Volkan Cevher

TMLR Journal 2026 Journal Article

Multi-Step Alignment as Markov Games: An Optimistic Online Mirror Descent Approach with Convergence Guarantees

• Yongtao Wu
• Luca Viano
• Kimon Antonakopoulos
• Yihang Chen
• Zhenyu Zhu
• Quanquan Gu
• Volkan Cevher

Reinforcement Learning from Human Feedback (RLHF) has been highly successful in aligning large language models with human preferences. While prevalent methods like DPO have demonstrated strong performance, they frame interactions with the language model as a bandit problem, which limits their applicability in real-world scenarios where multi-turn conversations are common. Additionally, DPO relies on the Bradley-Terry model assumption, which does not adequately capture the non-transitive nature of human preferences. In this paper, we address these challenges by modeling the alignment problem as a two-player constant-sum Markov game, where each player seeks to maximize their winning rate against the other across all steps of the conversation. Our approach Optimistic Multi-step Preference Optimization (OMPO) is built upon the optimistic online mirror descent algorithm~\citep{rakhlin2013online,joulani17a}. Theoretically, we provide a rigorous analysis for the convergence of OMPO and show that OMPO requires $\mathcal{O}(\epsilon^{-1})$ policy updates to converge to an $\epsilon$-approximate Nash equilibrium. We also validate the effectiveness of our method on multi-turn conversations dataset and math reasoning dataset.

TMLR Journal 2025 Journal Article

A Proximal Operator for Inducing 2:4-Sparsity

• Jonas M. Kübler
• Yu-Xiang Wang
• Shoham Sabach
• Navid Ansari
• Matthäus Kleindessner
• Kailash Budhathoki
• Volkan Cevher
• George Karypis

Recent hardware advancements in AI Accelerators and GPUs allow to efficiently compute sparse matrix multiplications, especially when 2 out of 4 consecutive weights are set to zero. However, this so-called 2:4 sparsity usually comes at a decreased accuracy of the model. We derive a regularizer that exploits the local correlation of features to find better sparsity masks in trained models. We minimize the regularizer jointly with a local squared loss by deriving the proximal operator for which we show that it has an efficient solution in the 2:4-sparse case. After optimizing the mask, we introduce masked-gradient updates to further minimize the local squared loss. We illustrate our method on toy problems and apply it to pruning entire large language models up to 70B parameters. On models up to 13B we improve over previous state of the art algorithms, whilst on 70B models we match their performance.

ICML Conference 2025 Conference Paper

Accelerating Spectral Clustering under Fairness Constraints

• Francesco Tonin
• Alex Lambert
• Johan A. K. Suykens
• Volkan Cevher

Fairness of decision-making algorithms is an increasingly important issue. In this paper, we focus on spectral clustering with group fairness constraints, where every demographic group is represented in each cluster proportionally as in the general population. We present a new efficient method for fair spectral clustering (Fair SC) by casting the Fair SC problem within the difference of convex functions (DC) framework. To this end, we introduce a novel variable augmentation strategy and employ an alternating direction method of multipliers type of algorithm adapted to DC problems. We show that each associated subproblem can be solved efficiently, resulting in higher computational efficiency compared to prior work, which required a computationally expensive eigendecomposition. Numerical experiments demonstrate the effectiveness of our approach on both synthetic and real-world benchmarks, showing significant speedups in computation time over prior art, especially as the problem size grows. This work thus represents a considerable step forward towards the adoption of fair clustering in real-world applications.

ICLR Conference 2025 Conference Paper

Addressing Label Shift in Distributed Learning via Entropy Regularization​

• Zhiyuan Wu
• Changkyu Choi
• Xiangcheng Cao
• Volkan Cevher
• Ali Ramezani-Kebrya

We address the challenge of minimizing "true risk" in multi-node distributed learning.\footnote{We use the term node to refer to a client, FPGA, APU, CPU, GPU, or worker.} These systems are frequently exposed to both inter-node and intra-node "label shifts", which present a critical obstacle to effectively optimizing model performance while ensuring that data remains confined to each node. To tackle this, we propose the Versatile Robust Label Shift (VRLS) method, which enhances the maximum likelihood estimation of the test-to-train label importance ratio. VRLS incorporates Shannon entropy-based regularization and adjusts the importance ratio during training to better handle label shifts at the test time. In multi-node learning environments, VRLS further extends its capabilities by learning and adapting importance ratios across nodes, effectively mitigating label shifts and improving overall model performance. Experiments conducted on MNIST, Fashion MNIST, and CIFAR-10 demonstrate the effectiveness of VRLS, outperforming baselines by up to 20\% in imbalanced settings. These results highlight the significant improvements VRLS offers in addressing label shifts. Our theoretical analysis further supports this by establishing high-probability bounds on estimation errors.

ICLR Conference 2025 Conference Paper

Adversarial Training for Defense Against Label Poisoning Attacks

• Melis Ilayda Bal
• Volkan Cevher
• Michael Muehlebach

As machine learning models grow in complexity and increasingly rely on publicly sourced data, such as the human-annotated labels used in training large language models, they become more vulnerable to label poisoning attacks. These attacks, in which adversaries subtly alter the labels within a training dataset, can severely degrade model performance, posing significant risks in critical applications. In this paper, we propose $\textbf{Floral}$, a novel adversarial training defense strategy based on support vector machines (SVMs) to counter these threats. Utilizing a bilevel optimization framework, we cast the training process as a non-zero-sum Stackelberg game between an $\textit{attacker}$, who strategically poisons critical training labels, and the $\textit{model}$, which seeks to recover from such attacks. Our approach accommodates various model architectures and employs a projected gradient descent algorithm with kernel SVMs for adversarial training. We provide a theoretical analysis of our algorithm’s convergence properties and empirically evaluate $\textbf{Floral}$'s effectiveness across diverse classification tasks. Compared to robust baselines and foundation models such as RoBERTa, $\textbf{Floral}$ consistently achieves higher robust accuracy under increasing attacker budgets. These results underscore the potential of $\textbf{Floral}$ to enhance the resilience of machine learning models against label poisoning threats, thereby ensuring robust classification in adversarial settings.

NeurIPS Conference 2025 Conference Paper

Ascent Fails to Forget

• Ioannis Mavrothalassitis
• Pol Puigdemont
• Noam Levi
• Volkan Cevher

Contrary to common belief, we show that gradient ascent-based unconstrained optimization methods frequently fail to perform machine unlearning, a phenomenon we attribute to the inherent statistical dependence between the forget and retain data sets. This dependence, which can manifest itself even as simple correlations, undermines the misconception that these sets can be independently manipulated during unlearning. We provide empirical and theoretical evidence showing these methods often fail precisely due to this overlooked relationship. For random forget sets, this dependence means that degrading forget set metrics (which, for a retrained model, should mirror test set metrics) inevitably harms overall test performance. Going beyond random sets, we consider logistic regression as an instructive example where a critical failure mode emerges: inter-set dependence causes gradient descent-ascent iterations to progressively diverge from the ideal retrained model. Strikingly, these methods can converge to solutions that are not only far from the retrained ideal but are potentially even further from it than the original model itself, rendering the unlearning process actively detrimental. A toy example further illustrates how this dependence can trap models in inferior local minima, inescapable via finetuning. Our findings highlight that the presence of such statistical dependencies, even when manifest only as correlations, can be sufficient for ascent-based unlearning to fail. Our theoretical insights are corroborated by experiments on complex neural networks, demonstrating that these methods do not perform as expected in practice due to this unaddressed statistical interplay.

ICML Conference 2025 Conference Paper

Best of Both Worlds: Regret Minimization versus Minimax Play

• Adrian Müller 0002
• Jon Schneider
• Stratis Skoulakis
• Luca Viano
• Volkan Cevher

In this paper, we investigate the existence of online learning algorithms with bandit feedback that simultaneously guarantee $O(1)$ regret compared to a given comparator strategy, and $\tilde{O}(\sqrt{T})$ regret compared to any fixed strategy, where $T$ is the number of rounds. We provide the first affirmative answer to this question whenever the comparator strategy supports every action. In the context of zero-sum games with min-max value zero, both in normal- and extensive form, we show that our results allow us to guarantee to risk at most $O(1)$ loss while being able to gain $\Omega(T)$ from exploitable opponents, thereby combining the benefits of both no-regret algorithms and minimax play.

EWRL Workshop 2025 Workshop Paper

Best of Both Worlds: Regret Minimization versus Minimax Play

• Adrian Müller
• Jon Schneider
• Stratis Skoulakis
• Luca Viano
• Volkan Cevher

In this paper, we investigate the existence of online learning algorithms with bandit feedback that simultaneously guarantee $O(1)$ regret compared to a given comparator strategy, and $\tilde{O}(\sqrt{T})$ regret compared to any fixed strategy, where $T$ is the number of rounds. We provide the first affirmative answer to this question whenever the comparator strategy supports every action. In the context of zero-sum games with min-max value zero, both in normal- and extensive form, we show that our results allow us to guarantee to risk at most $O(1)$ loss while being able to gain $\Omega(T)$ from exploitable opponents, thereby combining the benefits of both no-regret algorithms and minimax play.

ICLR Conference 2025 Conference Paper

Certified Robustness Under Bounded Levenshtein Distance

• Elías Abad-Rocamora
• Grigorios Chrysos 0002
• Volkan Cevher

Text classifiers suffer from small perturbations, that if chosen adversarially, can dramatically change the output of the model. Verification methods can provide robustness certificates against such adversarial perturbations, by computing a sound lower bound on the robust accuracy. Nevertheless, existing verification methods incur in prohibitive costs and cannot practically handle Levenshtein distance constraints. We propose the first method for computing the Lipschitz constant of convolutional classifiers with respect to the Levenshtein distance. We use these Lipschitz constant estimates for training 1-Lipschitz classifiers. This enables computing the certified radius of a classifier in a single forward pass. Our method, LipsLev, is able to obtain $38.80$% and $13.93$% verified accuracy at distance $1$ and $2$ respectively in the AG-News dataset, while being $4$ orders of magnitude faster than existing approaches. We believe our work can open the door to more efficient verification in the text domain.

