient stochastic optimization algorithm to optimize the resulting objective. Our empirical studies, which compare our method to current leading adversarial training baselines and other robust AP maximization strategies, demonstrate the effectiveness of the proposed approach. Notably, our methods outperform a state-of-the-art method (TRADES) by more than 4\% in terms of robust AP against PGD attacks while achieving 7\% higher AP on clean data simultaneously on CIFAR10 and CIFAR100. The code is available at: https: //github. com/GangLii/Adversarial-AP

JMLR Journal 2023 Journal Article

Memory-Based Optimization Methods for Model-Agnostic Meta-Learning and Personalized Federated Learning

• Bokun Wang
• Zhuoning Yuan
• Yiming Ying
• Tianbao Yang

In recent years, model-agnostic meta-learning (MAML) has become a popular research area. However, the stochastic optimization of MAML is still underdeveloped. Existing MAML algorithms rely on the “episode” idea by sampling a few tasks and data points to update the meta-model at each iteration. Nonetheless, these algorithms either fail to guarantee convergence with a constant mini-batch size or require processing a large number of tasks at every iteration, which is unsuitable for continual learning or cross-device federated learning where only a small number of tasks are available per iteration or per round. To address these issues, this paper proposes memory-based stochastic algorithms for MAML that converge with vanishing error. The proposed algorithms require sampling a constant number of tasks and data samples per iteration, making them suitable for the continual learning scenario. Moreover, we introduce a communication-efficient memory-based MAML algorithm for personalized federated learning in cross-device (with client sampling) and cross-silo (without client sampling) settings. Our theoretical analysis improves the optimization theory for MAML, and our empirical results corroborate our theoretical findings. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2023. ( edit, beta )

NeurIPS Conference 2023 Conference Paper

Non-Smooth Weakly-Convex Finite-sum Coupled Compositional Optimization

• Quanqi Hu
• Dixian Zhu
• Tianbao Yang

This paper investigates new families of compositional optimization problems, called non-smooth weakly-convex finite-sum coupled compositional optimization (NSWC FCCO). There has been a growing interest in FCCO due to its wide-ranging applications in machine learning and AI, as well as its ability to address the shortcomings of stochastic algorithms based on empirical risk minimization. However, current research on FCCO presumes that both the inner and outer functions are smooth, limiting their potential to tackle a more diverse set of problems. Our research expands on this area by examining non-smooth weakly-convex FCCO, where the outer function is weakly convex and non-decreasing, and the inner function is weakly-convex. We analyze a single-loop algorithm and establish its complexity for finding an $\epsilon$-stationary point of the Moreau envelop of the objective function. Additionally, we also extend the algorithm for solving novel non-smooth weakly-convex tri-level finite-sum coupled compositional optimization problems, which feature a nested arrangement of three functions. Lastly, we explore the applications of our algorithms in deep learning for two-way partial AUC maximization and multi-instance two-way partial AUC maximization, using empirical studies to showcase the effectiveness of the proposed algorithms.

ICML Conference 2023 Conference Paper

Not All Semantics are Created Equal: Contrastive Self-supervised Learning with Automatic Temperature Individualization

• Zi-Hao Qiu
• Quanqi Hu
• Zhuoning Yuan
• Denny Zhou
• Lijun Zhang 0005
• Tianbao Yang

In this paper, we aim to optimize a contrastive loss with individualized temperatures in a principled manner. The common practice of using a global temperature parameter $\tau$ ignores the fact that “not all semantics are created equal", meaning that different anchor data may have different numbers of samples with similar semantics, especially when data exhibits long-tails. First, we propose a new robust contrastive loss inspired by distributionally robust optimization (DRO), providing us an intuition about the effect of $\tau$ and a mechanism for automatic temperature individualization. Then, we propose an efficient stochastic algorithm for optimizing the robust contrastive loss with a provable convergence guarantee without using large mini-batch sizes. Theoretical and experimental results show that our algorithm automatically learns a suitable $\tau$ for each sample. Specifically, samples with frequent semantics use large temperatures to keep local semantic structures, while samples with rare semantics use small temperatures to induce more separable features. Our method not only outperforms prior strong baselines (e. g. , SimCLR, CLIP) on unimodal and bimodal tasks with larger improvements on imbalanced data but also is less sensitive to hyper-parameters. To our best knowledge, this is the first methodical approach to optimizing a contrastive loss with individualized temperatures. Our proposed algorithms are implemented in the LibAUC library at https: //libauc. org.

ICML Conference 2023 Conference Paper

Provable Multi-instance Deep AUC Maximization with Stochastic Pooling

• Dixian Zhu
• Bokun Wang
• Zhi Chen 0025
• Yaxing Wang
• Milan Sonka
• Xiaodong Wu 0001
• Tianbao Yang

This paper considers a novel application of deep AUC maximization (DAM) for multi-instance learning (MIL), in which a single class label is assigned to a bag of instances (e. g. , multiple 2D slices of a CT scan for a patient). We address a neglected yet non-negligible computational challenge of MIL in the context of DAM, i. e. , bag size is too large to be loaded into GPU memory for backpropagation, which is required by the standard pooling methods of MIL. To tackle this challenge, we propose variance-reduced stochastic pooling methods in the spirit of stochastic optimization by formulating the loss function over the pooled prediction as a multi-level compositional function. By synthesizing techniques from stochastic compositional optimization and non-convex min-max optimization, we propose a unified and provable muli-instance DAM (MIDAM) algorithm with stochastic smoothed-max pooling or stochastic attention-based pooling, which only samples a few instances for each bag to compute a stochastic gradient estimator and to update the model parameter. We establish a similar convergence rate of the proposed MIDAM algorithm as the state-of-the-art DAM algorithms. Our extensive experiments on conventional MIL datasets and medical datasets demonstrate the superiority of our MIDAM algorithm. The method is open-sourced at https: //libauc. org/.

NeurIPS Conference 2023 Conference Paper

SpatialRank: Urban Event Ranking with NDCG Optimization on Spatiotemporal Data

• Bang An
• Xun Zhou
• YONGJIAN ZHONG
• Tianbao Yang

The problem of urban event ranking aims at predicting the top-$k$ most risky locations of future events such as traffic accidents and crimes. This problem is of fundamental importance to public safety and urban administration especially when limited resources are available. The problem is, however, challenging due to complex and dynamic spatio-temporal correlations between locations, uneven distribution of urban events in space, and the difficulty to correctly rank nearby locations with similar features. Prior works on event forecasting mostly aim at accurately predicting the actual risk score or counts of events for all the locations. Rankings obtained as such usually have low quality due to prediction errors. Learning-to-rank methods directly optimize measures such as Normalized Discounted Cumulative Gain (NDCG), but cannot handle the spatiotemporal autocorrelation existing among locations. Due to the common assumption that items are independent. In this paper, we bridge the gap by proposing a novel spatial event ranking approach named SpatialRank. SpatialRank features adaptive graph convolution layers that dynamically learn the spatiotemporal dependencies across locations from data. In addition, the model optimizes through surrogates a hybrid NDCG loss with a spatial component to better rank neighboring spatial locations. We design an importance-sampling with a spatial filtering algorithm to effectively evaluate the loss during training. Comprehensive experiments on three real-world datasets demonstrate that SpatialRank can effectively identify the top riskiest locations of crimes and traffic accidents and outperform state-of-art methods in terms of NDCG by up to 12. 7%.

NeurIPS Conference 2023 Conference Paper

Stochastic Approximation Approaches to Group Distributionally Robust Optimization

• Lijun Zhang
• Peng Zhao
• Zhen-Hua Zhuang
• Tianbao Yang
• Zhi-Hua Zhou

This paper investigates group distributionally robust optimization (GDRO), with the purpose to learn a model that performs well over $m$ different distributions. First, we formulate GDRO as a stochastic convex-concave saddle-point problem, and demonstrate that stochastic mirror descent (SMD), using $m$ samples in each iteration, achieves an $O(m (\log m)/\epsilon^2)$ sample complexity for finding an $\epsilon$-optimal solution, which matches the $\Omega(m/\epsilon^2)$ lower bound up to a logarithmic factor. Then, we make use of techniques from online learning to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample complexity. Specifically, we cast GDRO as a two-players game where one player simply performs SMD and the other executes an online algorithm for non-oblivious multi-armed bandits. Next, we consider a more practical scenario where the number of samples that can be drawn from each distribution is different, and propose a novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates. Denote by $n_i$ the sample budget for the $i$-th distribution, and assume $n_1 \geq n_2 \geq \cdots \geq n_m$. In the first approach, we incorporate non-uniform sampling into SMD such that the sample budget is satisfied in expectation, and prove that the excess risk of the $i$-th distribution decreases at an $O(\sqrt{n_1 \log m}/n_i)$ rate. In the second approach, we use mini-batches to meet the budget exactly and also reduce the variance in stochastic gradients, and then leverage stochastic mirror-prox algorithm, which can exploit small variances, to optimize a carefully designed weighted GDRO problem. Under appropriate conditions, it attains an $O((\log m)/\sqrt{n_i})$ convergence rate, which almost matches the optimal $O(\sqrt{1/n_i})$ rate of only learning from the $i$-th distribution with $n_i$ samples.

TMLR Journal 2023 Journal Article

Stochastic Constrained DRO with a Complexity Independent of Sample Size

• Qi Qi
• Jiameng Lyu
• Kung-Sik Chan
• Er-Wei Bai
• Tianbao Yang

Distributionally Robust Optimization (DRO), as a popular method to train robust models against distribution shift between training and test sets, has received tremendous attention in recent years. In this paper, we propose and analyze stochastic algorithms that apply to both non-convex and convex losses for solving Kullback–Leibler divergence constrained DRO problem. Compared with existing methods solving this problem, our stochastic algorithms not only enjoy competitive if not better complexity independent of sample size but also just require a constant batch size at every iteration, which is more practical for broad applications. We establish a nearly optimal complexity bound for finding an $\epsilon$-stationary solution for non-convex losses and an optimal complexity for finding an $\epsilon$-optimal solution for convex losses. Empirical studies demonstrate the effectiveness of the proposed algorithms for solving non-convex and convex constrained DRO problems.

ICML Conference 2022 Conference Paper

A Simple yet Universal Strategy for Online Convex Optimization

• Lijun Zhang 0005
• Guanghui Wang 0006
• Jinfeng Yi
• Tianbao Yang

Recently, several universal methods have been proposed for online convex optimization, and attain minimax rates for multiple types of convex functions simultaneously. However, they need to design and optimize one surrogate loss for each type of functions, making it difficult to exploit the structure of the problem and utilize existing algorithms. In this paper, we propose a simple strategy for universal online convex optimization, which avoids these limitations. The key idea is to construct a set of experts to process the original online functions, and deploy a meta-algorithm over the linearized losses to aggregate predictions from experts. Specifically, the meta-algorithm is required to yield a second-order bound with excess losses, so that it can leverage strong convexity and exponential concavity to control the meta-regret. In this way, our strategy inherits the theoretical guarantee of any expert designed for strongly convex functions and exponentially concave functions, up to a double logarithmic factor. As a result, we can plug in off-the-shelf online solvers as black-box experts to deliver problem-dependent regret bounds. For general convex functions, it maintains the minimax optimality and also achieves a small-loss bound.

ICLR Conference 2022 Conference Paper

Compositional Training for End-to-End Deep AUC Maximization

• Zhuoning Yuan
• Zhishuai Guo
• Nitesh V. Chawla
• Tianbao Yang

Recently, deep AUC maximization (DAM) has achieved great success in different domains (e.g., medical image classification). However, the end-to-end training for deep AUC maximization still remains a challenging problem. Previous studies employ an ad-hoc two-stage approach that first trains the network by optimizing a traditional loss (e.g., cross-entropy loss) and then finetunes the network by optimizing an AUC loss. This is because that training a deep neural network from scratch by maximizing an AUC loss usually does not yield a satisfactory performance. This phenomenon can be attributed to the degraded feature representations learned by maximizing the AUC loss from scratch. To address this issue, we propose a novel compositional training framework for end-to-end DAM, namely compositional DAM. The key idea of compositional training is to minimize a compositional objective function, where the outer function corresponds to an AUC loss and the inner function represents a gradient descent step for minimizing a traditional loss, e.g., the cross-entropy (CE) loss. To optimize the non-standard compositional objective, we propose an efficient and provable stochastic optimization algorithm. The proposed algorithm enhances the capabilities of both robust feature learning and robust classifier learning by alternatively taking a gradient descent step for the CE loss and for the AUC loss in a systematic way. We conduct extensive empirical studies on imbalanced benchmark and medical image datasets, which unanimously verify the effectiveness of the proposed method. Our results show that the compositional training approach dramatically improves both the feature representations and the testing AUC score compared with traditional deep learning approaches, and yields better performance than the two-stage approaches for DAM as well. The proposed method is implemented in our open-sourced library LibAUC (https://www.libauc.org) and code is available at https://github.com/Optimization-AI/LibAUC.