ICML Conference 2025 Conference Paper

Chameleon: A Flexible Data-mixing Framework for Language Model Pretraining and Finetuning

• Wanyun Xie
• Francesco Tonin
• Volkan Cevher

Training data mixtures greatly impact the generalization performance of large language models. Existing domain reweighting methods often rely on costly weight computations and require retraining when new data is introduced. To this end, we introduce a flexible and efficient data mixing framework, Chameleon, that employs leverage scores to quantify domain importance within a learned embedding space. We first construct a domain affinity matrix over domain embeddings. The induced leverage scores determine a mixture that upweights domains sharing common representations in embedding space. This formulation allows direct transfer to new data by computing the new domain embeddings. In experiments, we demonstrate improvements over three key scenarios: (i) our computed weights improve performance on pretraining domains with a fraction of the compute of existing methods; (ii) Chameleon can adapt to data changes without proxy retraining, boosting few-shot reasoning accuracies when transferred to new data; (iii) our method enables efficient domain reweighting in finetuning, consistently improving test perplexity on all finetuning domains over uniform mixture. Our code is available at https: //github. com/LIONS-EPFL/Chameleon.

ICML Conference 2025 Conference Paper

Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games

• Yi Feng
• Kaito Fujii
• Stratis Skoulakis
• Xiao Wang 0036
• Volkan Cevher

Since Polyak’s pioneering work, heavy ball (HB) momentum has been widely studied in minimization. However, its role in min-max games remains largely unexplored. As a key component of practical min-max algorithms like Adam, this gap limits their effectiveness. In this paper, we present a continuous-time analysis for HB with simultaneous and alternating update schemes in min-max games. Locally, we prove smaller momentum enhances algorithmic stability by enabling local convergence across a wider range of step sizes, with alternating updates generally converging faster. Globally, we study the implicit regularization of HB, and find smaller momentum guides algorithms trajectories towards shallower slope regions of the loss landscapes, with alternating updates amplifying this effect. Surprisingly, all these phenomena differ from those observed in minimization, where larger momentum yields similar effects. Our results reveal fundamental differences between HB in min-max games and minimization, and numerical experiments further validate our theoretical results.

ICLR Conference 2025 Conference Paper

Efficient Interpolation between Extragradient and Proximal Methods for Weak MVIs

• Thomas Pethick
• Ioannis Mavrothalassitis
• Volkan Cevher

We study nonmonotone games satisfying the weak Minty variational inequality (MVI) with parameter $\rho \in (-\tfrac{1}{L}, \infty)$, where $L$ is the Lipschitz constant of the gradient operator. An error corrected version of the inexact proximal point algorithm is proposed, with which we establish the first $\mathcal O(1/\epsilon)$ rate for the entire range $\rho \in (-\tfrac{1}{L}, \infty)$, thus removing a logarithmic factor compared with the complexity of existing methods. The scheme automatically selects the needed accuracy for the proximal computation, and can recover the relaxed extragradient method when $\rho > -\tfrac{1}{2L}$ and the relaxed proximal point algorithm (rPPA) when $\rho > -\tfrac{1}{L}$. Due to the error correction, the scheme inherits the strong properties of the _exact_ rPPA. Specifically, we show that linear convergence is automatically achieved under appropriate conditions. Tightness for the range of $\rho$ is established through a lower bound for rPPA. Central to the algorithmic construction is a halfspace projection, where the key insight is that the allowed error tolerance can both be used to correct for the proximal approximation and to enlarge the problem class.

NeurIPS Conference 2025 Conference Paper

Efficient Large Language Model Inference with Neural Block Linearization

• Mete Erdogan
• Francesco Tonin
• Volkan Cevher

The high inference demands of transformer-based Large Language Models (LLMs) pose substantial challenges in their deployment. To this end, we introduce Neural Block Linearization (NBL), a novel framework for accelerating transformer model inference by replacing self-attention layers with linear approximations derived from Linear Minimum Mean Squared Error estimators. NBL leverages Canonical Correlation Analysis to compute a theoretical upper bound on the approximation error. Then, we use this bound as a criterion for substitution, selecting the LLM layers with the lowest linearization error. NBL can be efficiently applied to pre-trained LLMs without the need for fine-tuning. In experiments, NBL achieves notable computational speed-ups while preserving competitive accuracy on multiple reasoning benchmarks. For instance, applying NBL to 12 self-attention layers in DeepSeek-R1-Distill-Llama-8B increases the inference speed by 32% with less than 1% accuracy trade-off, making it a flexible and promising solution to improve the inference efficiency of LLMs.

ICLR Conference 2025 Conference Paper

Faster Inference of Flow-Based Generative Models via Improved Data-Noise Coupling

• Aram Davtyan
• Leello Tadesse Dadi
• Volkan Cevher
• Paolo Favaro

Conditional Flow Matching (CFM), a simulation-free method for training continuous normalizing flows, provides an efficient alternative to diffusion models for key tasks like image and video generation. The performance of CFM in solving these tasks depends on the way data is coupled with noise. A recent approach uses minibatch optimal transport (OT) to reassign noise-data pairs in each training step to streamline sampling trajectories and thus accelerate inference. However, its optimization is restricted to individual minibatches, limiting its effectiveness on large datasets. To address this shortcoming, we introduce LOOM-CFM (Looking Out Of Minibatch-CFM), a novel method to extend the scope of minibatch OT by preserving and optimizing these assignments across minibatches over training time. Our approach demonstrates consistent improvements in the sampling speed-quality trade-off across multiple datasets. LOOM-CFM also enhances distillation initialization and supports high-resolution synthesis in latent space training.

ICML Conference 2025 Conference Paper

Generalization of noisy SGD in unbounded non-convex settings

• Leello Tadesse Dadi
• Volkan Cevher

We study generalization of iterative noisy gradient schemes on smooth non-convex losses. Formally, we establish time-independent information theoretic generalization bounds for Stochastic Gradient Langevin Dynamics (SGLD) that do not diverge as the iteration count increases. Our bounds are obtained through a stability argument: we analyze the difference between two SGLD sequences ran in parallel on two datasets sampled from the same distribution. Our result only requires an isoperimetric inequality to hold, which is merely a restriction on the tails of the loss. Our work relaxes the assumptions of prior work to establish that the iterates stay within a bounded KL divergence from each other. Under an additional dissipativity assumption, we show that the stronger Renyi divergence also stays bounded by establishing a uniform log-Sobolev constant of the iterates. Without dissipativity, we sidestep the need for local log-Sobolev inequalities and instead exploit the regularizing properties of Gaussian convolution. These techniques allow us to show that strong convexity is not necessary for finite stability bounds. Our work shows that noisy SGD can have finite, iteration-independent, generalization and differential privacy bounds in unbounded non-convex settings.

NeurIPS Conference 2025 Conference Paper

Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness

• Thomas Pethick
• Wanyun Xie
• Mete Erdogan
• Kimon Antonakopoulos
• Antonio Silveti-Falls
• Volkan Cevher

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of ($L_0$, $L_1$)-smoothness. Weight decay is incorporated in a principled manner by identifying a connection to the Frank-Wolfe short step. In the stochastic case, we show an order optimal $O(n^{-1/4})$ convergence rate by leveraging a momentum based gradient estimator. We discuss how to instantiate the algorithms for deep learning, which we dub Clipped Scion, and demonstrate their properties on image classification and language modeling. The code is available at https: //github. com/LIONS-EPFL/ClippedScion.

ICLR Conference 2025 Conference Paper

How Gradient descent balances features: A dynamical analysis for two-layer neural networks

• Zhenyu Zhu
• Fanghui Liu 0001
• Volkan Cevher

This paper investigates the fundamental regression task of learning $k$ neurons (\emph{a.k.a.} teachers) from Gaussian input, using two-layer ReLU neural networks with width $m$ (\emph{a.k.a.} students) and $m, k= \mathcal{O}(1)$, trained via gradient descent under proper initialization and a small step-size. Our analysis follows a three-phase structure: \emph{alignment} after weak recovery, \emph{tangential growth}, and \emph{local convergence}, providing deeper insights into the learning dynamics of gradient descent (GD). We prove the global convergence at the rate of $\mathcal{O}(T^{-3})$ for the zero loss of excess risk. Additionally, our results show that GD automatically groups and balances student neurons, revealing an implicit bias toward achieving the minimum ``balanced'' $\ell_2$-norm in the solution. Our work extends beyond previous studies in exact-parameterization setting ($m = k = 1$, (Yehudai and Ohad, 2020)) and single-neuron setting ($m \geq k = 1$, (Xu and Du, 2023)). The key technical challenge lies in handling the interactions between multiple teachers and students during training, which we address by refining the alignment analysis in Phase 1 and introducing a new dynamic system analysis for tangential components in Phase 2. Our results pave the way for further research on optimizing neural network training dynamics and understanding implicit biases in more complex architectures.

EWRL Workshop 2025 Workshop Paper

IL-SOAR: Imitation Learning with soft optimistic actor critic

• Stefano Viel
• Luca Viano
• Volkan Cevher

This paper introduces the SOAR framework for imitation learning. SOAR is an algorithmic template that learns a policy from expert demonstrations with a primal dual style algorithm that alternates cost and policy updates. Within the policy updates, the SOAR framework uses an actor critic method with multiple critics to estimate the critic uncertainty and build an optimistic critic fundamental to drive exploration. When instantiated in the tabular setting, we get a provable algorithm with guarantees that matches the best known results in the desired accuracy parameter $\epsilon$. Practically, the SOAR template can boost the performance of \emph{any} imitation learning algorithm based on Soft Actror Critic (SAC). As an example, we show that SOAR can boost consistently the performance of the following SAC-based imitation learning algorithms: $f$-IRL, ML-IRL and CSIL. Overall, thanks to SOAR, the required number of episodes to achieve the same performance is reduced by half.