ICML Conference 2022 Conference Paper

Finite-Sum Coupled Compositional Stochastic Optimization: Theory and Applications

• Bokun Wang
• Tianbao Yang

This paper studies stochastic optimization for a sum of compositional functions, where the inner-level function of each summand is coupled with the corresponding summation index. We refer to this family of problems as finite-sum coupled compositional optimization (FCCO). It has broad applications in machine learning for optimizing non-convex or convex compositional measures/objectives such as average precision (AP), p-norm push, listwise ranking losses, neighborhood component analysis (NCA), deep survival analysis, deep latent variable models, etc. , which deserves finer analysis. Yet, existing algorithms and analyses are restricted in one or other aspects. The contribution of this paper is to provide a comprehensive convergence analysis of a simple stochastic algorithm for both non-convex and convex objectives. Our key result is the improved oracle complexity with the parallel speed-up by using the moving-average based estimator with mini-batching. Our theoretical analysis also exhibits new insights for improving the practical implementation by sampling the batches of equal size for the outer and inner levels. Numerical experiments on AP maximization, NCA, and p-norm push corroborate some aspects of the theory.

ICML Conference 2022 Conference Paper

GraphFM: Improving Large-Scale GNN Training via Feature Momentum

• Haiyang Yu 0005
• Limei Wang
• Bokun Wang
• Meng Liu 0015
• Tianbao Yang
• Shuiwang Ji

Training of graph neural networks (GNNs) for large-scale node classification is challenging. A key difficulty lies in obtaining accurate hidden node representations while avoiding the neighborhood explosion problem. Here, we propose a new technique, named feature momentum (FM), that uses a momentum step to incorporate historical embeddings when updating feature representations. We develop two specific algorithms, known as GraphFM-IB and GraphFM-OB, that consider in-batch and out-of-batch data, respectively. GraphFM-IB applies FM to in-batch sampled data, while GraphFM-OB applies FM to out-of-batch data that are 1-hop neighborhood of in-batch data. We provide a convergence analysis for GraphFM-IB and some theoretical insight for GraphFM-OB. Empirically, we observe that GraphFM-IB can effectively alleviate the neighborhood explosion problem of existing methods. In addition, GraphFM-OB achieves promising performance on multiple large-scale graph datasets.

NeurIPS Conference 2022 Conference Paper

Large-scale Optimization of Partial AUC in a Range of False Positive Rates

• Yao Yao
• Qihang Lin
• Tianbao Yang

The area under the ROC curve (AUC) is one of the most widely used performance measures for classification models in machine learning. However, it summarizes the true positive rates (TPRs) over all false positive rates (FPRs) in the ROC space, which may include the FPRs with no practical relevance in some applications. The partial AUC, as a generalization of the AUC, summarizes only the TPRs over a specific range of the FPRs and is thus a more suitable performance measure in many real-world situations. Although partial AUC optimization in a range of FPRs had been studied, existing algorithms are not scalable to big data and not applicable to deep learning. To address this challenge, we cast the problem into a non-smooth difference-of-convex (DC) program for any smooth predictive functions (e. g. , deep neural networks), which allowed us to develop an efficient approximated gradient descent method based on the Moreau envelope smoothing technique, inspired by recent advances in non-smooth DC optimization. To increase the efficiency of large data processing, we used an efficient stochastic block coordinate update in our algorithm. Our proposed algorithm can also be used to minimize the sum of ranked range loss, which also lacks efficient solvers. We established a complexity of $\tilde O(1/\epsilon^6)$ for finding a nearly $\epsilon$-critical solution. Finally, we numerically demonstrated the effectiveness of our proposed algorithms in training both linear models and deep neural networks for partial AUC maximization and sum of ranked range loss minimization.

ICML Conference 2022 Conference Paper

Large-scale Stochastic Optimization of NDCG Surrogates for Deep Learning with Provable Convergence

• Zi-Hao Qiu
• Quanqi Hu
• Yongjian Zhong
• Lijun Zhang 0005
• Tianbao Yang

NDCG, namely Normalized Discounted Cumulative Gain, is a widely used ranking metric in information retrieval and machine learning. However, efficient and provable stochastic methods for maximizing NDCG are still lacking, especially for deep models. In this paper, we propose a principled approach to optimize NDCG and its top-$K$ variant. First, we formulate a novel compositional optimization problem for optimizing the NDCG surrogate, and a novel bilevel compositional optimization problem for optimizing the top-$K$ NDCG surrogate. Then, we develop efficient stochastic algorithms with provable convergence guarantees for the non-convex objectives. Different from existing NDCG optimization methods, the per-iteration complexity of our algorithms scales with the mini-batch size instead of the number of total items. To improve the effectiveness for deep learning, we further propose practical strategies by using initial warm-up and stop gradient operator. Experimental results on multiple datasets demonstrate that our methods outperform prior ranking approaches in terms of NDCG. To the best of our knowledge, this is the first time that stochastic algorithms are proposed to optimize NDCG with a provable convergence guarantee. Our proposed methods are implemented in the LibAUC library at https: //libauc. org.

NeurIPS Conference 2022 Conference Paper

Multi-block Min-max Bilevel Optimization with Applications in Multi-task Deep AUC Maximization

• Quanqi Hu
• YONGJIAN ZHONG
• Tianbao Yang

In this paper, we study multi-block min-max bilevel optimization problems, where the upper level is non-convex strongly-concave minimax objective and the lower level is a strongly convex objective, and there are multiple blocks of dual variables and lower level problems. Due to the intertwined multi-block min-max bilevel structure, the computational cost at each iteration could be prohibitively high, especially with a large number of blocks. To tackle this challenge, we present two single-loop randomized stochastic algorithms, which require updates for only a constant number of blocks at each iteration. Under some mild assumptions on the problem, we establish their sample complexity of $\mathcal{O}(1/\epsilon^4)$ for finding an $\epsilon$-stationary point. This matches the optimal complexity order for solving stochastic nonconvex optimization under a general unbiased stochastic oracle model. Moreover, we provide two applications of the proposed method in multi-task deep AUC (area under ROC curve) maximization. Experimental results validate our theory and demonstrate the effectiveness of our method.

NeurIPS Conference 2022 Conference Paper

Multi-block-Single-probe Variance Reduced Estimator for Coupled Compositional Optimization

• Wei Jiang
• Gang Li
• Yibo Wang
• Lijun Zhang
• Tianbao Yang

Variance reduction techniques such as SPIDER/SARAH/STORM have been extensively studied to improve the convergence rates of stochastic non-convex optimization, which usually maintain and update a sequence of estimators for a single function across iterations. What if we need to track multiple functional mappings across iterations but only with access to stochastic samples of $\mathcal{O}(1)$ functional mappings at each iteration? There is an important application in solving an emerging family of coupled compositional optimization problems in the form of $\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where $g_i$ is accessible through a stochastic oracle. The key issue is to track and estimate a sequence of $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ across iterations, where $\mathbf g(\mathbf{w})$ has $m$ blocks and it is only allowed to probe $\mathcal{O}(1)$ blocks to attain their stochastic values and Jacobians. To improve the complexity for solving these problems, we propose a novel stochastic method named Multi-block-Single-probe Variance Reduced (MSVR) estimator to track the sequence of $\mathbf g(\mathbf{w})$. It is inspired by STORM but introduces a customized error correction term to alleviate the noise not only in stochastic samples for the selected blocks but also in those blocks that are not sampled. With the help of the MSVR estimator, we develop several algorithms for solving the aforementioned compositional problems with improved complexities across a spectrum of settings with non-convex/convex/strongly convex/Polyak-Lojasiewicz (PL) objectives. Our results improve upon prior ones in several aspects, including the order of sample complexities and dependence on the strong convexity parameter. Empirical studies on multi-task deep AUC maximization demonstrate the better performance of using the new estimator.

ICML Conference 2022 Conference Paper

Optimal Algorithms for Stochastic Multi-Level Compositional Optimization

• Wei Jiang 0029
• Bokun Wang
• Yibo Wang 0005
• Lijun Zhang 0005
• Tianbao Yang

In this paper, we investigate the problem of stochastic multi-level compositional optimization, where the objective function is a composition of multiple smooth but possibly non-convex functions. Existing methods for solving this problem either suffer from sub-optimal sample complexities or need a huge batch size. To address this limitation, we propose a Stochastic Multi-level Variance Reduction method (SMVR), which achieves the optimal sample complexity of $\mathcal{O}\left(1 / \epsilon^{3}\right)$ to find an $\epsilon$-stationary point for non-convex objectives. Furthermore, when the objective function satisfies the convexity or Polyak-{Ł}ojasiewicz (PL) condition, we propose a stage-wise variant of SMVR and improve the sample complexity to $\mathcal{O}\left(1 / \epsilon^{2}\right)$ for convex functions or $\mathcal{O}\left(1 /(\mu\epsilon)\right)$ for non-convex functions satisfying the $\mu$-PL condition. The latter result implies the same complexity for $\mu$-strongly convex functions. To make use of adaptive learning rates, we also develop Adaptive SMVR, which achieves the same optimal complexities but converges faster in practice. All our complexities match the lower bounds not only in terms of $\epsilon$ but also in terms of $\mu$ (for PL or strongly convex functions), without using a large batch size in each iteration.

ICML Conference 2022 Conference Paper

Provable Stochastic Optimization for Global Contrastive Learning: Small Batch Does Not Harm Performance

• Zhuoning Yuan
• Yuexin Wu
• Zi-Hao Qiu
• Xianzhi Du
• Lijun Zhang 0005
• Denny Zhou
• Tianbao Yang

In this paper, we study contrastive learning from an optimization perspective, aiming to analyze and address a fundamental issue of existing contrastive learning methods that either rely on a large batch size or a large dictionary of feature vectors. We consider a global objective for contrastive learning, which contrasts each positive pair with all negative pairs for an anchor point. From the optimization perspective, we explain why existing methods such as SimCLR require a large batch size in order to achieve a satisfactory result. In order to remove such requirement, we propose a memory-efficient Stochastic Optimization algorithm for solving the Global objective of Contrastive Learning of Representations, named SogCLR. We show that its optimization error is negligible under a reasonable condition after a sufficient number of iterations or is diminishing for a slightly different global contrastive objective. Empirically, we demonstrate that SogCLR with small batch size (e. g. , 256) can achieve similar performance as SimCLR with large batch size (e. g. , 8192) on self-supervised learning task on ImageNet-1K. We also attempt to show that the proposed optimization technique is generic and can be applied to solving other contrastive losses, e. g. , two-way contrastive losses for bimodal contrastive learning. The proposed method is implemented in our open-sourced library LibAUC (www. libauc. org).

NeurIPS Conference 2022 Conference Paper

Smoothed Online Convex Optimization Based on Discounted-Normal-Predictor

• Lijun Zhang
• Wei Jiang
• Jinfeng Yi
• Tianbao Yang

In this paper, we investigate an online prediction strategy named as Discounted-Normal-Predictor [Kapralov and Panigrahy, 2010] for smoothed online convex optimization (SOCO), in which the learner needs to minimize not only the hitting cost but also the switching cost. In the setting of learning with expert advice, Daniely and Mansour [2019] demonstrate that Discounted-Normal-Predictor can be utilized to yield nearly optimal regret bounds over any interval, even in the presence of switching costs. Inspired by their results, we develop a simple algorithm for SOCO: Combining online gradient descent (OGD) with different step sizes sequentially by Discounted-Normal-Predictor. Despite its simplicity, we prove that it is able to minimize the adaptive regret with switching cost, i. e. , attaining nearly optimal regret with switching cost on every interval. By exploiting the theoretical guarantee of OGD for dynamic regret, we further show that the proposed algorithm can minimize the dynamic regret with switching cost in every interval.

ICML Conference 2022 Conference Paper

When AUC meets DRO: Optimizing Partial AUC for Deep Learning with Non-Convex Convergence Guarantee

• Dixian Zhu
• Gang Li
• Bokun Wang
• Xiaodong Wu
• Tianbao Yang

In this paper, we propose systematic and efficient gradient-based methods for both one-way and two-way partial AUC (pAUC) maximization that are applicable to deep learning. We propose new formulations of pAUC surrogate objectives by using the distributionally robust optimization (DRO) to define the loss for each individual positive data. We consider two formulations of DRO, one of which is based on conditional-value-at-risk (CVaR) that yields a non-smooth but exact estimator for pAUC, and another one is based on a KL divergence regularized DRO that yields an inexact but smooth (soft) estimator for pAUC. For both one-way and two-way pAUC maximization, we propose two algorithms and prove their convergence for optimizing their two formulations, respectively. Experiments demonstrate the effectiveness of the proposed algorithms for pAUC maximization for deep learning on various datasets.

NeurIPS Conference 2021 Conference Paper

An Online Method for A Class of Distributionally Robust Optimization with Non-convex Objectives

• Qi Qi
• Zhishuai Guo
• Yi Xu
• Rong Jin
• Tianbao Yang

In this paper, we propose a practical online method for solving a class of distributional robust optimization (DRO) with non-convex objectives, which has important applications in machine learning for improving the robustness of neural networks. In the literature, most methods for solving DRO are based on stochastic primal-dual methods. However, primal-dual methods for DRO suffer from several drawbacks: (1) manipulating a high-dimensional dual variable corresponding to the size of data is time expensive; (2) they are not friendly to online learning where data is coming sequentially. To address these issues, we consider a class of DRO with an KL divergence regularization on the dual variables, transform the min-max problem into a compositional minimization problem, and propose practical duality-free online stochastic methods without requiring a large mini-batch size. We establish the state-of-the-art complexities of the proposed methods with and without a Polyak-Łojasiewicz (PL) condition of the objective. Empirical studies on large-scale deep learning tasks (i) demonstrate that our method can speed up the training by more than 2 times than baseline methods and save days of training time on a large-scale dataset with ∼ 265K images, and (ii) verify the supreme performance of DRO over Empirical Risk Minimization (ERM) on imbalanced datasets. Of independent interest, the proposed method can be also used for solving a family of stochastic compositional problems with state-of-the-art complexities.