ICML Conference 2025 Conference Paper

IL-SOAR: Imitation Learning with Soft Optimistic Actor cRitic

• Stefano Viel
• Luca Viano
• Volkan Cevher

This paper introduces the SOAR framework for imitation learning. SOAR is an algorithmic template which learns a policy from expert demonstrations with a primal-dual style algorithm which alternates cost and policy updates. Within the policy updates the SOAR framework prescribe to use an actor critic method with multiple critics to estimate the critic uncertainty and therefore build an optimistic critic fundamental to drive exploration. When instantiated to the tabular setting, we get a provable algorithms dubbed FRA with guarantees matching the best known results in $\epsilon$. Practically, the SOAR template is shown to boost consistently the performance of primal dual IL algorithms building on actor critic routines for the policy updates. Approximately, thanks to SOAR, the required number of episodes to achieve the same performance is reduced by a half.

ICML Conference 2025 Conference Paper

Layer-wise Quantization for Quantized Optimistic Dual Averaging

• Anh Duc Nguyen
• Ilia Markov
• Frank Zhengqing Wu
• Ali Ramezani-Kebrya
• Kimon Antonakopoulos
• Dan Alistarh
• Volkan Cevher

Modern deep neural networks exhibit heterogeneity across numerous layers of various types such as residuals, multi-head attention, etc. , due to varying structures (dimensions, activation functions, etc.), distinct representation characteristics, which impact predictions. We develop a general layer-wise quantization framework with tight variance and code-length bounds, adapting to the heterogeneities over the course of training. We then apply a new layer-wise quantization technique within distributed variational inequalities (VIs), proposing a novel Quantized Optimistic Dual Averaging (QODA) algorithm with adaptive learning rates, which achieves competitive convergence rates for monotone VIs. We empirically show that QODA achieves up to a $150$% speedup over the baselines in end-to-end training time for training Wasserstein GAN on $12+$ GPUs.

EWRL Workshop 2025 Workshop Paper

Learning Equilibria from Data: Provably Efficient Multi-Agent Imitation Learning

• Till Freihaut
• Luca Viano
• Volkan Cevher
• Matthieu Geist
• Giorgia Ramponi

This paper provides the first expert sample complexity characterization for learning a Nash equilibrium from expert data in Markov Games. We show that a new quantity named the \emph{single policy deviation concentrability coefficient} is unavoidable in the non-interactive imitation learning setting, and we provide an upper bound for behavioral cloning (BC) featuring such coefficient. BC exhibits substantial regret in games with high concentrability coefficient, leading us to utilize expert queries to develop and introduce two novel solution algorithms: MAIL-BRO and MURMAIL. The former employs a best response oracle and learns an $\varepsilon$-Nash equilibrium with $\mathcal{O}(\varepsilon^{-4})$ expert and oracle queries. The latter bypasses completely the best response oracle at the cost of a worse expert query complexity of order $\mathcal{O}(\varepsilon^{-8})$. Finally, we provide numerical evidence, confirming our theoretical findings.

NeurIPS Conference 2025 Conference Paper

Learning Equilibria from Data: Provably Efficient Multi-Agent Imitation Learning

• Till Freihaut
• Luca Viano
• Volkan Cevher
• Matthieu Geist
• Giorgia Ramponi

This paper provides the first expert sample complexity characterization for learning a Nash equilibrium from expert data in Markov Games. We show that a new quantity named the *single policy deviation concentrability coefficient* is unavoidable in the non-interactive imitation learning setting, and we provide an upper bound for behavioral cloning (BC) featuring such coefficient. BC exhibits substantial regret in games with high concentrability coefficient, leading us to utilize expert queries to develop and introduce two novel solution algorithms: MAIL-BRO and MURMAIL. The former employs a best response oracle and learns an $\varepsilon$-Nash equilibrium with $\mathcal{O}(\varepsilon^{-4})$ expert and oracle queries. The latter bypasses completely the best response oracle at the cost of a worse expert query complexity of order $\mathcal{O}(\varepsilon^{-8})$. Finally, we provide numerical evidence, confirming our theoretical findings.

NeurIPS Conference 2025 Conference Paper

Linear Attention for Efficient Bidirectional Sequence Modeling

• Arshia Afzal
• Elias Abad Rocamora
• Leyla Candogan
• Pol Puigdemont
• Francesco Tonin
• Yongtao Wu
• Mahsa Shoaran
• Volkan Cevher

Linear Transformers and State Space Models have emerged as efficient alternatives to softmax Transformers for causal sequence modeling, enabling parallel training via matrix multiplication and efficient RNN-style inference. However, despite their success in causal tasks, no unified framework exists for applying Linear Transformers to bidirectional sequence modeling. We introduce LION, the first framework to systematically extend Linear Transformers to the bidirectional setting. LION generalizes three core representations commonly used in the causal case—full Linear Attention, bidirectional RNN, and chunkwise parallel form—to the bidirectional setting. These forms are theoretically equivalent and enable models to exploit the strengths of each during training and inference. We prove that a broad class of Linear Transformers can be extended using LION and validate our framework via three core examples based on the choice of decay type: LION-LIT, the bidirectional extension of [25]; LION-D, based on [44]; and LION-S, a variant using selective decay [34, 13]. Across standard bidirectional tasks, LION enables models to match or exceed the performance of softmax Transformers, while offering significantly faster training and more efficient inference than existing State Space Models.

ICLR Conference 2025 Conference Paper

Quantum-PEFT: Ultra parameter-efficient fine-tuning

• Toshiaki Koike-Akino
• Francesco Tonin
• Yongtao Wu
• Frank Zhengqing Wu
• Leyla Naz Candogan
• Volkan Cevher

This paper introduces Quantum-PEFT that leverages quantum computations for parameter-efficient fine-tuning (PEFT). Unlike other additive PEFT methods, such as low-rank adaptation (LoRA), Quantum-PEFT exploits an underlying full-rank yet surprisingly parameter efficient _quantum unitary parameterization_. With the use of Pauli parameterization, the number of trainable parameters grows only logarithmically with the ambient dimension, as opposed to linearly as in LoRA-based PEFT methods. Quantum-PEFT achieves vanishingly smaller number of trainable parameters than the lowest-rank LoRA as dimensions grow, enhancing parameter efficiency while maintaining a competitive performance. We apply Quantum-PEFT to several transfer learning benchmarks in language and vision, demonstrating significant advantages in parameter efficiency.

NeurIPS Conference 2025 Conference Paper

Robustness in Both Domains: CLIP Needs a Robust Text Encoder

• Elias Abad Rocamora
• Christian Schlarmann
• Naman Deep Singh
• Yongtao Wu
• Matthias Hein
• Volkan Cevher

Adversarial input attacks can cause a significant shift of CLIP embeddings. This can affect the downstream robustness of models incorporating CLIP in the pipeline, such as text-to-image generative models or large vision language models. While some efforts have been done towards making the CLIP image encoders robust, the robustness of text encoders remains unexplored. In this work, we cover this gap in the literature. We propose LEAF: an efficient adversarial finetuning method for the text domain, with the ability to scale to large CLIP models. Our models significantly improve the zero-shot adversarial accuracy in the text domain, while maintaining the vision performance provided by robust image encoders. When combined with text-to-image diffusion models, we can improve the generation quality under adversarial noise. In multimodal retrieval tasks, LEAF improves the recall under adversarial noise over standard CLIP models. Finally, we show that robust text encoders facilitate better reconstruction of input text from its embedding via direct optimization. We open-source our code and models.

TMLR Journal 2025 Journal Article

Single-pass Detection of Jailbreaking Input in Large Language Models

• Leyla Naz Candogan
• Yongtao Wu
• Elias Abad Rocamora
• Grigorios Chrysos
• Volkan Cevher

Defending aligned Large Language Models (LLMs) against jailbreaking attacks is a challenging problem, with existing approaches requiring multiple requests or even queries to auxiliary LLMs, making them computationally heavy. Instead, we focus on detecting jailbreaking input in a single forward pass. Our method, called SPD, leverages the information carried by the logits to predict whether the output sentence will be harmful. This allows us to defend in just a forward pass. SPD can not only detect attacks effectively on open-source models, but also minimizes the misclassification of harmless inputs. Furthermore, we show that SPD remains effective even without complete logit access in GPT-3.5 and GPT-4. We believe that our proposed method offers a promising approach to efficiently safeguard LLMs against adversarial attacks.

ICML Conference 2025 Conference Paper

Training Deep Learning Models with Norm-Constrained LMOs

• Thomas Pethick
• Wanyun Xie
• Kimon Antonakopoulos
• Zhenyu Zhu
• Antonio Silveti-Falls
• Volkan Cevher

In this work, we study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new stochastic family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems. The resulting update rule unifies several existing optimization methods under a single framework. Furthermore, we propose an explicit choice of norm for deep architectures, which, as a side benefit, leads to the transferability of hyperparameters across model sizes. Experimentally, we demonstrate significant speedups on nanoGPT training without any reliance on Adam. The proposed method is memory-efficient, requiring only one set of model weights and one set of gradients, which can be stored in half-precision.

TMLR Journal 2025 Journal Article

νSAM: Memory-Efficient Sharpness-Aware Minimization via Nuclear Norm Constraints

• Thomas Pethick
• Parameswaran Raman
• Lenon Minorics
• Mingyi Hong
• Shoham Sabach
• Volkan Cevher

Sharpness-aware minimization (SAM) has been shown to improve the generalization of neural networks. However, the method comes at the expense of storing a perturbation of the model parameters, which can be restrictive when memory bound. We design a variant of SAM, called $\nu$SAM, which obtains a low-rank perturbation by modifying the perturbation constraint. The update almost entirely removes the memory footprint of the perturbation without increasing the computational complexity, thus achieving close to a $1/3$ memory saving regarding the parameters when using SGD as the base optimizer. We demonstrate comparable performance of $\nu$SAM with SAM on vision transformers both when training models from scratch and for fine-tuning. Interestingly, $\nu$SAM seems to significantly improve performance for MLP-Mixer architectures across both settings. The results are corroborated theoretically, where we show that SAM with an \emph{arbitrary} norm choice (which includes $\nu$SAM) can converge even with fixed perturbation radius.

NeurIPS Conference 2024 Conference Paper

$\boldsymbol{\mu}\mathbf{P^2}$: Effective Sharpness Aware Minimization Requires Layerwise Perturbation Scaling

• Moritz Haas
• Jin Xu
• Volkan Cevher
• Leena Chennuru Vankadara

Sharpness Aware Minimization (SAM) enhances performance across various neural architectures and datasets. As models are continually scaled up to improve performance, a rigorous understanding of SAM’s scaling behaviour is paramount. To this end, we study the infinite-width limit of neural networks trained with SAM, using the Tensor Programs framework. Our findings reveal that the dynamics of standard SAM effectively reduce to applying SAM solely in the last layer in wide neural networks, even with optimal hyperparameters. In contrast, we identify a stable parameterization with layerwise perturbation scaling, which we call *Maximal Update and Perturbation Parameterization* ($\mu$P$^2$), that ensures all layers are both feature learning and effectively perturbed in the limit. Through experiments with MLPs, ResNets and Vision Transformers, we empirically demonstrate that $\mu$P$^2$ is the first parameterization to achieve hyperparameter transfer of the joint optimum of learning rate and perturbation radius across model scales. Moreover, we provide an intuitive condition to derive $\mu$P$^2$ for other perturbation rules like Adaptive SAM and SAM-ON, also ensuring balanced perturbation effects across all layers.

ICLR Conference 2024 Conference Paper

Advancing the Lower Bounds: an Accelerated, Stochastic, Second-order Method with Optimal Adaptation to Inexactness

• Artem Agafonov
• Dmitry Kamzolov
• Alexander V. Gasnikov
• Ali Kavis
• Kimon Antonakopoulos
• Volkan Cevher
• Martin Takác 0001

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, typical in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

ICLR Conference 2024 Conference Paper

Adversarial Training Should Be Cast as a Non-Zero-Sum Game

• Alexander Robey
• Fabian Latorre
• George J. Pappas
• Seyed Hamed Hassani
• Volkan Cevher

One prominent approach toward resolving the adversarial vulnerability of deep neural networks is the two-player zero-sum paradigm of adversarial training, in which predictors are trained against adversarially chosen perturbations of data. Despite the promise of this approach, algorithms based on this paradigm have not engendered sufficient levels of robustness and suffer from pathological behavior like robust overfitting. To understand this shortcoming, we first show that the commonly used surrogate-based relaxation used in adversarial training algorithms voids all guarantees on the robustness of trained classifiers. The identification of this pitfall informs a novel non-zero-sum bilevel formulation of adversarial training, wherein each player optimizes a different objective function. Our formulation yields a simple algorithmic framework that matches and in some cases outperforms state-of-the-art attacks, attains comparable levels of robustness to standard adversarial training algorithms, and does not suffer from robust overfitting.

ICLR Conference 2024 Conference Paper

Efficient Continual Finite-Sum Minimization

• Ioannis Mavrothalassitis
• Stratis Skoulakis
• Leello Tadesse Dadi
• Volkan Cevher

Given a sequence of functions $f_1,\ldots,f_n$ with $f_i:\mathcal{D}\mapsto \mathbb{R}$, finite-sum minimization seeks a point ${x}^\star \in \mathcal{D}$ minimizing $\sum_{j=1}^nf_j(x)/n$. In this work, we propose a key twist into the finite-sum minimization, dubbed as *continual finite-sum minimization*, that asks for a sequence of points $x_1^\star, \ldots, x_n^\star \in D$ such that each ${x}^\star_i \in D$ minimizes the prefix-sum $\sum_{j=1}^if_j(x)/i$. Assuming that each prefix-sum is strongly convex, we develop a first-order continual stochastic variance reduction gradient method ($\mathrm{CSVRG}$) producing an $\epsilon$-optimal sequence with $\tilde{\mathcal{O}}(n/\epsilon^{1/3} + 1/\sqrt{\epsilon})$ overall *first-order oracles* (FO). An FO corresponds to the computation of a single gradient $\nabla f_j(x)$ at a given $x \in \mathcal{D}$ for some $j \in [n]$. Our approach significantly improves upon the $\mathcal{O}(n/\epsilon)$ FOs that $\mathrm{StochasticGradientDescent}$ requires and the $\mathcal{O}(n^2 \log (1/\epsilon))$ FOs that state-of-the-art variance reduction methods such as $\mathrm{Katyusha}$ require. We also prove that there is no natural first-order method with $\mathcal{O}\left(n/\epsilon^\alpha\right)$ gradient complexity for $\alpha < 1/4$, establishing that the first-order complexity of our method is nearly tight.

ICLR Conference 2024 Conference Paper

Efficient local linearity regularization to overcome catastrophic overfitting

• Elías Abad-Rocamora
• Fanghui Liu 0001
• Grigorios Chrysos 0002
• Pablo M. Olmos
• Volkan Cevher

Catastrophic overfitting (CO) in single-step adversarial training (AT) results in abrupt drops in the adversarial test accuracy (even down to $0$%). For models trained with multi-step AT, it has been observed that the loss function behaves locally linearly with respect to the input, this is however lost in single-step AT. To address CO in single-step AT, several methods have been proposed to enforce local linearity of the loss via regularization. However, these regularization terms considerably slow down training due to *Double Backpropagation*. Instead, in this work, we introduce a regularization term, called ELLE, to mitigate CO *effectively* and *efficiently* in classical AT evaluations, as well as some more difficult regimes, e.g., large adversarial perturbations and long training schedules. Our regularization term can be theoretically linked to curvature of the loss function and is computationally cheaper than previous methods by avoiding *Double Backpropagation*. Our thorough experimental validation demonstrates that our work does not suffer from CO, even in challenging settings where previous works suffer from it. We also notice that adapting our regularization parameter during training (ELLE-A) greatly improves the performance, specially in large $\epsilon$ setups. Our implementation is available in https://github.com/LIONS-EPFL/ELLE.

ICLR Conference 2024 Conference Paper

Generalization of Scaled Deep ResNets in the Mean-Field Regime

• Yihang Chen 0003
• Fanghui Liu 0001
• Yiping Lu
• Grigorios Chrysos 0002
• Volkan Cevher

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of which the gradient flow is described by a partial differential equation in the large-neural network limit, i.e., the mean-field regime. To derive the generalization bounds under this setting, our analysis necessitates a shift from the conventional time-invariant Gram matrix employed in the lazy training regime to a time-variant, distribution-dependent version. To this end, we provide a global lower bound on the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides, for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we establish the linear convergence of the empirical error and estimate the upper bound of the KL divergence over parameters distribution. Finally, we build the uniform convergence for generalization bound via Rademacher complexity. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime and contribute to advancing the understanding of the fundamental properties of deep neural networks.

ICML Conference 2024 Conference Paper

Going beyond Compositions, DDPMs Can Produce Zero-Shot Interpolations

• Justin Deschenaux
• Igor Krawczuk
• Grigorios Chrysos 0002
• Volkan Cevher

Denoising Diffusion Probabilistic Models (DDPMs) exhibit remarkable capabilities in image generation, with studies suggesting that they can generalize by composing latent factors learned from the training data. In this work, we go further and study DDPMs trained on strictly separate subsets of the data distribution with large gaps on the support of the latent factors. We show that such a model can effectively generate images in the unexplored, intermediate regions of the distribution. For instance, when trained on clearly smiling and non-smiling faces, we demonstrate a sampling procedure which can generate slightly smiling faces without reference images (zero-shot interpolation). We replicate these findings for other attributes as well as other datasets. Our code is available on GitHub.

ICML Conference 2024 Conference Paper

High-Dimensional Kernel Methods under Covariate Shift: Data-Dependent Implicit Regularization

• Yihang Chen 0003
• Fanghui Liu 0001
• Taiji Suzuki
• Volkan Cevher

This paper studies kernel ridge regression in high dimensions under covariate shifts and analyzes the role of importance re-weighting. We first derive the asymptotic expansion of high dimensional kernels under covariate shifts. By a bias-variance decomposition, we theoretically demonstrate that the re-weighting strategy allows for decreasing the variance. For bias, we analyze the regularization of the arbitrary or well-chosen scale, showing that the bias can behave very differently under different regularization scales. In our analysis, the bias and variance can be characterized by the spectral decay of a data-dependent regularized kernel: the original kernel matrix associated with an additional re-weighting matrix, and thus the re-weighting strategy can be regarded as a data-dependent regularization for better understanding. Besides, our analysis provides asymptotic expansion of kernel functions/vectors under covariate shift, which has its own interest.

EWRL Workshop 2024 Workshop Paper

Imitation Learning in Discounted Linear MDP without exploration assumptions

• Luca Viano
• Stratis Skoulakis
• Volkan Cevher

We present a new algorithm for imitation learning in infinite horizon linear MDPs dubbed ILARL which greatly improves the bound on the number of trajectories that the learner needs to sample from the environment. In particular, we remove exploration assumptions required in previous works and we improve the dependence on the desired accuracy $\epsilon$ from $\mathcal{O}\br{\epsilon^{-5}}$ to $\mathcal{O}\br{\epsilon^{-4}}$. Our result relies on a connection between imitation learning and online learning in MDPs with adversarial losses. For the latter setting, we present the first result for infinite horizon linear MDP which may be of independent interest. Moreover, we are able to provide a strengthen result for the finite horizon case where we achieve $\mathcal{O}\br{\epsilon^{-2}}$. Numerical experiments with linear function approximation shows that ILARL outperforms other commonly used algorithms.

ICML Conference 2024 Conference Paper

Imitation Learning in Discounted Linear MDPs without exploration assumptions

• Luca Viano
• Stratis Skoulakis
• Volkan Cevher

We present a new algorithm for imitation learning in infinite horizon linear MDPs dubbed ILARL which greatly improves the bound on the number of trajectories that the learner needs to sample from the environment. In particular, we remove exploration assumptions required in previous works and we improve the dependence on the desired accuracy $\epsilon$ from $\mathcal{O}(\epsilon^{-5})$ to $\mathcal{O} (\epsilon^{-4})$. Our result relies on a connection between imitation learning and online learning in MDPs with adversarial losses. For the latter setting, we present the first result for infinite horizon linear MDP which may be of independent interest. Moreover, we are able to provide a strengthen result for the finite horizon case where we achieve $\mathcal{O}(\epsilon^{-2})$. Numerical experiments with linear function approximation shows that ILARL outperforms other commonly used algorithms.