ICML Conference 2021 Conference Paper

Federated Deep AUC Maximization for Hetergeneous Data with a Constant Communication Complexity

• Zhuoning Yuan
• Zhishuai Guo
• Yi Xu 0008
• Yiming Ying
• Tianbao Yang

Deep AUC (area under the ROC curve) Maximization (DAM) has attracted much attention recently due to its great potential for imbalanced data classification. However, the research on Federated Deep AUC Maximization (FDAM) is still limited. Compared with standard federated learning (FL) approaches that focus on decomposable minimization objectives, FDAM is more complicated due to its minimization objective is non-decomposable over individual examples. In this paper, we propose improved FDAM algorithms for heterogeneous data by solving the popular non-convex strongly-concave min-max formulation of DAM in a distributed fashion, which can also be applied to a class of non-convex strongly-concave min-max problems. A striking result of this paper is that the communication complexity of the proposed algorithm is a constant independent of the number of machines and also independent of the accuracy level, which improves an existing result by orders of magnitude. The experiments have demonstrated the effectiveness of our FDAM algorithm on benchmark datasets, and on medical chest X-ray images from different organizations. Our experiment shows that the performance of FDAM using data from multiple hospitals can improve the AUC score on testing data from a single hospital for detecting life-threatening diseases based on chest radiographs.

JMLR Journal 2021 Journal Article

First-order Convergence Theory for Weakly-Convex-Weakly-Concave Min-max Problems

• Mingrui Liu
• Hassan Rafique
• Qihang Lin
• Tianbao Yang

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in machine learning including training Generative Adversarial Nets (GANs). We propose an algorithmic framework motivated by the inexact proximal point method, where the weakly monotone variational inequality (VI) corresponding to the original min-max problem is solved through approximately solving a sequence of strongly monotone VIs constructed by adding a strongly monotone mapping to the original gradient mapping. We prove first-order convergence to a nearly stationary solution of the original min-max problem of the generic algorithmic framework and establish different rates by employing different algorithms for solving each strongly monotone VI. Experiments verify the convergence theory and also demonstrate the effectiveness of the proposed methods on training GANs. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

NeurIPS Conference 2021 Conference Paper

Online Convex Optimization with Continuous Switching Constraint

• Guanghui Wang
• Yuanyu Wan
• Tianbao Yang
• Lijun Zhang

In many sequential decision making applications, the change of decision would bring an additional cost, such as the wear-and-tear cost associated with changing server status. To control the switching cost, we introduce the problem of online convex optimization with continuous switching constraint, where the goal is to achieve a small regret given a budget on the \emph{overall} switching cost. We first investigate the hardness of the problem, and provide a lower bound of order $\Omega(\sqrt{T})$ when the switching cost budget $S=\Omega(\sqrt{T})$, and $\Omega(\min\{\frac{T}{S}, T\})$ when $S=O(\sqrt{T})$, where $T$ is the time horizon. The essential idea is to carefully design an adaptive adversary, who can adjust the loss function according to the cumulative switching cost of the player incurred so far based on the orthogonal technique. We then develop a simple gradient-based algorithm which enjoys the minimax optimal regret bound. Finally, we show that, for strongly convex functions, the regret bound can be improved to $O(\log T)$ for $S=\Omega(\log T)$, and $O(\min\{T/\exp(S)+S, T\})$ for $S=O(\log T)$.

NeurIPS Conference 2021 Conference Paper

Revisiting Smoothed Online Learning

• Lijun Zhang
• Wei Jiang
• Shiyin Lu
• Tianbao Yang

In this paper, we revisit the problem of smoothed online learning, in which the online learner suffers both a hitting cost and a switching cost, and target two performance metrics: competitive ratio and dynamic regret with switching cost. To bound the competitive ratio, we assume the hitting cost is known to the learner in each round, and investigate the simple idea of balancing the two costs by an optimization problem. Surprisingly, we find that minimizing the hitting cost alone is $\max(1, \frac{2}{\alpha})$-competitive for $\alpha$-polyhedral functions and $1 + \frac{4}{\lambda}$-competitive for $\lambda$-quadratic growth functions, both of which improve state-of-the-art results significantly. Moreover, when the hitting cost is both convex and $\lambda$-quadratic growth, we reduce the competitive ratio to $1 + \frac{2}{\sqrt{\lambda}}$ by minimizing the weighted sum of the hitting cost and the switching cost. To bound the dynamic regret with switching cost, we follow the standard setting of online convex optimization, in which the hitting cost is convex but hidden from the learner before making predictions. We modify Ader, an existing algorithm designed for dynamic regret, slightly to take into account the switching cost when measuring the performance. The proposed algorithm, named as Smoothed Ader, attains an optimal $O(\sqrt{T(1+P_T)})$ bound for dynamic regret with switching cost, where $P_T$ is the path-length of the comparator sequence. Furthermore, if the hitting cost is accessible in the beginning of each round, we obtain a similar guarantee without the bounded gradient condition, and establish an $\Omega(\sqrt{T(1+P_T)})$ lower bound to confirm the optimality.

NeurIPS Conference 2021 Conference Paper

Simple Stochastic and Online Gradient Descent Algorithms for Pairwise Learning

• Zhenhuan Yang
• Yunwen Lei
• Puyu Wang
• Tianbao Yang
• Yiming Ying

Pairwise learning refers to learning tasks where the loss function depends on a pair of instances. It instantiates many important machine learning tasks such as bipartite ranking and metric learning. A popular approach to handle streaming data in pairwise learning is an online gradient descent (OGD) algorithm, where one needs to pair the current instance with a buffering set of previous instances with a sufficiently large size and therefore suffers from a scalability issue. In this paper, we propose simple stochastic and online gradient descent methods for pairwise learning. A notable difference from the existing studies is that we only pair the current instance with the previous one in building a gradient direction, which is efficient in both the storage and computational complexity. We develop novel stability results, optimization, and generalization error bounds for both convex and nonconvex as well as both smooth and nonsmooth problems. We introduce novel techniques to decouple the dependency of models and the previous instance in both the optimization and generalization analysis. Our study resolves an open question on developing meaningful generalization bounds for OGD using a buffering set with a very small fixed size. We also extend our algorithms and stability analysis to develop differentially private SGD algorithms for pairwise learning which significantly improves the existing results.

ICML Conference 2021 Conference Paper

Stability and Generalization of Stochastic Gradient Methods for Minimax Problems

• Yunwen Lei
• Zhenhuan Yang
• Tianbao Yang
• Yiming Ying

Many machine learning problems can be formulated as minimax problems such as Generative Adversarial Networks (GANs), AUC maximization and robust estimation, to mention but a few. A substantial amount of studies are devoted to studying the convergence behavior of their stochastic gradient-type algorithms. In contrast, there is relatively little work on understanding their generalization, i. e. , how the learning models built from training examples would behave on test examples. In this paper, we provide a comprehensive generalization analysis of stochastic gradient methods for minimax problems under both convex-concave and nonconvex-nonconcave cases through the lens of algorithmic stability. We establish a quantitative connection between stability and several generalization measures both in expectation and with high probability. For the convex-concave setting, our stability analysis shows that stochastic gradient descent ascent attains optimal generalization bounds for both smooth and nonsmooth minimax problems. We also establish generalization bounds for both weakly-convex-weakly-concave and gradient-dominated problems. We report preliminary experimental results to verify our theory.

NeurIPS Conference 2021 Conference Paper

Stochastic Optimization of Areas Under Precision-Recall Curves with Provable Convergence

• Qi Qi
• Youzhi Luo
• Zhao Xu
• Shuiwang Ji
• Tianbao Yang

Areas under ROC (AUROC) and precision-recall curves (AUPRC) are common metrics for evaluating classification performance for imbalanced problems. Compared with AUROC, AUPRC is a more appropriate metric for highly imbalanced datasets. While stochastic optimization of AUROC has been studied extensively, principled stochastic optimization of AUPRC has been rarely explored. In this work, we propose a principled technical method to optimize AUPRC for deep learning. Our approach is based on maximizing the averaged precision (AP), which is an unbiased point estimator of AUPRC. We cast the objective into a sum of dependent compositional functions with inner functions dependent on random variables of the outer level. We propose efficient adaptive and non-adaptive stochastic algorithms named SOAP with provable convergence guarantee under mild conditions by leveraging recent advances in stochastic compositional optimization. Extensive experimental results on image and graph datasets demonstrate that our proposed method outperforms prior methods on imbalanced problems in terms of AUPRC. To the best of our knowledge, our work represents the first attempt to optimize AUPRC with provable convergence. The SOAP has been implemented in the libAUC library at https: //libauc. org/.

JMLR Journal 2020 Journal Article

A Data Efficient and Feasible Level Set Method for Stochastic Convex Optimization with Expectation Constraints

• Qihang Lin
• Selvaprabu Nadarajah
• Negar Soheili
• Tianbao Yang

Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. The SOEC objective and constraints contain expectations defined with respect to complex distributions or large data sets, leading to high computational complexity when solved by the algorithms that use exact functions and their gradients. Recent stochastic first order methods exhibit low computational complexity when handling SOECs but guarantee near-feasibility and near-optimality only at convergence. These methods may thus return highly infeasible solutions when heuristically terminated, as is often the case, due to theoretical convergence criteria being highly conservative. This issue limits the use of first order methods in several applications where the SOEC constraints encode implementation requirements. We design a stochastic feasible level set method (SFLS) for SOECs that has low complexity and emphasizes feasibility before convergence. Specifically, our level-set method solves a root-finding problem by calling a novel first order oracle that computes a stochastic upper bound on the level-set function by extending mirror descent and online validation techniques. We establish that SFLS maintains a high-probability feasible solution at each root-finding iteration and exhibits favorable complexity compared to state-of-the-art deterministic feasible level set and stochastic subgradient methods. Numerical experiments on three diverse applications highlight how SFLS finds feasible solutions with small optimality gaps with lower complexity than the former approaches. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

NeurIPS Conference 2020 Conference Paper

A Decentralized Parallel Algorithm for Training Generative Adversarial Nets

• Mingrui Liu
• Wei Zhang
• Youssef Mroueh
• Xiaodong Cui
• Jarret Ross
• Tianbao Yang
• Payel Das

Generative Adversarial Networks (GANs) are a powerful class of generative models in the deep learning community. Current practice on large-scale GAN training utilizes large models and distributed large-batch training strategies, and is implemented on deep learning frameworks (e. g. , TensorFlow, PyTorch, etc. ) designed in a centralized manner. In the centralized network topology, every worker needs to either directly communicate with the central node or indirectly communicate with all other workers in every iteration. However, when the network bandwidth is low or network latency is high, the performance would be significantly degraded. Despite recent progress on decentralized algorithms for training deep neural networks, it remains unclear whether it is possible to train GANs in a decentralized manner. The main difficulty lies at handling the nonconvex-nonconcave min-max optimization and the decentralized communication simultaneously. In this paper, we address this difficulty by designing the \textbf{first gradient-based decentralized parallel algorithm} which allows workers to have multiple rounds of communications in one iteration and to update the discriminator and generator simultaneously, and this design makes it amenable for the convergence analysis of the proposed decentralized algorithm. Theoretically, our proposed decentralized algorithm is able to solve a class of non-convex non-concave min-max problems with provable non-asymptotic convergence to first-order stationary point. Experimental results on GANs demonstrate the effectiveness of the proposed algorithm.

AAAI Conference 2020 Conference Paper

Adversarial Localized Energy Network for Structured Prediction

• Pingbo Pan
• Ping Liu
• Yan Yan
• Tianbao Yang
• Yi Yang

This paper focuses on energy model based structured output prediction. Though inheriting the benefits from energybased models to handle the sophisticated cases, previous deep energy-based methods suffered from the substantial computation cost introduced by the enormous amounts of gradient steps in the inference process. To boost the efficiency and accuracy of the energy-based models on structured output prediction, we propose a novel method analogous to the adversarial learning framework. Specifically, in our proposed framework, the generator consists of an inference network while the discriminator is comprised of an energy network. The two sub-modules, i. e. , the inference network and the energy network, can benefit each other mutually during the whole computation process. On the one hand, our modified inference network can boost the efficiency by predicting good initializations and reducing the searching space for the inference process; On the other hand, inheriting the benefits of the energy network, the energy module in our network can evaluate the quality of the generated output from the inference network and correspondingly provides a resourceful guide to the training of the inference network. In the ideal case, the adversarial learning strategy makes sure the two sub-modules can achieve an equilibrium state after steps. We conduct extensive experiments to verify the effectiveness and efficiency of our proposed method.