ICML Conference 2024 Conference Paper

Improving SAM Requires Rethinking its Optimization Formulation

• Wanyun Xie
• Fabian Latorre
• Kimon Antonakopoulos
• Thomas Pethick
• Volkan Cevher

This paper rethinks Sharpness-Aware Minimization (SAM), which is originally formulated as a zero-sum game where the weights of a network and a bounded perturbation try to minimize/maximize, respectively, the same differentiable loss. To fundamentally improve this design, we argue that SAM should instead be reformulated using the 0-1 loss. As a continuous relaxation, we follow the simple conventional approach where the minimizing (maximizing) player uses an upper bound (lower bound) surrogate to the 0-1 loss. This leads to a novel formulation of SAM as a bilevel optimization problem, dubbed as BiSAM. BiSAM with newly designed lower-bound surrogate loss indeed constructs stronger perturbation. Through numerical evidence, we show that BiSAM consistently results in improved performance when compared to the original SAM and variants, while enjoying similar computational complexity. Our code is available at https: //github. com/LIONS-EPFL/BiSAM.

ICML Conference 2024 Conference Paper

Learning to Remove Cuts in Integer Linear Programming

• Pol Puigdemont
• Stratis Skoulakis
• Grigorios Chrysos 0002
• Volkan Cevher

Cutting plane methods are a fundamental approach for solving integer linear programs (ILPs). In each iteration of such methods, additional linear constraints (cuts) are introduced to the constraint set with the aim of excluding the previous fractional optimal solution while not affecting the optimal integer solution. In this work, we explore a novel approach within cutting plane methods: instead of only adding new cuts, we also consider the removal of previous cuts introduced at any of the preceding iterations of the method under a learnable parametric criteria. We demonstrate that in fundamental combinatorial optimization settings such cut removal policies can lead to significant improvements over both human-based and machine learning-guided cut addition policies even when implemented with simple models.

JMLR Journal 2024 Journal Article

Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks

• Fanghui Liu
• Leello Dadi
• Volkan Cevher

Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks as the curse of dimensionality (CoD) cannot be evaded when trying to approximate even a single ReLU neuron. In this paper, we study a suitable function space for over-parameterized two-layer neural networks with bounded norms (e.g., the path norm, the Barron norm) in the perspective of sample complexity and generalization properties. First, we show that the path norm (as well as the Barron norm) is able to obtain width-independence sample complexity bounds, which allows for uniform convergence guarantees. Based on this result, we derive the improved result of metric entropy for $\epsilon$-covering up to $O(\epsilon^{-\frac{2d}{d+2}})$ ($d$ is the input dimension and the depending constant is at most polynomial order of $d$) via the convex hull technique, which demonstrates the separation with kernel methods with $\Omega(\epsilon^{-d})$ to learn the target function in a Barron space. Second, this metric entropy result allows for building a sharper generalization bound under a general moment hypothesis setting, achieving the rate at $O(n^{-\frac{d+2}{2d+2}})$. Our analysis is novel in that it offers a sharper and refined estimation for metric entropy (with a clear dependence relationship on the dimension $d$) and unbounded sampling in the estimation of the sample error and the output error. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

ICML Conference 2024 Conference Paper

MADA: Meta-Adaptive Optimizers Through Hyper-Gradient Descent

• Kaan Ozkara
• Can Karakus
• Parameswaran Raman
• Mingyi Hong 0001
• Shoham Sabach
• Branislav Kveton
• Volkan Cevher

Following the introduction of Adam, several novel adaptive optimizers for deep learning have been proposed. These optimizers typically excel in some tasks but may not outperform Adam uniformly across all tasks. In this work, we introduce Meta-Adaptive Optimizers (MADA), a unified optimizer framework that can generalize several known optimizers and dynamically learn the most suitable one during training. The key idea in MADA is to parameterize the space of optimizers and dynamically search through it using hyper-gradient descent during training. We empirically compare MADA to other popular optimizers on vision and language tasks, and find that MADA consistently outperforms Adam and other popular optimizers, and is robust against sub-optimally tuned hyper-parameters. MADA achieves a greater validation performance improvement over Adam compared to other popular optimizers during GPT-2 training and fine-tuning. We also propose AVGrad, a modification of AMSGrad that replaces the maximum operator with averaging, which is more suitable for hyper-gradient optimization. Finally, we provide a convergence analysis to show that parameterized interpolations of optimizers can improve their error bounds (up to constants), hinting at an advantage for meta-optimizers.

NeurIPS Conference 2024 Conference Paper

Membership Inference Attacks against Large Vision-Language Models

• Zhan Li
• Yongtao Wu
• Yihang Chen
• Francesco Tonin
• Elias Abad Rocamora
• Volkan Cevher

Large vision-language models (VLLMs) exhibit promising capabilities for processing multi-modal tasks across various application scenarios. However, their emergence also raises significant data security concerns, given the potential inclusion of sensitive information, such as private photos and medical records, in their training datasets. Detecting inappropriately used data in VLLMs remains a critical and unresolved issue, mainly due to the lack of standardized datasets and suitable methodologies. In this study, we introduce the first membership inference attack (MIA) benchmark tailored for various VLLMs to facilitate training data detection. Then, we propose a novel MIA pipeline specifically designed for token-level image detection. Lastly, we present a new metric called MaxRényi-K%, which is based on the confidence of the model output and applies to both text and image data. We believe that our work can deepen the understanding and methodology of MIAs in the context of VLLMs. Our code and datasets are available at https: //github. com/LIONS-EPFL/VL-MIA.

TMLR Journal 2024 Journal Article

Mixed Nash for Robust Federated Learning

• Wanyun Xie
• Thomas Pethick
• Ali Ramezani-Kebrya
• Volkan Cevher

We study robust federated learning (FL) within a game theoretic framework to alleviate the server vulnerabilities to even an informed adversary who can tailor training-time attacks. Specifically, we introduce RobustTailor, a simulation-based framework that prevents the adversary from being omniscient and derives its convergence guarantees. RobustTailor improves robustness to training-time attacks significantly while preserving almost the same privacy guarantees as standard robust aggregation schemes in FL. Empirical results under challenging attacks show that RobustTailor performs close to an upper bound with perfect knowledge of honest clients.

ICLR Conference 2024 Conference Paper

Multilinear Operator Networks

• Yixin Cheng
• Grigorios Chrysos 0002
• Markos Georgopoulos
• Volkan Cevher

Despite the remarkable capabilities of deep neural networks in image recognition, the dependence on activation functions remains a largely unexplored area and has yet to be eliminated. On the other hand, Polynomial Networks is a class of models that does not require activation functions, but have yet to perform on par with modern architectures. In this work, we aim close this gap and propose MONet, which relies *solely* on multilinear operators. The core layer of MONet, called Mu-Layer, captures multiplicative interactions of the elements of the input token. MONet captures high-degree interactions of the input elements and we demonstrate the efficacy of our approach on a series of image recognition and scientific computing benchmarks. The proposed model outperforms prior polynomial networks and performs on par with modern architectures. We believe that MONet can inspire further research on models that use entirely multilinear operations.

NeurIPS Conference 2024 Conference Paper

On Feature Learning in Structured State Space Models

• Leena Chennuru Vankadara
• Jin Xu
• Moritz Haas
• Volkan Cevher

This paper studies the scaling behavior of state-space models (SSMs) and their structured variants, such as Mamba, that have recently arisen in popularity as alternatives to transformer-based neural network architectures. Specifically, we focus on the capability of SSMs to learn features as their network width approaches infinity. Our findings reveal that established scaling rules, such as the Maximal Update Parameterization, fail to support feature learning as these models cannot be represented in the form of Tensor Programs. Additionally, we demonstrate that spectral scaling conditions, shown to be effective for feature learning in a host of other architectures, do not hold the same implications for SSMs. Through a detailed signal propagation analysis in SSMs, both forward and backward, we identify the appropriate scaling necessary for non-trivial feature evolution in the infinite-width limit. Our proposed scaling shows behavior akin to the Maximal Update Parameterization, such as improved stability, better generalization, and transferability of optimal hyper-parameters from small to large scale SSMs.

JMLR Journal 2024 Journal Article

On the Generalization of Stochastic Gradient Descent with Momentum

• Ali Ramezani-Kebrya
• Kimon Antonakopoulos
• Volkan Cevher
• Ashish Khisti
• Ben Liang

While momentum-based accelerated variants of stochastic gradient descent (SGD) are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In this work, we first show that there exists a convex loss function for which the stability gap for multiple epochs of SGD with standard heavy-ball momentum (SGDM) becomes unbounded. Then, for smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM) under a broad range of step-sizes, and show that it can train machine learning models for multiple epochs with a guarantee for generalization. Finally, for the special case of strongly convex loss functions, we find a range of momentum such that multiple epochs of standard SGDM, as a special form of SGDEM, also generalizes. Extending our results on generalization, we also develop an upper bound on the expected true risk, in terms of the number of training steps, sample size, and momentum. Our experimental evaluations verify the consistency between the numerical results and our theoretical bounds. SGDEM improves the generalization error of SGDM when training ResNet-18 on ImageNet in practical distributed settings. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

NeurIPS Conference 2024 Conference Paper

Randomized algorithms and PAC bounds for inverse reinforcement learning in continuous spaces

• Angeliki Kamoutsi
• Peter Schmitt-Förster
• Tobias Sutter
• Volkan Cevher
• John Lygeros

This work studies discrete-time discounted Markov decision processes with continuous state and action spaces and addresses the inverse problem of inferring a cost function from observed optimal behavior. We first consider the case in which we have access to the entire expert policy and characterize the set of solutions to the inverse problem by using occupation measures, linear duality, and complementary slackness conditions. To avoid trivial solutions and ill-posedness, we introduce a natural linear normalization constraint. This results in an infinite-dimensional linear feasibility problem, prompting a thorough analysis of its properties. Next, we use linear function approximators and adopt a randomized approach, namely the scenario approach and related probabilistic feasibility guarantees, to derive $\varepsilon$-optimal solutions for the inverse problem. We further discuss the sample complexity for a desired approximation accuracy. Finally, we deal with the more realistic case where we only have access to a finite set of expert demonstrations and a generative model and provide bounds on the error made when working with samples.