ICML Conference 2020 Conference Paper

Communication-Efficient Distributed Stochastic AUC Maximization with Deep Neural Networks

• Zhishuai Guo
• Mingrui Liu
• Zhuoning Yuan
• Li Shen 0005
• Wei Liu 0005
• Tianbao Yang

In this paper, we study distributed algorithms for large-scale AUC maximization with a deep neural network as a predictive model. Although distributed learning techniques have been investigated extensively in deep learning, they are not directly applicable to stochastic AUC maximization with deep neural networks due to its striking differences from standard loss minimization problems (e. g. , cross-entropy). Towards addressing this challenge, we propose and analyze a communication-efficient distributed optimization algorithm based on a \emph{non-convex concave} reformulation of the AUC maximization, in which the communication of both the primal variable and the dual variable between each worker and the parameter server only occurs after multiple steps of gradient-based updates in each worker. Compared with the naive parallel version of an existing algorithm that computes stochastic gradients at individual machines and averages them for updating the model parameter, our algorithm requires a much less number of communication rounds and still achieves linear speedup in theory. To the best of our knowledge, this is the \textbf{first} work that solves the \emph{non-convex concave min-max} problem for AUC maximization with deep neural networks in a communication-efficient distributed manner while still maintaining the linear speedup property in theory. Our experiments on several benchmark datasets show the effectiveness of our algorithm and also confirm our theory.

AAAI Conference 2020 Conference Paper

Deep Unsupervised Binary Coding Networks for Multivariate Time Series Retrieval

• Dixian Zhu
• Dongjin Song
• Yuncong Chen
• Cristian Lumezanu
• Wei Cheng
• Bo Zong
• Jingchao Ni
• Takehiko Mizoguchi

Multivariate time series data are becoming increasingly ubiquitous in varies real-world applications such as smart city, power plant monitoring, wearable devices, etc. Given the current time series segment, how to retrieve similar segments within the historical data in an efficient and effective manner is becoming increasingly important. As it can facilitate underlying applications such as system status identification, anomaly detection, etc. Despite the fact that various binary coding techniques can be applied to this task, few of them are specially designed for multivariate time series data in an unsupervised setting. To this end, we present Deep Unsupervised Binary Coding Networks (DUBCNs) to perform multivariate time series retrieval. DUBCNs employ the Long Short-Term Memory (LSTM) encoder-decoder framework to capture the temporal dynamics within the input segment and consist of three key components, i. e. , a temporal encoding mechanism to capture the temporal order of different segments within a mini-batch, a clustering loss on the hidden feature space to capture the hidden feature structure, and an adversarial loss based upon Generative Adversarial Networks (GANs) to enhance the generalization capability of the generated binary codes. Thoroughly empirical studies on three public datasets demonstrated that the proposed DUBCNs can outperform state-of-the-art unsupervised binary coding techniques.

NeurIPS Conference 2020 Conference Paper

Improved Schemes for Episodic Memory-based Lifelong Learning

• Yunhui Guo
• Mingrui Liu
• Tianbao Yang
• Tajana Rosing

Current deep neural networks can achieve remarkable performance on a single task. However, when the deep neural network is continually trained on a sequence of tasks, it seems to gradually forget the previous learned knowledge. This phenomenon is referred to as catastrophic forgetting and motivates the field called lifelong learning. Recently, episodic memory based approaches such as GEM and A-GEM have shown remarkable performance. In this paper, we provide the first unified view of episodic memory based approaches from an optimization's perspective. This view leads to two improved schemes for episodic memory based lifelong learning, called MEGA-\rom{1} and MEGA-\rom{2}. MEGA-\rom{1} and MEGA-\rom{2} modulate the balance between old tasks and the new task by integrating the current gradient with the gradient computed on the episodic memory. Notably, we show that GEM and A-GEM are degenerate cases of MEGA-\rom{1} and MEGA-\rom{2} which consistently put the same emphasis on the current task, regardless of how the loss changes over time. Our proposed schemes address this issue by using novel loss-balancing updating rules, which drastically improve the performance over GEM and A-GEM. Extensive experimental results show that the proposed schemes significantly advance the state-of-the-art on four commonly used lifelong learning benchmarks, reducing the error by up to 18%.

NeurIPS Conference 2020 Conference Paper

Optimal Epoch Stochastic Gradient Descent Ascent Methods for Min-Max Optimization

• Yan Yan
• Yi Xu
• Qihang Lin
• Wei Liu
• Tianbao Yang

Epoch gradient descent method (a. k. a. Epoch-GD) proposed by (Hazan and Kale, 2011) was deemeda breakthrough for stochastic strongly convex minimization, which achieves theoptimal convergence rate of O(1/T) with T iterative updates for the objective gap. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whethera fast rate ofO(1/T)for theduality gapis achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some re-cent studies have proposed stochastic algorithms with fast convergence rates formin-max problems, they require additional assumptions about the problem, e. g. ,smoothness, bi-linear structure, etc. In this paper, we bridge this gap by providinga sharp analysis of epoch-wise stochastic gradient descent ascent method (referredto as Epoch-GDA) for solving strongly convex strongly concave (SCSC) min-maxproblems, without imposing any additional assumption about smoothness or the function’s structure. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the optimal rate ofO(1/T)for the duality gapof general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can also be used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to a nearly optimal complexity without resorting to smoothness or other structural conditions.

ICML Conference 2020 Conference Paper

Quadratically Regularized Subgradient Methods for Weakly Convex Optimization with Weakly Convex Constraints

• Runchao Ma
• Qihang Lin
• Tianbao Yang

Optimization models with non-convex constraints arise in many tasks in machine learning, e. g. , learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed with theoretical convergence guarantees for non-convex unconstrained problems, it remains a challenge to design provably efficient algorithms for problems with non-convex functional constraints. This paper proposes a class of subgradient methods for constrained optimization where the objective function and the constraint functions are weakly convex and nonsmooth. Our methods solve a sequence of strongly convex subproblems, where a quadratic regularization term is added to both the objective function and each constraint function. Each subproblem can be solved by various algorithms for strongly convex optimization. Under a uniform Slater’s condition, we establish the computation complexities of our methods for finding a nearly stationary point.

ICLR Conference 2020 Conference Paper

Stochastic AUC Maximization with Deep Neural Networks

• Mingrui Liu
• Zhuoning Yuan
• Yiming Ying
• Tianbao Yang

Stochastic AUC maximization has garnered an increasing interest due to better fit to imbalanced data classification. However, existing works are limited to stochastic AUC maximization with a linear predictive model, which restricts its predictive power when dealing with extremely complex data. In this paper, we consider stochastic AUC maximization problem with a deep neural network as the predictive model. Building on the saddle point reformulation of a surrogated loss of AUC, the problem can be cast into a {\it non-convex concave} min-max problem. The main contribution made in this paper is to make stochastic AUC maximization more practical for deep neural networks and big data with theoretical insights as well. In particular, we propose to explore Polyak-\L{}ojasiewicz (PL) condition that has been proved and observed in deep learning, which enables us to develop new stochastic algorithms with even faster convergence rate and more practical step size scheme. An AdaGrad-style algorithm is also analyzed under the PL condition with adaptive convergence rate. Our experimental results demonstrate the effectiveness of the proposed algorithms.

ICML Conference 2020 Conference Paper

Stochastic Optimization for Non-convex Inf-Projection Problems

• Yan Yan 0006
• Yi Xu 0008
• Lijun Zhang 0005
• Xiaoyu Wang 0002
• Tianbao Yang

In this paper, we study a family of non-convex and possibly non-smooth inf-projection minimization problems, where the target objective function is equal to minimization of a joint function over another variable. This problem include difference of convex (DC) functions and a family of bi-convex functions as special cases. We develop stochastic algorithms and establish their first-order convergence for finding a (nearly) stationary solution of the target non-convex function under different conditions of the component functions. To the best of our knowledge, this is the first work that comprehensively studies stochastic optimization of non-convex inf-projection minimization problems with provable convergence guarantee. Our algorithms enable efficient stochastic optimization of a family of non-decomposable DC functions and a family of bi-convex functions. To demonstrate the power of the proposed algorithms we consider an important application in variance-based regularization. Experiments verify the effectiveness of our inf-projection based formulation and the proposed stochastic algorithm in comparison with previous stochastic algorithms based on the min-max formulation for achieving the same effect.

ICLR Conference 2020 Conference Paper

Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets

• Mingrui Liu
• Youssef Mroueh
• Jerret Ross
• Wei Zhang 0022
• Xiaodong Cui
• Payel Das
• Tianbao Yang

Adaptive gradient algorithms perform gradient-based updates using the history of gradients and are ubiquitous in training deep neural networks. While adaptive gradient methods theory is well understood for minimization problems, the underlying factors driving their empirical success in min-max problems such as GANs remain unclear. In this paper, we aim at bridging this gap from both theoretical and empirical perspectives. First, we analyze a variant of Optimistic Stochastic Gradient (OSG) proposed in~\citep{daskalakis2017training} for solving a class of non-convex non-concave min-max problem and establish $O(\epsilon^{-4})$ complexity for finding $\epsilon$-first-order stationary point, in which the algorithm only requires invoking one stochastic first-order oracle while enjoying state-of-the-art iteration complexity achieved by stochastic extragradient method by~\citep{iusem2017extragradient}. Then we propose an adaptive variant of OSG named Optimistic Adagrad (OAdagrad) and reveal an \emph{improved} adaptive complexity $O\left(\epsilon^{-\frac{2}{1-\alpha}}\right)$, where $\alpha$ characterizes the growth rate of the cumulative stochastic gradient and $0\leq \alpha\leq 1/2$. To the best of our knowledge, this is the first work for establishing adaptive complexity in non-convex non-concave min-max optimization. Empirically, our experiments show that indeed adaptive gradient algorithms outperform their non-adaptive counterparts in GAN training. Moreover, this observation can be explained by the slow growth rate of the cumulative stochastic gradient, as observed empirically.

ICML Conference 2019 Conference Paper

Katalyst: Boosting Convex Katayusha for Non-Convex Problems with a Large Condition Number

• Zaiyi Chen
• Yi Xu 0008
• Haoyuan Hu
• Tianbao Yang

An important class of non-convex objectives that has wide applications in machine learning consists of a sum of $n$ smooth functions and a non-smooth convex function. Tremendous studies have been devoted to conquering these problems by leveraging one of the two types of variance reduction techniques, i. e. , SVRG-type that computes a full gradient occasionally and SAGA-type that maintains $n$ stochastic gradients at every iteration. In practice, SVRG-type is preferred to SAGA-type due to its potentially less memory costs. An interesting question that has been largely ignored is how to improve the complexity of variance reduction methods for problems with a large condition number that measures the degree to which the objective is close to a convex function. In this paper, we present a simple but non-trivial boosting of a state-of-the-art SVRG-type method for convex problems (namely Katyusha) to enjoy an improved complexity for solving non-convex problems with a large condition number (that is close to a convex function). To the best of our knowledge, its complexity has the best dependence on $n$ and the degree of non-convexity, and also matches that of a recent SAGA-type accelerated stochastic algorithm for a constrained non-convex smooth optimization problem.

UAI Conference 2019 Conference Paper

Learning with Non-Convex Truncated Losses by SGD

• Yi Xu 0008
• Shenghuo Zhu
• Sen Yang
• Chi Zhang 0012
• Rong Jin 0001
• Tianbao Yang

Learning with a convex loss function has been a dominating paradigm for many years. It remains an interesting question how non-convex loss functions help improve the generalization of learning with broad applicability. In this paper, we study a family of objective functions formed by truncating traditional loss functions, which is applicable to both shallow learning and deep learning. Truncating loss functions has potential to be less vulnerable and more robust to large noise in observations that could be adversarial. More importantly, it is a generic technique without assuming the knowledge of noise distribution. To justify non-convex learning with truncated losses, we establish excess risk bounds of empirical risk minimization based on truncated losses for heavy-tailed output, and statistical error of an approximate stationary point found by stochastic gradient descent (SGD) method. Our experiments for shallow and deep learning for regression with outliers, corrupted data and heavy-tailed noise further justify the proposed method.

NeurIPS Conference 2019 Conference Paper

Non-asymptotic Analysis of Stochastic Methods for Non-Smooth Non-Convex Regularized Problems

• Yi Xu
• Rong Jin
• Tianbao Yang

Stochastic Proximal Gradient (SPG) methods have been widely used for solving optimization problems with a simple (possibly non-smooth) regularizer in machine learning and statistics. However, to the best of our knowledge no non-asymptotic convergence analysis of SPG exists for non-convex optimization with a non-smooth and non-convex regularizer. All existing non-asymptotic analysis of SPG for solving non-smooth non-convex problems require the non-smooth regularizer to be a convex function, and hence are not applicable to a non-smooth non-convex regularized problem. This work initiates the analysis to bridge this gap and opens the door to non-asymptotic convergence analysis of non-smooth non-convex regularized problems. We analyze several variants of mini-batch SPG methods for minimizing a non-convex objective that consists of a smooth non-convex loss and a non-smooth non-convex regularizer. Our contributions are two-fold: (i) we show that they enjoy the same complexities as their counterparts for solving convex regularized non-convex problems in terms of finding an approximate stationary point; (ii) we develop more practical variants using dynamic mini-batch size instead of a fixed mini-batch size without requiring the target accuracy level of solution. The significance of our results is that they improve upon the-state-of-art results for solving non-smooth non-convex regularized problems. We also empirically demonstrate the effectiveness of the considered SPG methods in comparison with other peer stochastic methods.

IJCAI Conference 2019 Conference Paper

On the Convergence of (Stochastic) Gradient Descent with Extrapolation for Non-Convex Minimization

• Yi Xu
• Zhuoning Yuan
• Sen Yang
• Rong Jin
• Tianbao Yang

Extrapolation is a well-known technique for solving convex optimization and variational inequalities and recently attracts some attention for non-convex optimization. Several recent works have empirically shown its success in some machine learning tasks. However, it has not been analyzed for non-convex minimization and there still remains a gap between the theory and the practice. In this paper, we analyze gradient descent and stochastic gradient descent with extrapolation for finding an approximate first-order stationary point in smooth non-convex optimization problems. Our convergence upper bounds show that the algorithms with extrapolation can be accelerated than without extrapolation.

JMLR Journal 2019 Journal Article

Relative Error Bound Analysis for Nuclear Norm Regularized Matrix Completion

• Lijun Zhang
• Tianbao Yang
• Rong Jin
• Zhi-Hua Zhou

In this paper, we develop a relative error bound for nuclear norm regularized matrix completion, with the focus on the completion of full-rank matrices. Under the assumption that the top eigenspaces of the target matrix are incoherent, we derive a relative upper bound for recovering the best low-rank approximation of the unknown matrix. Although multiple works have been devoted to analyzing the recovery error of full-rank matrix completion, their error bounds are usually additive, making it impossible to obtain the perfect recovery case and more generally difficult to leverage the skewed distribution of eigenvalues. Our analysis is built upon the optimality condition of the regularized formulation and existing guarantees for low-rank matrix completion. To the best of our knowledge, this is the first relative bound that has been proved for the regularized formulation of matrix completion. [abs] [ pdf ][ bib ] &copy JMLR 2019. ( edit, beta )

NeurIPS Conference 2019 Conference Paper

Stagewise Training Accelerates Convergence of Testing Error Over SGD

• Zhuoning Yuan
• Yan Yan
• Rong Jin
• Tianbao Yang

Stagewise training strategy is widely used for learning neural networks, which runs a stochastic algorithm (e. g. , SGD) starting with a relatively large step size (aka learning rate) and geometrically decreasing the step size after a number of iterations. It has been observed that the stagewise SGD has much faster convergence than the vanilla SGD with a polynomially decaying step size in terms of both training error and testing error. {\it But how to explain this phenomenon has been largely ignored by existing studies. } This paper provides some theoretical evidence for explaining this faster convergence. In particular, we consider a stagewise training strategy for minimizing empirical risk that satisfies the Polyak-\L ojasiewicz (PL) condition, which has been observed/proved for neural networks and also holds for a broad family of convex functions. For convex loss functions and two classes of ``nice-behaviored" non-convex objectives that are close to a convex function, we establish faster convergence of stagewise training than the vanilla SGD under the PL condition on both training error and testing error. Experiments on stagewise learning of deep residual networks exhibits that it satisfies one type of non-convexity assumption and therefore can be explained by our theory.

ICML Conference 2019 Conference Paper

Stochastic Optimization for DC Functions and Non-smooth Non-convex Regularizers with Non-asymptotic Convergence

• Yi Xu 0008
• Qi Qi 0006
• Qihang Lin
• Rong Jin 0001
• Tianbao Yang

Difference of convex (DC) functions cover a broad family of non-convex and possibly non-smooth and non-differentiable functions, and have wide applications in machine learning and statistics. Although deterministic algorithms for DC functions have been extensively studied, stochastic optimization that is more suitable for learning with big data remains under-explored. In this paper, we propose new stochastic optimization algorithms and study their first-order convergence theories for solving a broad family of DC functions. We improve the existing algorithms and theories of stochastic optimization for DC functions from both practical and theoretical perspectives. Moreover, we extend the proposed stochastic algorithms for DC functions to solve problems with a general non-convex non-differentiable regularizer, which does not necessarily have a DC decomposition but enjoys an efficient proximal mapping. To the best of our knowledge, this is the first work that gives the first non-asymptotic convergence for solving non-convex optimization whose objective has a general non-convex non-differentiable regularizer.

IJCAI Conference 2018 Conference Paper

A Generic Approach for Accelerating Stochastic Zeroth-Order Convex Optimization

• XiaoTian Yu
• Irwin King
• Michael R. Lyu
• Tianbao Yang

In this paper, we propose a generic approach for accelerating the convergence of existing algorithms to solve the problem of stochastic zeroth-order convex optimization (SZCO). Standard techniques for accelerating the convergence of stochastic zeroth-order algorithms are by exploring multiple functional evaluations (e. g. , two-point evaluations), or by exploiting global conditions of the problem (e. g. , smoothness and strong convexity). Nevertheless, these classic acceleration techniques are necessarily restricting the applicability of newly developed algorithms. The key of our proposed generic approach is to explore a local growth condition (or called local error bound condition) of the objective function in SZCO. The benefits of the proposed acceleration technique are: (i) it is applicable to both settings with one-point evaluation and two-point evaluations; (ii) it does not necessarily require strong convexity or smoothness condition of the objective function; (iii) it yields an improvement on convergence for a broad family of problems. Empirical studies in various settings demonstrate the effectiveness of the proposed acceleration approach.

IJCAI Conference 2018 Conference Paper

A Unified Analysis of Stochastic Momentum Methods for Deep Learning

• Yan Yan
• Tianbao Yang
• Zhe Li
• Qihang Lin
• Yi Yang

Stochastic momentum methods have been widely adopted in training deep neural networks. However, their theoretical analysis of convergence of the training objective and the generalization error for prediction is still under-explored. This paper aims to bridge the gap between practice and theory by analyzing the stochastic gradient (SG) method, and the stochastic momentum methods including two famous variants, i. e. , the stochastic heavy-ball (SHB) method and the stochastic variant of Nesterov? s accelerated gradient (SNAG) method. We propose a framework that unifies the three variants. We then derive the convergence rates of the norm of gradient for the non-convex optimization problem, and analyze the generalization performance through the uniform stability approach. Particularly, the convergence analysis of the training objective exhibits that SHB and SNAG have no advantage over SG. However, the stability analysis shows that the momentum term can improve the stability of the learned model and hence improve the generalization performance. These theoretical insights verify the common wisdom and are also corroborated by our empirical analysis on deep learning.

NeurIPS Conference 2018 Conference Paper

Adaptive Negative Curvature Descent with Applications in Non-convex Optimization

• Mingrui Liu
• Zhe Li
• Xiaoyu Wang
• Jinfeng Yi
• Tianbao Yang

Negative curvature descent (NCD) method has been utilized to design deterministic or stochastic algorithms for non-convex optimization aiming at finding second-order stationary points or local minima. In existing studies, NCD needs to approximate the smallest eigen-value of the Hessian matrix with a sufficient precision (e. g. , $\epsilon_2\ll 1$) in order to achieve a sufficiently accurate second-order stationary solution (i. e. , $\lambda_{\min}(\nabla^2 f(\x))\geq -\epsilon_2)$. One issue with this approach is that the target precision $\epsilon_2$ is usually set to be very small in order to find a high quality solution, which increases the complexity for computing a negative curvature. To address this issue, we propose an adaptive NCD to allow for an adaptive error dependent on the current gradient's magnitude in approximating the smallest eigen-value of the Hessian, and to encourage competition between a noisy NCD step and gradient descent step. We consider the applications of the proposed adaptive NCD for both deterministic and stochastic non-convex optimization, and demonstrate that it can help reduce the the overall complexity in computing the negative curvatures during the course of optimization without sacrificing the iteration complexity.

ICML Conference 2018 Conference Paper

Dynamic Regret of Strongly Adaptive Methods

• Lijun Zhang 0005
• Tianbao Yang
• Rong Jin 0001
• Zhi-Hua Zhou

To cope with changing environments, recent developments in online learning have introduced the concepts of adaptive regret and dynamic regret independently. In this paper, we illustrate an intrinsic connection between these two concepts by showing that the dynamic regret can be expressed in terms of the adaptive regret and the functional variation. This observation implies that strongly adaptive algorithms can be directly leveraged to minimize the dynamic regret. As a result, we present a series of strongly adaptive algorithms that have small dynamic regrets for convex functions, exponentially concave functions, and strongly convex functions, respectively. To the best of our knowledge, this is the first time that exponential concavity is utilized to upper bound the dynamic regret. Moreover, all of those adaptive algorithms do not need any prior knowledge of the functional variation, which is a significant advantage over previous specialized methods for minimizing dynamic regret.

NeurIPS Conference 2018 Conference Paper

Fast Rates of ERM and Stochastic Approximation: Adaptive to Error Bound Conditions

• Mingrui Liu
• Xiaoxuan Zhang
• Lijun Zhang
• Rong Jin
• Tianbao Yang

Error bound conditions (EBC) are properties that characterize the growth of an objective function when a point is moved away from the optimal set. They have recently received increasing attention in the field of optimization for developing optimization algorithms with fast convergence. However, the studies of EBC in statistical learning are hitherto still limited. The main contributions of this paper are two-fold. First, we develop fast and intermediate rates of empirical risk minimization (ERM) under EBC for risk minimization with Lipschitz continuous, and smooth convex random functions. Second, we establish fast and intermediate rates of an efficient stochastic approximation (SA) algorithm for risk minimization with Lipschitz continuous random functions, which requires only one pass of $n$ samples and adapts to EBC. For both approaches, the convergence rates span a full spectrum between $\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the power constant in EBC, and could be even faster than $O(1/n)$ in special cases for ERM. Moreover, these convergence rates are automatically adaptive without using any knowledge of EBC. Overall, this work not only strengthens the understanding of ERM for statistical learning but also brings new fast stochastic algorithms for solving a broad range of statistical learning problems.

ICML Conference 2018 Conference Paper

Fast Stochastic AUC Maximization with O(1/n)-Convergence Rate

• Mingrui Liu
• Xiaoxuan Zhang
• Zaiyi Chen
• Xiaoyu Wang
• Tianbao Yang

In this paper, we consider statistical learning with AUC (area under ROC curve) maximization in the classical stochastic setting where one random data drawn from an unknown distribution is revealed at each iteration for updating the model. Although consistent convex surrogate losses for AUC maximization have been proposed to make the problem tractable, it remains an challenging problem to design fast optimization algorithms in the classical stochastic setting due to that the convex surrogate loss depends on random pairs of examples from positive and negative classes. Building on a saddle point formulation for a consistent square loss, this paper proposes a novel stochastic algorithm to improve the standard $O(1/\sqrt{n})$ convergence rate to $\widetilde O(1/n)$ convergence rate without strong convexity assumption or any favorable statistical assumptions (e. g. , low noise), where $n$ is the number of random samples. To the best of our knowledge, this is the first stochastic algorithm for AUC maximization with a statistical convergence rate as fast as $O(1/n)$ up to a logarithmic factor. Extensive experiments on eight large-scale benchmark data sets demonstrate the superior performance of the proposed algorithm comparing with existing stochastic or online algorithms for AUC maximization.

NeurIPS Conference 2018 Conference Paper

Faster Online Learning of Optimal Threshold for Consistent F-measure Optimization

• Xiaoxuan Zhang
• Mingrui Liu
• Xun Zhou
• Tianbao Yang

In this paper, we consider online F-measure optimization (OFO). Unlike traditional performance metrics (e. g. , classification error rate), F-measure is non-decomposable over training examples and is a non-convex function of model parameters, making it much more difficult to be optimized in an online fashion. Most existing results of OFO usually suffer from high memory/computational costs and/or lack statistical consistency guarantee for optimizing F-measure at the population level. To advance OFO, we propose an efficient online algorithm based on simultaneously learning a posterior probability of class and learning an optimal threshold by minimizing a stochastic strongly convex function with unknown strong convexity parameter. A key component of the proposed method is a novel stochastic algorithm with low memory and computational costs, which can enjoy a convergence rate of $\widetilde O(1/\sqrt{n})$ for learning the optimal threshold under a mild condition on the convergence of the posterior probability, where $n$ is the number of processed examples. It is provably faster than its predecessor based on a heuristic for updating the threshold. The experiments verify the efficiency of the proposed algorithm in comparison with state-of-the-art OFO algorithms.

NeurIPS Conference 2018 Conference Paper

First-order Stochastic Algorithms for Escaping From Saddle Points in Almost Linear Time

• Yi Xu
• Rong Jin
• Tianbao Yang

(This is a theory paper) In this paper, we consider first-order methods for solving stochastic non-convex optimization problems. The key building block of the proposed algorithms is first-order procedures to extract negative curvature from the Hessian matrix through a principled sequence starting from noise, which are referred to {\it NEgative-curvature-Originated-from-Noise or NEON} and are of independent interest. Based on this building block, we design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in the problem's dimensionality). In particular, we develop a general framework of {\it first-order stochastic algorithms} with a second-order convergence guarantee based on our new technique and existing algorithms that may only converge to a first-order stationary point. For finding a nearly {\it second-order stationary point} $\x$ such that $\|\nabla F(\x)\|\leq \epsilon$ and $\nabla^2 F(\x)\geq -\sqrt{\epsilon}I$ (in high probability), the best time complexity of the presented algorithms is $\widetilde O(d/\epsilon^{3. 5})$, where $F(\cdot)$ denotes the objective function and $d$ is the dimensionality of the problem. To the best of our knowledge, this is the first theoretical result of first-order stochastic algorithms with an almost linear time in terms of problem's dimensionality for finding second-order stationary points, which is even competitive with existing stochastic algorithms hinging on the second-order information.