ICML Conference 2024 Conference Paper

REST: Efficient and Accelerated EEG Seizure Analysis through Residual State Updates

• Arshia Afzal
• Grigorios Chrysos 0002
• Volkan Cevher
• Mahsa Shoaran

EEG-based seizure detection models face challenges in terms of inference speed and memory efficiency, limiting their real-time implementation in clinical devices. This paper introduces a novel graph-based residual state update mechanism (REST) for real-time EEG signal analysis in applications such as epileptic seizure detection. By leveraging a combination of graph neural networks and recurrent structures, REST efficiently captures both non-Euclidean geometry and temporal dependencies within EEG data. Our model demonstrates high accuracy in both seizure detection and classification tasks. Notably, REST achieves a remarkable 9-fold acceleration in inference speed compared to state-of-the-art models, while simultaneously demanding substantially less memory than the smallest model employed for this task. These attributes position REST as a promising candidate for real-time implementation in clinical devices, such as Responsive Neurostimulation or seizure alert systems.

ICML Conference 2024 Conference Paper

Revisiting Character-level Adversarial Attacks for Language Models

• Elías Abad-Rocamora
• Yongtao Wu
• Fanghui Liu 0001
• Grigorios Chrysos 0002
• Volkan Cevher

Adversarial attacks in Natural Language Processing apply perturbations in the character or token levels. Token-level attacks, gaining prominence for their use of gradient-based methods, are susceptible to altering sentence semantics, leading to invalid adversarial examples. While character-level attacks easily maintain semantics, they have received less attention as they cannot easily adopt popular gradient-based methods, and are thought to be easy to defend. Challenging these beliefs, we introduce Charmer, an efficient query-based adversarial attack capable of achieving high attack success rate (ASR) while generating highly similar adversarial examples. Our method successfully targets both small (BERT) and large (Llama 2) models. Specifically, on BERT with SST-2, Charmer improves the ASR in $4. 84$% points and the USE similarity in $8$% points with respect to the previous art. Our implementation is available in https: //github. com/LIONS-EPFL/Charmer.

ICLR Conference 2024 Conference Paper

Robust NAS under adversarial training: benchmark, theory, and beyond

• Yongtao Wu
• Fanghui Liu 0001
• Carl-Johann Simon-Gabriel
• Grigorios Chrysos 0002
• Volkan Cevher

Recent developments in neural architecture search (NAS) emphasize the significance of considering robust architectures against malicious data. However, there is a notable absence of benchmark evaluations and theoretical guarantees for searching these robust architectures, especially when adversarial training is considered. In this work, we aim to address these two challenges, making twofold contributions. First, we release a comprehensive data set that encompasses both clean accuracy and robust accuracy for a vast array of adversarially trained networks from the NAS-Bench-201 search space on image datasets. Then, leveraging the neural tangent kernel (NTK) tool from deep learning theory, we establish a generalization theory for searching architecture in terms of clean accuracy and robust accuracy under multi-objective adversarial training. We firmly believe that our benchmark and theoretical insights will significantly benefit the NAS community through reliable reproducibility, efficient assessment, and theoretical foundation, particularly in the pursuit of robust architectures.

NeurIPS Conference 2024 Conference Paper

SAMPa: Sharpness-aware Minimization Parallelized

• Wanyun Xie
• Thomas Pethick
• Volkan Cevher

Sharpness-aware minimization (SAM) has been shown to improve the generalization of neural networks. However, each SAM update requires sequentially computing two gradients, effectively doubling the per-iteration cost compared to base optimizers like SGD. We propose a simple modification of SAM, termed SAMPa, which allows us to fully parallelize the two gradient computations. SAMPa achieves a twofold speedup of SAM under the assumption that communication costs between devices are negligible. Empirical results show that SAMPa ranks among the most efficient variants of SAM in terms of computational time. Additionally, our method consistently outperforms SAM across both vision and language tasks. Notably, SAMPa theoretically maintains convergence guarantees even for fixed perturbation sizes, which is established through a novel Lyapunov function. We in fact arrive at SAMPa by treating this convergence guarantee as a hard requirement---an approach we believe is promising for developing SAM-based methods in general. Our code is available at https: //github. com/LIONS-EPFL/SAMPa.

ICML Conference 2024 Conference Paper

Truly No-Regret Learning in Constrained MDPs

• Adrian Müller 0002
• Pragnya Alatur
• Volkan Cevher
• Giorgia Ramponi
• Niao He

Constrained Markov decision processes (CMDPs) are a common way to model safety constraints in reinforcement learning. State-of-the-art methods for efficiently solving CMDPs are based on primal-dual algorithms. For these algorithms, all currently known regret bounds allow for error cancellations — one can compensate for a constraint violation in one round with a strict constraint satisfaction in another. This makes the online learning process unsafe since it only guarantees safety for the final (mixture) policy but not during learning. As Efroni et al. (2020) pointed out, it is an open question whether primal-dual algorithms can provably achieve sublinear regret if we do not allow error cancellations. In this paper, we give the first affirmative answer. We first generalize a result on last-iterate convergence of regularized primal-dual schemes to CMDPs with multiple constraints. Building upon this insight, we propose a model-based primal-dual algorithm to learn in an unknown CMDP. We prove that our algorithm achieves sublinear regret without error cancellations.

EWRL Workshop 2024 Workshop Paper

Truly No-Regret Learning in Constrained MDPs

• Adrian Müller
• Pragnya Alatur
• Volkan Cevher
• Giorgia Ramponi
• Niao He

Constrained Markov decision processes (CMDPs) are a common way to model safety constraints in reinforcement learning. State-of-the-art methods for efficiently solving CMDPs are based on primal-dual algorithms. For these algorithms, all currently known regret bounds allow for *error cancellations* --- one can compensate for a constraint violation in one round with a strict constraint satisfaction in another. This makes the online learning process unsafe since it only guarantees safety for the final (mixture) policy but not during learning. As Efroni et al. (2020) pointed out, it is an open question whether primal-dual algorithms can provably achieve sublinear regret if we do not allow error cancellations. In this paper, we give the first affirmative answer. We first generalize a result on last-iterate convergence of regularized primal-dual schemes to CMDPs with multiple constraints. Building upon this insight, we propose a model-based primal-dual algorithm to learn in an unknown CMDP. We prove that our algorithm achieves sublinear regret without error cancellations.

ICML Conference 2024 Conference Paper

Universal Gradient Methods for Stochastic Convex Optimization

• Anton Rodomanov
• Ali Kavis
• Yongtao Wu
• Kimon Antonakopoulos
• Volkan Cevher

We develop universal gradient methods for Stochastic Convex Optimization (SCO). Our algorithms automatically adapt not only to the oracle’s noise but also to the Hölder smoothness of the objective function without a priori knowledge of the particular setting. The key ingredient is a novel strategy for adjusting step-size coefficients in the Stochastic Gradient Method (SGD). Unlike AdaGrad, which accumulates gradient norms, our Universal Gradient Method accumulates appropriate combinations of gradientand iterate differences. The resulting algorithm has state-of-the-art worst-case convergence rate guarantees for the entire Hölder class including, in particular, both nonsmooth functions and those with Lipschitz continuous gradient. We also present the Universal Fast Gradient Method for SCO enjoying optimal efficiency estimates.

NeurIPS Conference 2023 Conference Paper

Alternation makes the adversary weaker in two-player games

• Volkan Cevher
• Ashok Cutkosky
• Ali Kavis
• Georgios Piliouras
• Stratis Skoulakis
• Luca Viano

Motivated by alternating game-play in two-player games, we study an altenating variant of the \textit{Online Linear Optimization} (OLO). In alternating OLO, a \textit{learner} at each round $t \in [n]$ selects a vector $x^t$ and then an \textit{adversary} selects a cost-vector $c^t \in [-1, 1]^n$. The learner then experiences cost $(c^t + c^{t-1})^\top x^t$ instead of $(c^t)^\top x^t$ as in standard OLO. We establish that under this small twist, the $\Omega(\sqrt{T})$ lower bound on the regret is no longer valid. More precisely, we present two online learning algorithms for alternating OLO that respectively admit $\mathcal{O}((\log n)^{4/3} T^{1/3})$ regret for the $n$-dimensional simplex and $\mathcal{O}(\rho \log T)$ regret for the ball of radius $\rho>0$. Our results imply that in alternating game-play, an agent can always guarantee $\mathcal{\tilde{O}}((\log n)^{4/3} T^{1/3})$ regardless the strategies of the other agent while the regret bound improves to $\mathcal{O}(\log T)$ in case the agent admits only two actions.

ICML Conference 2023 Conference Paper

Benign Overfitting in Deep Neural Networks under Lazy Training

• Zhenyu Zhu
• Fanghui Liu 0001
• Grigorios Chrysos 0002
• Francesco Locatello
• Volkan Cevher

This paper focuses on over-parameterized deep neural networks (DNNs) with ReLU activation functions and proves that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification while obtaining (nearly) zero-training error under the lazy training regime. For this purpose, we unify three interrelated concepts of overparameterization, benign overfitting, and the Lipschitz constant of DNNs. Our results indicate that interpolating with smoother functions leads to better generalization. Furthermore, we investigate the special case where interpolating smooth ground-truth functions is performed by DNNs under the Neural Tangent Kernel (NTK) regime for generalization. Our result demonstrates that the generalization error converges to a constant order that only depends on label noise and initialization noise, which theoretically verifies benign overfitting. Our analysis provides a tight lower bound on the normalized margin under non-smooth activation functions, as well as the minimum eigenvalue of NTK under high-dimensional settings, which has its own interest in learning theory.