ICML Conference 2018 Conference Paper

Level-Set Methods for Finite-Sum Constrained Convex Optimization

• Qihang Lin
• Runchao Ma
• Tianbao Yang

We consider the constrained optimization where the objective function and the constraints are defined as summation of finitely many loss functions. This model has applications in machine learning such as Neyman-Pearson classification. We consider two level-set methods to solve this class of problems, an existing inexact Newton method and a new feasible level-set method. To update the level parameter towards the optimality, both methods require an oracle that generates upper and lower bounds as well as an affine-minorant of the level function. To construct the desired oracle, we reformulate the level function as the value of a saddle-point problem using the conjugate and perspective of the loss functions. Then a stochastic variance-reduced gradient method with a special Bregman divergence is proposed as the oracle for solving that saddle-point problem. The special divergence ensures the proximal mapping in each iteration can be solved in a closed form. The total complexity of both level-set methods using the proposed oracle are analyzed.

JMLR Journal 2018 Journal Article

RSG: Beating Subgradient Method without Smoothness and Strong Convexity

• Tianbao Yang
• Qihang Lin

In this paper, we study the efficiency of a R estarted S ub G radient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an $\epsilon$-optimal solution with a lower complexity than the SG method. In particular, we first show that RSG can reduce the dependence of SG's iteration complexity on the distance between the initial solution and the optimal set to that between the $\epsilon$-level set and the optimal set {multiplied by a logarithmic factor}. Moreover, we show the advantages of RSG over SG in solving a broad family of problems that satisfy a local error bound condition, and also demonstrate its advantages for three specific families of convex optimization problems with different power constants in the local error bound condition. (a) For the problems whose epigraph is a polyhedron, RSG is shown to converge linearly. (b) For the problems with local quadratic growth property in the $\epsilon$-sublevel set, RSG has an $O(\frac{1}{\epsilon}\log(\frac{1}{\epsilon}))$ iteration complexity. (c) For the problems that admit a local Kurdyka-Łojasiewicz property with a power constant of $\beta\in[0,1)$, RSG has an $O(\frac{1}{\epsilon^{2\beta}}\log(\frac{1}{\epsilon}))$ iteration complexity. The novelty of our analysis lies at exploiting the lower bound of the first-order optimality residual at the $\epsilon$-level set. It is this novelty that allows us to explore the local properties of functions (e.g., local quadratic growth property, local Kurdyka-\L ojasiewicz property, more generally local error bound conditions) to develop the improved convergence of RSG. We also develop a practical variant of RSG enjoying faster convergence than the SG method, which can be run without knowing the involved parameters in the local error bound condition. We demonstrate the effectiveness of the proposed algorithms on several machine learning tasks including regression, classification and matrix completion. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

ICML Conference 2018 Conference Paper

SADAGRAD: Strongly Adaptive Stochastic Gradient Methods

• Zaiyi Chen
• Yi Xu 0008
• Enhong Chen
• Tianbao Yang

Although the convergence rates of existing variants of ADAGRAD have a better dependence on the number of iterations under the strong convexity condition, their iteration complexities have a explicitly linear dependence on the dimensionality of the problem. To alleviate this bad dependence, we propose a simple yet novel variant of ADAGRAD for stochastic (weakly) strongly convex optimization. Different from existing variants, the proposed variant (referred to as SADAGRAD) uses an adaptive restarting scheme in which (i) ADAGRAD serves as a sub-routine and is restarted periodically; (ii) the number of iterations for restarting ADAGRAD depends on the history of learning that incorporates knowledge of the geometry of the data. In addition to the adaptive proximal functions and adaptive number of iterations for restarting, we also develop a variant that is adaptive to the (implicit) strong convexity from the data, which together makes the proposed algorithm strongly adaptive. In terms of iteration complexity, in the worst case SADAGRAD has an O(1/\epsilon) for finding an \epsilon-optimal solution similar to other variants. However, it could enjoy faster convergence and much better dependence on the problem’s dimensionality when stochastic gradients are sparse. Extensive experiments on large-scale data sets demonstrate the efficiency of the proposed algorithms in comparison with several variants of ADAGRAD and stochastic gradient method.

AAAI Conference 2017 Conference Paper

A Framework of Online Learning with Imbalanced Streaming Data

• Yan Yan
• Tianbao Yang
• Yi Yang
• Jianhui Chen

A challenge for mining large-scale streaming data overlooked by most existing studies on online learning is the skewdistribution of examples over different classes. Many previous works have considered cost-sensitive approaches in an online setting for streaming data, where fixed costs are assigned to different classes, or ad-hoc costs are adapted based on the distribution of data received so far. However, it is not necessary for them to achieve optimal performance in terms of the measures suited for imbalanced data, such as Fmeasure, area under ROC curve (AUROC), area under precision and recall curve (AUPRC). This work proposes a general framework for online learning with imbalanced streaming data, where examples are coming sequentially and models are updated accordingly on-the-fly. By simultaneously learning multiple classifiers with different cost vectors, the proposed method can be adopted for different target measures for imbalanced data, including F-measure, AUROC and AUPRC. Moreover, we present a rigorous theoretical justification of the proposed framework for the F-measure maximization. Our empirical studies demonstrate the competitive if not better performance of the proposed method compared to previous cost-sensitive and resampling based online learning algorithms and those that are designed for optimizing certain measures.

ICML Conference 2017 Conference Paper

A Richer Theory of Convex Constrained Optimization with Reduced Projections and Improved Rates

• Tianbao Yang
• Qihang Lin
• Lijun Zhang 0005

This paper focuses on convex constrained optimization problems, where the solution is subject to a convex inequality constraint. In particular, we aim at challenging problems for which both projection into the constrained domain and a linear optimization under the inequality constraint are time-consuming, which render both projected gradient methods and conditional gradient methods (a. k. a. the Frank-Wolfe algorithm) expensive. In this paper, we develop projection reduced optimization algorithms for both smooth and non-smooth optimization with improved convergence rates under a certain regularity condition of the constraint function. We first present a general theory of optimization with only one projection. Its application to smooth optimization with only one projection yields $O(1/\epsilon)$ iteration complexity, which improves over the $O(1/\epsilon^2)$ iteration complexity established before for non-smooth optimization and can be further reduced under strong convexity. Then we introduce a local error bound condition and develop faster algorithms for non-strongly convex optimization at the price of a logarithmic number of projections. In particular, we achieve an iteration complexity of $\widetilde O(1/\epsilon^{2(1-\theta)})$ for non-smooth optimization and $\widetilde O(1/\epsilon^{1-\theta})$ for smooth optimization, where $\theta\in(0, 1]$ appearing the local error bound condition characterizes the functional local growth rate around the optimal solutions. Novel applications in solving the constrained $\ell_1$ minimization problem and a positive semi-definite constrained distance metric learning problem demonstrate that the proposed algorithms achieve significant speed-up compared with previous algorithms.

AAAI Conference 2017 Conference Paper

A Two-Stage Approach for Learning a Sparse Model with Sharp Excess Risk Analysis

• Zhe Li
• Tianbao Yang
• Lijun Zhang
• Rong Jin

This paper aims to provide a sharp excess risk guarantee for learning a sparse linear model without any assumptions about the strong convexity of the expected loss and the sparsity of the optimal solution in hindsight. Given a target level _ for the excess risk, an interesting question to ask is how many examples and how large the support set of the solution are enough for learning a good model with the target excess risk. To answer these questions, we present a two-stage algorithm that (i) in the first stage an epoch based stochastic optimization algorithm is exploited with an established O(1/_) bound on the sample complexity; and (ii) in the second stage a distribution dependent randomized sparsification is presented with an O(1/_) bound on the sparsity (referred to as support complexity) of the resulting model. Compared to previous works, our contributions lie at (i) we reduce the order of the sample complexity from O(1/_2) to O(1/_) without the strong convexity assumption; and (ii) we reduce the constant in O(1/_) for the sparsity by exploring the distribution dependent sampling.

NeurIPS Conference 2017 Conference Paper

Adaptive Accelerated Gradient Converging Method under H\"{o}lderian Error Bound Condition

• Mingrui Liu
• Tianbao Yang

Recent studies have shown that proximal gradient (PG) method and accelerated gradient method (APG) with restarting can enjoy a linear convergence under a weaker condition than strong convexity, namely a quadratic growth condition (QGC). However, the faster convergence of restarting APG method relies on the potentially unknown constant in QGC to appropriately restart APG, which restricts its applicability. We address this issue by developing a novel adaptive gradient converging methods, i. e. , leveraging the magnitude of proximal gradient as a criterion for restart and termination. Our analysis extends to a much more general condition beyond the QGC, namely the H\"{o}lderian error bound (HEB) condition. {\it The key technique} for our development is a novel synthesis of {\it adaptive regularization and a conditional restarting scheme}, which extends previous work focusing on strongly convex problems to a much broader family of problems. Furthermore, we demonstrate that our results have important implication and applications in machine learning: (i) if the objective function is coercive and semi-algebraic, PG's convergence speed is essentially $o(\frac{1}{t})$, where $t$ is the total number of iterations; (ii) if the objective function consists of an $\ell_1$, $\ell_\infty$, $\ell_{1, \infty}$, or huber norm regularization and a convex smooth piecewise quadratic loss (e. g. , square loss, squared hinge loss and huber loss), the proposed algorithm is parameter-free and enjoys a {\it faster linear convergence} than PG without any other assumptions (e. g. , restricted eigen-value condition). It is notable that our linear convergence results for the aforementioned problems are global instead of local. To the best of our knowledge, these improved results are first shown in this work.

NeurIPS Conference 2017 Conference Paper

Adaptive SVRG Methods under Error Bound Conditions with Unknown Growth Parameter

• Yi Xu
• Qihang Lin
• Tianbao Yang

Error bound, an inherent property of an optimization problem, has recently revived in the development of algorithms with improved global convergence without strong convexity. The most studied error bound is the quadratic error bound, which generalizes strong convexity and is satisfied by a large family of machine learning problems. Quadratic error bound have been leveraged to achieve linear convergence in many first-order methods including the stochastic variance reduced gradient (SVRG) method, which is one of the most important stochastic optimization methods in machine learning. However, the studies along this direction face the critical issue that the algorithms must depend on an unknown growth parameter (a generalization of strong convexity modulus) in the error bound. This parameter is difficult to estimate exactly and the algorithms choosing this parameter heuristically do not have theoretical convergence guarantee. To address this issue, we propose novel SVRG methods that automatically search for this unknown parameter on the fly of optimization while still obtain almost the same convergence rate as when this parameter is known. We also analyze the convergence property of SVRG methods under H\"{o}lderian error bound, which generalizes the quadratic error bound.

NeurIPS Conference 2017 Conference Paper

ADMM without a Fixed Penalty Parameter: Faster Convergence with New Adaptive Penalization

• Yi Xu
• Mingrui Liu
• Qihang Lin
• Tianbao Yang

Alternating direction method of multipliers (ADMM) has received tremendous interest for solving numerous problems in machine learning, statistics and signal processing. However, it is known that the performance of ADMM and many of its variants is very sensitive to the penalty parameter of a quadratic penalty applied to the equality constraints. Although several approaches have been proposed for dynamically changing this parameter during the course of optimization, they do not yield theoretical improvement in the convergence rate and are not directly applicable to stochastic ADMM. In this paper, we develop a new ADMM and its linearized variant with a new adaptive scheme to update the penalty parameter. Our methods can be applied under both deterministic and stochastic optimization settings for structured non-smooth objective function. The novelty of the proposed scheme lies at that it is adaptive to a local sharpness property of the objective function, which marks the key difference from previous adaptive scheme that adjusts the penalty parameter per-iteration based on certain conditions on iterates. On theoretical side, given the local sharpness characterized by an exponent $\theta\in(0, 1]$, we show that the proposed ADMM enjoys an improved iteration complexity of $\widetilde O(1/\epsilon^{1-\theta})$\footnote{$\widetilde O()$ suppresses a logarithmic factor. } in the deterministic setting and an iteration complexity of $\widetilde O(1/\epsilon^{2(1-\theta)})$ in the stochastic setting without smoothness and strong convexity assumptions. The complexity in either setting improves that of the standard ADMM which only uses a fixed penalty parameter. On the practical side, we demonstrate that the proposed algorithms converge comparably to, if not much faster than, ADMM with a fine-tuned fixed penalty parameter.

JMLR Journal 2017 Journal Article

Distributed Stochastic Variance Reduced Gradient Methods by Sampling Extra Data with Replacement

• Jason D. Lee
• Qihang Lin
• Tengyu Ma
• Tianbao Yang

We study the round complexity of minimizing the average of convex functions under a new setting of distributed optimization where each machine can receive two subsets of functions. The first subset is from a random partition and the second subset is randomly sampled with replacement. Under this setting, we define a broad class of distributed algorithms whose local computation can utilize both subsets and design a distributed stochastic variance reduced gradient method belonging to in this class. When the condition number of the problem is small, our method achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. When the condition number is relatively large, a lower bound is provided for the number of rounds of communication needed by any algorithm in this class. Then, we present an accelerated version of our method whose the rounds of communication matches the lower bound up to logarithmic terms, which establishes that this accelerated algorithm has the lowest round complexity among all algorithms in our class under this new setting. [abs] [ pdf ][ bib ] &copy JMLR 2017. ( edit, beta )

AAAI Conference 2017 Conference Paper

Efficient Non-Oblivious Randomized Reduction for Risk Minimization with Improved Excess Risk Guarantee

• Yi Xu
• Haiqin Yang
• Lijun Zhang
• Tianbao Yang

In this paper, we address learning problems for high dimensional data. Previously, oblivious random projection based approaches that project high dimensional features onto a random subspace have been used in practice for tackling highdimensionality challenge in machine learning. Recently, various non-oblivious randomized reduction methods have been developed and deployed for solving many numerical problems such as matrix product approximation, low-rank matrix approximation, etc. However, they are less explored for the machine learning tasks, e. g. , classification. More seriously, the theoretical analysis of excess risk bounds for risk minimization, an important measure of generalization performance, has not been established for non-oblivious randomized reduction methods. It therefore remains an open problem what is the benefit of using them over previous oblivious random projection based approaches. To tackle these challenges, we propose an algorithmic framework for employing non-oblivious randomized reduction method for general empirical risk minimizing in machine learning tasks, where the original high-dimensional features are projected onto a random subspace that is derived from the data with a small matrix approximation error. We then derive the first excess risk bound for the proposed non-oblivious randomized reduction approach without requiring strong assumptions on the training data. The established excess risk bound exhibits that the proposed approach provides much better generalization performance and it also sheds more insights about different randomized reduction approaches. Finally, we conduct extensive experiments on both synthetic and real-world benchmark datasets, whose dimension scales to O(107 ), to demonstrate the efficacy of our proposed approach.

NeurIPS Conference 2017 Conference Paper

Improved Dynamic Regret for Non-degenerate Functions

• Lijun Zhang
• Tianbao Yang
• Jinfeng Yi
• Rong Jin
• Zhi-Hua Zhou

Recently, there has been a growing research interest in the analysis of dynamic regret, which measures the performance of an online learner against a sequence of local minimizers. By exploiting the strong convexity, previous studies have shown that the dynamic regret can be upper bounded by the path-length of the comparator sequence. In this paper, we illustrate that the dynamic regret can be further improved by allowing the learner to query the gradient of the function multiple times, and meanwhile the strong convexity can be weakened to other non-degenerate conditions. Specifically, we introduce the squared path-length, which could be much smaller than the path-length, as a new regularity of the comparator sequence. When multiple gradients are accessible to the learner, we first demonstrate that the dynamic regret of strongly convex functions can be upper bounded by the minimum of the path-length and the squared path-length. We then extend our theoretical guarantee to functions that are semi-strongly convex or self-concordant. To the best of our knowledge, this is the first time that semi-strong convexity and self-concordance are utilized to tighten the dynamic regret.

ICML Conference 2017 Conference Paper

Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence

• Yi Xu 0008
• Qihang Lin
• Tianbao Yang

In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal solutions. In particular, if the objective function $F(\mathbf{w})$ in the $\epsilon$-sublevel set grows as fast as $\|\mathbf{w} - \mathbf{w}_*\|_2^{1/\theta}$, where $\mathbf{w}_*$ represents the closest optimal solution to $\mathbf{w}$ and $\theta\in(0, 1]$ quantifies the local growth rate, the iteration complexity of first-order stochastic optimization for achieving an $\epsilon$-optimal solution can be $\widetilde O(1/\epsilon^{2(1-\theta)})$, which is optimal at most up to a logarithmic factor. This result is fundamentally better in contrast with the previous works that either assume a global growth condition in the entire domain or achieve a local faster convergence under the local faster growth condition. To achieve the faster global convergence, we develop two different accelerated stochastic subgradient methods by iteratively solving the original problem approximately in a local region around a historical solution with the size of the local region gradually decreasing as the solution approaches the optimal set. Besides the theoretical improvements, this work also include new contributions towards making the proposed algorithms practical: (i) we present practical variants of accelerated stochastic subgradient methods that can run without the knowledge of multiplicative growth constant and even the growth rate $\theta$; (ii) we consider a broad family of problems in machine learning to demonstrate that the proposed algorithms enjoy faster convergence than traditional stochastic subgradient method. For example, when applied to the $\ell_1$ regularized empirical polyhedral loss minimization (e. g. , hinge loss, absolute loss), the proposed stochastic methods have a logarithmic iteration complexity.

IJCAI Conference 2017 Conference Paper

SVD-free Convex-Concave Approaches for Nuclear Norm Regularization

• Yichi Xiao
• Zhe Li
• Tianbao Yang
• Lijun Zhang

Minimizing a convex function of matrices regularized by the nuclear norm arises in many applications such as collaborative filtering and multi-task learning. In this paper, we study the general setting where the convex function could be non-smooth. When the size of the data matrix, denoted by m x n, is very large, existing optimization methods are inefficient because in each iteration, they need to perform a singular value decomposition (SVD) which takes O(m^2 n) time. To reduce the computation cost, we exploit the dual characterization of the nuclear norm to introduce a convex-concave optimization problem and design a subgradient-based algorithm without performing SVD. In each iteration, the proposed algorithm only computes the largest singular vector, reducing the time complexity from O(m^2 n) to O(mn). To the best of our knowledge, this is the first SVD-free convex optimization approach for nuclear-norm regularized problems that does not rely on the smoothness assumption. Theoretical analysis shows that the proposed algorithm converges at an optimal O(1/\sqrt{T}) rate where T is the number of iterations. We also extend our algorithm to the stochastic case where only stochastic subgradients of the convex function are available and a special case that contains an additional non-smooth regularizer (e. g. , L1 norm regularizer). We conduct experiments on robust low-rank matrix approximation and link prediction to demonstrate the efficiency of our algorithms.

AAAI Conference 2016 Conference Paper

Fast and Accurate Refined Nyström-Based Kernel SVM

• Zhe Li
• Tianbao Yang
• Lijun Zhang
• Rong Jin

NeurIPS Conference 2016 Conference Paper

Homotopy Smoothing for Non-Smooth Problems with Lower Complexity than $O(1/\epsilon)$

• Yi Xu
• Yan Yan
• Qihang Lin
• Tianbao Yang

In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving such non-smooth optimization problems is $O(1/\epsilon)$ without any assumption on the strong convexity. In this work, we will show that the proposed HOPS achieved a lower iteration complexity of $\tilde O(1/\epsilon^{1-\theta})$ with $\theta\in(0, 1]$ capturing the local sharpness of the objective function around the optimal solutions. To the best of our knowledge, this is the lowest iteration complexity achieved so far for the considered non-smooth optimization problems without strong convexity assumption. The HOPS algorithm employs Nesterov's smoothing technique and Nesterov's accelerated gradient method and runs in stages, which gradually decreases the smoothing parameter in a stage-wise manner until it yields a sufficiently good approximation of the original function. We show that HOPS enjoys a linear convergence for many well-known non-smooth problems (e. g. , empirical risk minimization with a piece-wise linear loss function and $\ell_1$ norm regularizer, finding a point in a polyhedron, cone programming, etc). Experimental results verify the effectiveness of HOPS in comparison with Nesterov's smoothing algorithm and the primal-dual style of first-order methods.

NeurIPS Conference 2016 Conference Paper

Improved Dropout for Shallow and Deep Learning

• Zhe Li
• Boqing Gong
• Tianbao Yang

Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e. g. , logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i. e. , sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow learning with multinomial dropout and establish the risk bound for stochastic optimization. By minimizing a sampling dependent factor in the risk bound, we obtain a distribution-dependent dropout with sampling probabilities dependent on the second order statistics of the data distribution. To tackle the issue of evolving distribution of neurons in deep learning, we propose an efficient adaptive dropout (named \textbf{evolutional dropout}) that computes the sampling probabilities on-the-fly from a mini-batch of examples. Empirical studies on several benchmark datasets demonstrate that the proposed dropouts achieve not only much faster convergence and but also a smaller testing error than the standard dropout. For example, on the CIFAR-100 data, the evolutional dropout achieves relative improvements over 10\% on the prediction performance and over 50\% on the convergence speed compared to the standard dropout.

ICML Conference 2016 Conference Paper

Online Stochastic Linear Optimization under One-bit Feedback

• Lijun Zhang 0005
• Tianbao Yang
• Rong Jin 0001
• Yichi Xiao
• Zhi-Hua Zhou

In this paper, we study a special bandit setting of online stochastic linear optimization, where only one-bit of information is revealed to the learner at each round. This problem has found many applications including online advertisement and online recommendation. We assume the binary feedback is a random variable generated from the logit model, and aim to minimize the regret defined by the unknown linear function. Although the existing method for generalized linear bandit can be applied to our problem, the high computational cost makes it impractical for real-world applications. To address this challenge, we develop an efficient online learning algorithm by exploiting particular structures of the observation model. Specifically, we adopt online Newton step to estimate the unknown parameter and derive a tight confidence region based on the exponential concavity of the logistic loss. Our analysis shows that the proposed algorithm achieves a regret bound of O(d\sqrtT), which matches the optimal result of stochastic linear bandits.

UAI Conference 2016 Conference Paper

Optimal Stochastic Strongly Convex Optimization with a Logarithmic Number of Projections

• Jianhui Chen
• Tianbao Yang
• Qihang Lin
• Lijun Zhang 0005
• Yi Chang 0001

We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent (SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent (Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Given a strongly convex optimization and for a total number of T iterations, Epro-SGD requires only log(T ) projections, and meanwhile attains an optimal convergence rate of O(1/T ), both in expectation and with a high probability. To exploit the structure of the optimization problem, we propose a proximal variant of EproSGD, namely Epro-ORDA, based on the optimal regularized dual averaging method. We apply the proposed methods on real-world applications; the empirical results demonstrate the effectiveness of our methods.

AAAI Conference 2016 Conference Paper

Stochastic Optimization for Kernel PCA

• Lijun Zhang
• Tianbao Yang
• Jinfeng Yi
• Rong Jin
• Zhi-Hua Zhou

Kernel Principal Component Analysis (PCA) is a popular extension of PCA which is able to find nonlinear patterns from data. However, the application of kernel PCA to largescale problems remains a big challenge, due to its quadratic space complexity and cubic time complexity in the number of examples. To address this limitation, we utilize techniques from stochastic optimization to solve kernel PCA with linear space and time complexities per iteration. Specifically, we formulate it as a stochastic composite optimization problem, where a nuclear norm regularizer is introduced to promote low-rankness, and then develop a simple algorithm based on stochastic proximal gradient descent. During the optimization process, the proposed algorithm always maintains a low-rank factorization of iterates that can be conveniently held in memory. Compared to previous iterative approaches, a remarkable property of our algorithm is that it is equipped with an explicit rate of convergence. Theoretical analysis shows that the solution of our algorithm converges to the optimal one at an O(1/T) rate, where T is the number of iterations.

ICML Conference 2016 Conference Paper

Tracking Slowly Moving Clairvoyant: Optimal Dynamic Regret of Online Learning with True and Noisy Gradient

• Tianbao Yang
• Lijun Zhang 0005
• Rong Jin 0001
• Jinfeng Yi

This work focuses on dynamic regret of online convex optimization that compares the performance of online learning to a clairvoyant who knows the sequence of loss functions in advance and hence selects the minimizer of the loss function at each step. By assuming that the clairvoyant moves slowly (i. e. , the minimizers change slowly), we present several improved variation-based upper bounds of the dynamic regret under the true and noisy gradient feedback, which are \it optimal in light of the presented lower bounds. The key to our analysis is to explore a regularity metric that measures the temporal changes in the clairvoyant’s minimizers, to which we refer as path variation. Firstly, we present a general lower bound in terms of the path variation, and then show that under full information or gradient feedback we are able to achieve an optimal dynamic regret. Secondly, we present a lower bound with noisy gradient feedback and then show that we can achieve optimal dynamic regrets under a stochastic gradient feedback and two-point bandit feedback. Moreover, for a sequence of smooth loss functions that admit a small variation in the gradients, our dynamic regret under the two-point bandit feedback matches that is achieved with full information.

ICML Conference 2015 Conference Paper

An Explicit Sampling Dependent Spectral Error Bound for Column Subset Selection

• Tianbao Yang
• Lijun Zhang 0005
• Rong Jin 0001
• Shenghuo Zhu

In this paper, we consider the problem of column subset selection. We present a novel analysis of the spectral norm reconstruction for a simple randomized algorithm and establish a new bound that depends explicitly on the sampling probabilities. The sampling dependent error bound (i) allows us to better understand the tradeoff in the reconstruction error due to sampling probabilities, (ii) exhibits more insights than existing error bounds that exploit specific probability distributions, and (iii) implies better sampling distributions. In particular, we show that a sampling distribution with probabilities proportional to the square root of the statistical leverage scores is better than uniform sampling, and is better than leverage-based sampling when the statistical leverage scores are very nonuniform. And by solving a constrained optimization problem related to the error bound with an efficient bisection search we are able to achieve better performance than using either the leverage-based distribution or that proportional to the square root of the statistical leverage scores. Numerical simulations demonstrate the benefits of the new sampling distributions for low-rank matrix approximation and least square approximation compared to state-of-the art algorithms.