ICLR Conference 2023 Conference Paper

DiGress: Discrete Denoising diffusion for graph generation

• Clément Vignac
• Igor Krawczuk
• Antoine Siraudin
• Bohan Wang
• Volkan Cevher
• Pascal Frossard

This work introduces DiGress, a discrete denoising diffusion model for generating graphs with categorical node and edge attributes. Our model utilizes a discrete diffusion process that progressively edits graphs with noise, through the process of adding or removing edges and changing the categories. A graph transformer network is trained to revert this process, simplifying the problem of distribution learning over graphs into a sequence of node and edge classification tasks. We further improve sample quality by introducing a Markovian noise model that preserves the marginal distribution of node and edge types during diffusion, and by incorporating auxiliary graph-theoretic features. A procedure for conditioning the generation on graph-level features is also proposed. DiGress achieves state-of-the-art performance on molecular and non-molecular datasets, with up to 3x validity improvement on a planar graph dataset. It is also the first model to scale to the large GuacaMol dataset containing 1.3M drug-like molecules without the use of molecule-specific representations.

ICLR Conference 2023 Conference Paper

Distributed Extra-gradient with Optimal Complexity and Communication Guarantees

• Ali Ramezani-Kebrya
• Kimon Antonakopoulos
• Igor Krawczuk
• Justin Deschenaux
• Volkan Cevher

We consider monotone variational inequality (VI) problems in multi-GPU settings where multiple processors/workers/clients have access to local stochastic dual vectors. This setting includes a broad range of important problems from distributed convex minimization to min-max and games. Extra-gradient, which is a de facto algorithm for monotone VI problems, has not been designed to be communication-efficient. To this end, we propose a quantized generalized extra-gradient (Q-GenX), which is an unbiased and adaptive compression method tailored to solve VIs. We provide an adaptive step-size rule, which adapts to the respective noise profiles at hand and achieve a fast rate of ${\cal O}(1/T)$ under relative noise, and an order-optimal ${\cal O}(1/\sqrt{T})$ under absolute noise and show distributed training accelerates convergence. Finally, we validate our theoretical results by providing real-world experiments and training generative adversarial networks on multiple GPUs.

NeurIPS Conference 2023 Conference Paper

Efficient Online Clustering with Moving Costs

• Dimitrios Christou
• Stratis Skoulakis
• Volkan Cevher

In this work we consider an online learning problem, called Online $k$-Clustering with Moving Costs, at which a learner maintains a set of $k$ facilities over $T$ rounds so as to minimize the connection cost of an adversarially selected sequence of clients. The learner is informed on the positions of the clients at each round $t$ only after its facility-selection and can use this information to update its decision in the next round. However, updating the facility positions comes with an additional moving cost based on the moving distance of the facilities. We present the first $\mathcal{O}(\log n)$-regret polynomial-time online learning algorithm guaranteeing that the overall cost (connection $+$ moving) is at most $\mathcal{O}(\log n)$ times the time-averaged connection cost of the best fixed solution. Our work improves on the recent result of (Fotakis et al. , 2021) establishing $\mathcal{O}(k)$-regret guarantees only on the connection cost.

NeurIPS Conference 2023 Conference Paper

Exponential Lower Bounds for Fictitious Play in Potential Games

• Ioannis Panageas
• Nikolas Patris
• Stratis Skoulakis
• Volkan Cevher

Fictitious Play (FP) is a simple and natural dynamic for repeated play with many applications in game theory and multi-agent reinforcement learning. It was introduced by Brown and its convergence properties for two-player zero-sum games was established later by Robinson. Potential games [Monderer and Shapley 1996] is another class of games which exhibit the FP property [Monderer and Shapley 1996], i. e. , FP dynamics converges to a Nash equilibrium if all agents follows it. Nevertheless, except for two-player zero-sum games and for specific instances of payoff matrices [Abernethy et. al. 2021] or for adversarial tie-breaking rules [Daskalakis and Pan, 2014], the \textit{convergence rate} of FP is unknown. In this work, we focus on the rate of convergence of FP when applied to potential games and more specifically identical payoff games. We prove that FP can take exponential time (in the number of strategies) to reach a Nash equilibrium, even if the game is restricted to \textit{two agents}. To prove this, we recursively construct a two-player coordination game with a unique Nash equilibrium. Moreover, every approximate Nash equilibrium in the constructed game must be close to the pure Nash equilibrium in $\ell_1$-distance.

TMLR Journal 2023 Journal Article

Federated Learning under Covariate Shifts with Generalization Guarantees

• Ali Ramezani-Kebrya
• Fanghui Liu
• Thomas Pethick
• Grigorios Chrysos
• Volkan Cevher

This paper addresses intra-client and inter-client covariate shifts in federated learning (FL) with a focus on the overall generalization performance. To handle covariate shifts, we formulate a new global model training paradigm and propose Federated Importance-Weighted Empirical Risk Minimization (FTW-ERM) along with improving density ratio matching methods without requiring perfect knowledge of the supremum over true ratios. We also propose the communication-efficient variant FITW-ERM with the same level of privacy guarantees as those of classical ERM in FL. We theoretically show that FTW-ERM achieves smaller generalization error than classical ERM under certain settings. Experimental results demonstrate the superiority of FTW-ERM over existing FL baselines in challenging imbalanced federated settings in terms of data distribution shifts across clients.

ICLR Conference 2023 Conference Paper

Finding Actual Descent Directions for Adversarial Training

• Fabian Latorre
• Igor Krawczuk
• Leello Tadesse Dadi
• Thomas Pethick
• Volkan Cevher

Adversarial Training using a strong first-order adversary (PGD) is the gold standard for training Deep Neural Networks that are robust to adversarial examples. We show that, contrary to the general understanding of the method, the gradient at an optimal adversarial example may increase, rather than decrease, the adversarially robust loss. This holds independently of the learning rate. More precisely, we provide a counterexample to a corollary of Danskin's Theorem presented in the seminal paper of Madry et al. (2018) which states that a solution of the inner maximization problem can yield a descent direction for the adversarially robust loss. Based on a correct interpretation of Danskin's Theorem, we propose Danskin's Descent Direction (DDi) and we verify experimentally that it provides better directions than those obtained by a PGD adversary. Using the CIFAR10 dataset we further provide a real world example showing that our method achieves a steeper increase in robustness levels in the early stages of training, and is more stable than the PGD baseline. As a limitation, PGD training of ReLU+BatchNorm networks still performs better, but current theory is unable to explain this.

NeurIPS Conference 2023 Conference Paper

Initialization Matters: Privacy-Utility Analysis of Overparameterized Neural Networks

• Jiayuan Ye
• Zhenyu Zhu
• Fanghui Liu
• Reza Shokri
• Volkan Cevher

We analytically investigate how over-parameterization of models in randomized machine learning algorithms impacts the information leakage about their training data. Specifically, we prove a privacy bound for the KL divergence between model distributions on worst-case neighboring datasets, and explore its dependence on the initialization, width, and depth of fully connected neural networks. We find that this KL privacy bound is largely determined by the expected squared gradient norm relative to model parameters during training. Notably, for the special setting of linearized network, our analysis indicates that the squared gradient norm (and therefore the escalation of privacy loss) is tied directly to the per-layer variance of the initialization distribution. By using this analysis, we demonstrate that privacy bound improves with increasing depth under certain initializations (LeCun and Xavier), while degrades with increasing depth under other initializations (He and NTK). Our work reveals a complex interplay between privacy and depth that depends on the chosen initialization distribution. We further prove excess empirical risk bounds under a fixed KL privacy budget, and show that the interplay between privacy utility trade-off and depth is similarly affected by the initialization.

NeurIPS Conference 2023 Conference Paper

Maximum Independent Set: Self-Training through Dynamic Programming

• Lorenzo Brusca
• Lars C. P. M. Quaedvlieg
• Stratis Skoulakis
• Grigorios Chrysos
• Volkan Cevher

This work presents a graph neural network (GNN) framework for solving the maximum independent set (MIS) problem, inspired by dynamic programming (DP). Specifically, given a graph, we propose a DP-like recursive algorithm based on GNNs that firstly constructs two smaller sub-graphs, predicts the one with the larger MIS, and then uses it in the next recursive call. To train our algorithm, we require annotated comparisons of different graphs concerning their MIS size. Annotating the comparisons with the output of our algorithm leads to a self-training process that results in more accurate self-annotation of the comparisons and vice versa. We provide numerical evidence showing the superiority of our method vs prior methods in multiple synthetic and real-world datasets.

NeurIPS Conference 2023 Conference Paper

On the Convergence of Encoder-only Shallow Transformers

• Yongtao Wu
• Fanghui Liu
• Grigorios Chrysos
• Volkan Cevher

In this paper, we aim to build the global convergence theory of encoder-only shallow Transformers under a realistic setting from the perspective of architectures, initialization, and scaling under a finite width regime. The difficulty lies in how to tackle the softmax in self-attention mechanism, the core ingredient of Transformer. In particular, we diagnose the scaling scheme, carefully tackle the input/output of softmax, and prove that quadratic overparameterization is sufficient for global convergence of our shallow Transformers under commonly-used He/LeCun initialization in practice. Besides, neural tangent kernel (NTK) based analysis is also given, which facilitates a comprehensive comparison. Our theory demonstrates the separation on the importance of different scaling schemes and initialization. We believe our results can pave the way for a better understanding of modern Transformers, particularly on training dynamics.

EWRL Workshop 2023 Workshop Paper

Proximal Point Imitation Learning

• Luca Viano
• Angeliki Kamoutsi
• Gergely Neu
• Igor Krawczuk
• Volkan Cevher

This work develops new algorithms with rigorous efficiency guarantees for infinite horizon imitation learning (IL) with linear function approximation without restrictive coherence assumptions. We begin with the minimax formulation of the problem and then outline how to leverage classical tools from optimization, in particular, the proximal-point method (PPM) and dual smoothing, for online and offline IL, respectively. Thanks to PPM, we avoid nested policy evaluation and cost updates for online IL appearing in the prior literature. In particular, we do away with the conventional alternating updates by the optimization of a single convex and smooth objective over both cost and $Q$-functions. When solved inexactly, we relate the optimization errors to the suboptimality of the recovered policy. As an added bonus, by re-interpreting PPM as dual smoothing with the expert policy as a center point, we also obtain an offline IL algorithm enjoying theoretical guarantees in terms of required expert trajectories. Finally, we achieve convincing empirical performance for both linear and neural network function approximation.