AAAI Conference 2015 Conference Paper

Online Bandit Learning for a Special Class of Non-Convex Losses

• Lijun Zhang
• Tianbao Yang
• Rong Jin
• Zhi-Hua Zhou

In online bandit learning, the learner aims to minimize a sequence of losses, while only observing the value of each loss at a single point. Although various algorithms and theories have been developed for online bandit learning, most of them are limited to convex losses. In this paper, we investigate the problem of online bandit learning with non-convex losses, and develop an efficient algorithm with formal theoretical guarantees. To be specific, we consider a class of losses which is a composition of a non-increasing scalar function and a linear function. This setting models a wide range of supervised learning applications such as online classification with a non-convex loss. Theoretical analysis shows that our algorithm achieves an e O(poly(d)T2/3 ) regret bound when the variation of the loss function is small. To the best of our knowledge, this is the first work in online bandit learning that does not rely on convexity.

ICML Conference 2015 Conference Paper

Theory of Dual-sparse Regularized Randomized Reduction

• Tianbao Yang
• Lijun Zhang 0005
• Rong Jin 0001
• Shenghuo Zhu

In this paper, we study randomized reduction methods, which reduce high-dimensional features into low-dimensional space by randomized methods (e. g. , random projection, random hashing), for large-scale high-dimensional classification. Previous theoretical results on randomized reduction methods hinge on strong assumptions about the data, e. g. , low rank of the data matrix or a large separable margin of classification, which hinder their in broad domains. To address these limitations, we propose dual-sparse regularized randomized reduction methods that introduce a sparse regularizer into the reduced dual problem. Under a mild condition that the original dual solution is a (nearly) sparse vector, we show that the resulting dual solution is close to the original dual solution and concentrates on its support set. In numerical experiments, we present an empirical study to support the analysis and we also present a novel application of the dual-sparse randomized reduction methods to reducing the communication cost of distributed learning from large-scale high-dimensional data.

NeurIPS Conference 2014 Conference Paper

Extracting Certainty from Uncertainty: Transductive Pairwise Classification from Pairwise Similarities

• Tianbao Yang
• Rong Jin

In this work, we study the problem of transductive pairwise classification from pairwise similarities~\footnote{The pairwise similarities are usually derived from some side information instead of the underlying class labels. }. The goal of transductive pairwise classification from pairwise similarities is to infer the pairwise class relationships, to which we refer as pairwise labels, between all examples given a subset of class relationships for a small set of examples, to which we refer as labeled examples. We propose a very simple yet effective algorithm that consists of two simple steps: the first step is to complete the sub-matrix corresponding to the labeled examples and the second step is to reconstruct the label matrix from the completed sub-matrix and the provided similarity matrix. Our analysis exhibits that under several mild preconditions we can recover the label matrix with a small error, if the top eigen-space that corresponds to the largest eigenvalues of the similarity matrix covers well the column space of label matrix and is subject to a low coherence, and the number of observed pairwise labels is sufficiently enough. We demonstrate the effectiveness of the proposed algorithm by several experiments.

ICML Conference 2013 Conference Paper

O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions

• Lijun Zhang 0005
• Tianbao Yang
• Rong Jin 0001
• Xiaofei He 0001

Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as the positive semidefinite cone, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batches, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal O(1/T) rate of convergence with only O(logT) projections. Our empirical study verifies the theoretical result.

NeurIPS Conference 2013 Conference Paper

Stochastic Convex Optimization with Multiple Objectives

• Mehrdad Mahdavi
• Tianbao Yang
• Rong Jin

In this paper, we are interested in the development of efficient algorithms for convex optimization problems in the simultaneous presence of multiple objectives and stochasticity in the first-order information. We cast the stochastic multiple objective optimization problem into a constrained optimization problem by choosing one function as the objective and try to bound other objectives by appropriate thresholds. We first examine a two stages exploration-exploitation based algorithm which first approximates the stochastic objectives by sampling and then solves a constrained stochastic optimization problem by projected gradient method. This method attains a suboptimal convergence rate even under strong assumption on the objectives. Our second approach is an efficient primal-dual stochastic algorithm. It leverages on the theory of Lagrangian method in constrained optimization and attains the optimal convergence rate of $[O(1/ \sqrt{T})]$ in high probability for general Lipschitz continuous objectives.

NeurIPS Conference 2013 Conference Paper

Trading Computation for Communication: Distributed Stochastic Dual Coordinate Ascent

• Tianbao Yang

We present and study a distributed optimization algorithm by employing a stochastic dual coordinate ascent method. Stochastic dual coordinate ascent methods enjoy strong theoretical guarantees and often have better performances than stochastic gradient descent methods in optimizing regularized loss minimization problems. It still lacks of efforts in studying them in a distributed framework. We make a progress along the line by presenting a distributed stochastic dual coordinate ascent algorithm in a star network, with an analysis of the tradeoff between computation and communication. We verify our analysis by experiments on real data sets. Moreover, we compare the proposed algorithm with distributed stochastic gradient descent methods and distributed alternating direction methods of multipliers for optimizing SVMs in the same distributed framework, and observe competitive performances.

ICML Conference 2012 Conference Paper

A Simple Algorithm for Semi-supervised Learning with Improved Generalization Error Bound

• Ming Ji
• Tianbao Yang
• Binbin Lin
• Rong Jin 0001
• Jiawei Han 0001

ICML Conference 2012 Conference Paper

Multiple Kernel Learning from Noisy Labels by Stochastic Programming

• Tianbao Yang
• Mehrdad Mahdavi
• Rong Jin 0001
• Lijun Zhang 0005
• Yang Zhou 0033

NeurIPS Conference 2012 Conference Paper

Nyström Method vs Random Fourier Features: A Theoretical and Empirical Comparison

• Tianbao Yang
• Yu-Feng Li
• Mehrdad Mahdavi
• Rong Jin
• Zhi-Hua Zhou

Both random Fourier features and the Nyström method have been successfully applied to efficient kernel learning. In this work, we investigate the fundamental difference between these two approaches, and how the difference could affect their generalization performances. Unlike approaches based on random Fourier features where the basis functions (i. e. , cosine and sine functions) are sampled from a distribution {\it independent} from the training data, basis functions used by the Nyström method are randomly sampled from the training examples and are therefore {\it data dependent}. By exploring this difference, we show that when there is a large gap in the eigen-spectrum of the kernel matrix, approaches based the Nyström method can yield impressively better generalization error bound than random Fourier features based approach. We empirically verify our theoretical findings on a wide range of large data sets.

AAAI Conference 2012 Conference Paper

Online Kernel Selection: Algorithms and Evaluations

• Tianbao Yang
• Mehrdad Mahdavi
• Rong Jin
• Jinfeng Yi
• Steven Hoi

Kernel methods have been successfully applied to many machine learning problems. Nevertheless, since the performance of kernel methods depends heavily on the type of kernels being used, identifying good kernels among a set of given kernels is important to the success of kernel methods. A straightforward approach to address this problem is cross-validation by training a separate classifier for each kernel and choosing the best kernel classifier out of them. Another approach is Multiple Kernel Learning (MKL), which aims to learn a single kernel classifier from an optimal combination of multiple kernels. However, both approaches suffer from a high computational cost in computing the full kernel matrices and in training, especially when the number of kernels or the number of training examples is very large. In this paper, we tackle this problem by proposing an efficient online kernel selection algorithm. It incrementally learns a weight for each kernel classifier. The weight for each kernel classifier can help us to select a good kernel among a set of given kernels. The proposed approach is efficient in that (i) it is an online approach and therefore avoids computing all the full kernel matrices before training; (ii) it only updates a single kernel classifier each time by a sampling technique and therefore saves time on updating kernel classifiers with poor performance; (iii) it has a theoretically guaranteed performance compared to the best kernel predictor. Empirical studies on image classification tasks demonstrate the effectiveness of the proposed approach for selecting a good kernel among a set of kernels.

NeurIPS Conference 2012 Conference Paper

Semi-Crowdsourced Clustering: Generalizing Crowd Labeling by Robust Distance Metric Learning

• Jinfeng Yi
• Rong Jin
• Shaili Jain
• Tianbao Yang
• Anil Jain

One of the main challenges in data clustering is to define an appropriate similarity measure between two objects. Crowdclustering addresses this challenge by defining the pairwise similarity based on the manual annotations obtained through crowdsourcing. Despite its encouraging results, a key limitation of crowdclustering is that it can only cluster objects when their manual annotations are available. To address this limitation, we propose a new approach for clustering, called \textit{semi-crowdsourced clustering} that effectively combines the low-level features of objects with the manual annotations of a subset of the objects obtained via crowdsourcing. The key idea is to learn an appropriate similarity measure, based on the low-level features of objects, from the manual annotations of only a small portion of the data to be clustered. One difficulty in learning the pairwise similarity measure is that there is a significant amount of noise and inter-worker variations in the manual annotations obtained via crowdsourcing. We address this difficulty by developing a metric learning algorithm based on the matrix completion method. Our empirical study with two real-world image data sets shows that the proposed algorithm outperforms state-of-the-art distance metric learning algorithms in both clustering accuracy and computational efficiency.

NeurIPS Conference 2012 Conference Paper

Stochastic Gradient Descent with Only One Projection

• Mehrdad Mahdavi
• Tianbao Yang
• Rong Jin
• Shenghuo Zhu
• Jinfeng Yi

Although many variants of stochastic gradient descent have been proposed for large-scale convex optimization, most of them require projecting the solution at {\it each} iteration to ensure that the obtained solution stays within the feasible domain. For complex domains (e. g. , positive semidefinite cone), the projection step can be computationally expensive, making stochastic gradient descent unattractive for large-scale optimization problems. We address this limitation by developing a novel stochastic gradient descent algorithm that does not need intermediate projections. Instead, only one projection at the last iteration is needed to obtain a feasible solution in the given domain. Our theoretical analysis shows that with a high probability, the proposed algorithms achieve an $O(1/\sqrt{T})$ convergence rate for general convex optimization, and an $O(\ln T/T)$ rate for strongly convex optimization under mild conditions about the domain and the objective function.

JMLR Journal 2012 Journal Article

Trading Regret for Efficiency: Online Convex Optimization with Long Term Constraints

• Mehrdad Mahdavi
• Rong Jin
• Tianbao Yang

In this paper we propose efficient algorithms for solving constrained online convex optimization problems. Our motivation stems from the observation that most algorithms proposed for online convex optimization require a projection onto the convex set K from which the decisions are made. While the projection is straightforward for simple shapes (e.g., Euclidean ball), for arbitrary complex sets it is the main computational challenge and may be inefficient in practice. In this paper, we consider an alternative online convex optimization problem. Instead of requiring that decisions belong to K for all rounds, we only require that the constraints, which define the set K, be satisfied in the long run. By turning the problem into an online convex-concave optimization problem, we propose an efficient algorithm which achieves O(√T) regret bound and O(T 3/4 ) bound on the violation of constraints. Then, we modify the algorithm in order to guarantee that the constraints are satisfied in the long run. This gain is achieved at the price of getting O(T 3/4 ) regret bound. Our second algorithm is based on the mirror prox method (Nemirovski, 2005) to solve variational inequalities which achieves O(T 2/3 ) bound for both regret and the violation of constraints when the domain K can be described by a finite number of linear constraints. Finally, we extend the results to the setting where we only have partial access to the convex set K and propose a multipoint bandit feedback algorithm with the same bounds in expectation as our first algorithm. [abs] [ pdf ][ bib ] &copy JMLR 2012. ( edit, beta )

ICML Conference 2011 Conference Paper

Online AUC Maximization

• Peilin Zhao
• Steven C. H. Hoi
• Rong Jin 0001
• Tianbao Yang

ICML Conference 2010 Conference Paper

Learning from Noisy Side Information by Generalized Maximum Entropy Model

• Tianbao Yang
• Rong Jin 0001
• Anil K. Jain 0001

UAI Conference 2009 Conference Paper

A Bayesian Framework for Community Detection Integrating Content and Link

• Tianbao Yang
• Rong Jin 0001
• Yun Chi
• Shenghuo Zhu

This paper addresses the problem of community detection in networked data that combines link and content analysis. Most existing work combines link and content information by a generative model. There are two major shortcomings with the existing approaches. First, they assume that the probability of creating a link between two nodes is determined only by the community memberships of the nodes; however other factors(e. g. popularity) could also affect the link pattern. Second, they use generative models to model the content of individual nodes, whereas these generative models are vulnerable to the content attributes that are irrelevant to communities. We propose a Bayesian framework for combining link and content information for community detection that explicitly addresses these shortcomings. A new link model is presented that introduces a random variable to capture the node popularity when deciding the link between two nodes; a discriminative model is used to determine the community membership of a node by its content. An approximate inference algorithm is presented for efficient Bayesian inference. Our empirical study shows that the proposed framework outperforms several state-of-theart approaches in combining link and content information for community detection.