TMLR Journal 2023 Journal Article

Revisiting adversarial training for the worst-performing class

• Thomas Pethick
• Grigorios Chrysos
• Volkan Cevher

Despite progress in adversarial training (AT), there is a substantial gap between the top-performing and worst-performing classes in many datasets. For example, on CIFAR10, the accuracies for the best and worst classes are 74% and 23%, respectively. We argue that this gap can be reduced by explicitly optimizing for the worst-performing class, resulting in a min-max-max optimization formulation. Our method, called class focused online learning (CFOL), includes high probability convergence guarantees for the worst class loss and can be easily integrated into existing training setups with minimal computational overhead. We demonstrate an improvement to 32% in the worst class accuracy on CIFAR10, and we observe consistent behavior across CIFAR100 and STL10. Our study highlights the importance of moving beyond average accuracy, which is particularly important in safety-critical applications.

NeurIPS Conference 2023 Conference Paper

Sample Complexity Bounds for Score-Matching: Causal Discovery and Generative Modeling

• Zhenyu Zhu
• Francesco Locatello
• Volkan Cevher

This paper provides statistical sample complexity bounds for score-matching and its applications in causal discovery. We demonstrate that accurate estimation of the score function is achievable by training a standard deep ReLU neural network using stochastic gradient descent. We establish bounds on the error rate of recovering causal relationships using the score-matching-based causal discovery method of Rolland et al. [2022], assuming a sufficiently good estimation of the score function. Finally, we analyze the upper bound of score-matching estimation within the score-based generative modeling, which has been applied for causal discovery but is also of independent interest within the domain of generative models.

ICML Conference 2023 Conference Paper

Semi Bandit dynamics in Congestion Games: Convergence to Nash Equilibrium and No-Regret Guarantees

• Ioannis Panageas
• Stratis Skoulakis
• Luca Viano
• Xiao Wang 0036
• Volkan Cevher

In this work, we propose introduce a variant of online stochastic gradient descent and prove it converges to Nash equilibria and simultaneously it has sublinear regret for the class of congestion games in the semi-bandit feedback setting. Our proposed method admits convergence rates depending only polynomially on the number of players and the number of facilities, but not on the size of the action set, which can be exponentially large in terms of the number of facilities. Moreover, the running time of our method has polynomial-time dependence on the implicit description of the game. Our analysis exploits techniques from convex geometry, in particular Caratheodory’s theorem and recent advances in non-convex stochastic optimization. This work improves upon and answers an open question from (Cui et al 2022).

ICLR Conference 2023 Conference Paper

Solving stochastic weak Minty variational inequalities without increasing batch size

• Thomas Pethick
• Olivier Fercoq
• Puya Latafat
• Panagiotis Patrinos
• Volkan Cevher

This paper introduces a family of stochastic extragradient-type algorithms for a class of nonconvex-nonconcave problems characterized by the weak Minty variational inequality (MVI). Unlike existing results on extragradient methods in the monotone setting, employing diminishing stepsizes is no longer possible in the weak MVI setting. This has led to approaches such as increasing batch sizes per iteration which can however be prohibitively expensive. In contrast, our proposed methods involves two stepsizes and only requires one additional oracle evaluation per iteration. We show that it is possible to keep one fixed stepsize while it is only the second stepsize that is taken to be diminishing, making it interesting even in the monotone setting. Almost sure convergence is established and we provide a unified analysis for this family of schemes which contains a nonlinear generalization of the celebrated primal dual hybrid gradient algorithm.

NeurIPS Conference 2023 Conference Paper

Stable Nonconvex-Nonconcave Training via Linear Interpolation

• Thomas Pethick
• Wanyun Xie
• Volkan Cevher

This paper presents a theoretical analysis of linear interpolation as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear interpolation can help by leveraging the theory of nonexpansive operators. We construct a new optimization scheme called relaxed approximate proximal point (RAPP), which is the first 1-SCLI method to achieve last iterate convergence rates for $\rho$-comonotone problems while only requiring $\rho > -\tfrac{1}{2L}$. The construction extends to constrained and regularized settings. By replacing the inner optimizer in RAPP we rediscover the family of Lookahead algorithms for which we establish convergence in cohypomonotone problems even when the base optimizer is taken to be gradient descent ascent. The range of cohypomonotone problems in which Lookahead converges is further expanded by exploiting that Lookahead inherits the properties of the base optimizer. We corroborate the results with experiments on generative adversarial networks which demonstrates the benefits of the linear interpolation present in both RAPP and Lookahead.

EWRL Workshop 2023 Workshop Paper

Understanding Deep Neural Function Approximation in Reinforcement Learning via $\epsilon$-Greedy Exploration

• Fanghui Liu
• Luca Viano
• Volkan Cevher

This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $\epsilon$-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e. g. , width and depth) beyond the "linear'' regime. To be specific, we focus on the value based algorithm with the $\epsilon$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces, respectively, which aims at approximating an $\alpha$-smooth Q-function in a $d$-dimensional feature space. We prove that, with $T$ episodes, scaling the width $m = \widetilde{\mathcal{O}}(T^{\frac{d}{2\alpha + d}})$ and the depth $L=\mathcal{O}(\log T)$ of the neural network for deep RL is sufficient for learning with sublinear regret in Besov spaces. Moreover, for a two layer neural network endowed by the Barron space, scaling the width $\Omega(\sqrt{T})$ is sufficient. To achieve this, the key issue in our analysis is how to estimate the temporal difference error under deep neural function approximation as the $\epsilon$-greedy exploration is not enough to ensure ``optimism''. Our analysis reformulates the temporal difference error in an $L^2(\mathrm{d}\mu)$-integrable space over a certain averaged measure $\mu$, and transforms it to a generalization problem under the non-iid setting. This might have its own interest in RL theory for better understanding $\epsilon$-greedy exploration in deep RL.

ICML Conference 2023 Conference Paper

What can online reinforcement learning with function approximation benefit from general coverage conditions?

• Fanghui Liu 0001
• Luca Viano
• Volkan Cevher

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the $L^p$ variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e. g. , linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond $\widetilde{\mathcal{O}}(\sqrt{T})$ and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

ICML Conference 2023 Conference Paper

When do Minimax-fair Learning and Empirical Risk Minimization Coincide?

• Harvineet Singh
• Matthäus Kleindessner
• Volkan Cevher
• Rumi Chunara
• Chris Russell 0001

Minimax-fair machine learning minimizes the error for the worst-off group. However, empirical evidence suggests that when sophisticated models are trained with standard empirical risk minimization (ERM), they often have the same performance on the worst-off group as a minimax-trained model. Our work makes this counter-intuitive observation concrete. We prove that if the hypothesis class is sufficiently expressive and the group information is recoverable from the features, ERM and minimax-fairness learning formulations indeed have the same performance on the worst-off group. We provide additional empirical evidence of how this observation holds on a wide range of datasets and hypothesis classes. Since ERM is fundamentally easier than minimax optimization, our findings have implications on the practice of fair machine learning.

ICML Conference 2022 Conference Paper

A Natural Actor-Critic Framework for Zero-Sum Markov Games

• Ahmet Alacaoglu
• Luca Viano
• Niao He
• Volkan Cevher

We introduce algorithms based on natural actor-critic and analyze their sample complexity for solving two player zero-sum Markov games in the tabular case. Our results improve the best-known sample complexities of policy gradient/actor-critic methods for convergence to Nash equilibrium in the multi-agent setting. We use the error propagation scheme in approximate dynamic programming, recent advances for global convergence of policy gradient methods, temporal difference learning, and techniques from stochastic primal-dual optimization. Our algorithms feature two stages, requiring agents to agree on an etiquette before starting their interactions, which is feasible for instance in self-play. However, the agents only access to joint reward and joint next state and not to each other’s actions or policies. Our complexity results match the best-known results for global convergence of policy gradient algorithms for single agent RL. We provide numerical verification of our methods for a two player bandit environment and a two player game, Alesia. We observe improved empirical performance as compared to the recently proposed optimistic gradient descent-ascent variant for Markov games.

NeurIPS Conference 2022 Conference Paper

Adaptive Stochastic Variance Reduction for Non-convex Finite-Sum Minimization

• Ali Kavis
• Stratis Skoulakis
• Kimon Antonakopoulos
• Leello Tadesse Dadi
• Volkan Cevher

We propose an adaptive variance-reduction method, called AdaSpider, for minimization of $L$-smooth, non-convex functions with a finite-sum structure. In essence, AdaSpider combines an AdaGrad-inspired (Duchi et al. , 2011), but a fairly distinct, adaptive step-size schedule with the recursive \textit{stochastic path integrated estimator} proposed in (Fang et al. , 2018). To our knowledge, AdaSpider is the first parameter-free non-convex variance-reduction method in the sense that it does not require the knowledge of problem-dependent parameters, such as smoothness constant $L$, target accuracy $\epsilon$ or any bound on gradient norms. In doing so, we are able to compute an $\epsilon$-stationary point with $\tilde{O}\left(n + \sqrt{n}/\epsilon^2\right)$ oracle-calls, which matches the respective lower bound up to logarithmic factors.

ICLR Conference 2022 Conference Paper

Controlling the Complexity and Lipschitz Constant improves Polynomial Nets

• Zhenyu Zhu
• Fabian Latorre
• Grigorios Chrysos 0002
• Volkan Cevher

While the class of Polynomial Nets demonstrates comparable performance to neural networks (NN), it currently has neither theoretical generalization characterization nor robustness guarantees. To this end, we derive new complexity bounds for the set of Coupled CP-Decomposition (CCP) and Nested Coupled CP-decomposition (NCP) models of Polynomial Nets in terms of the $\ell_\infty$-operator-norm and the $\ell_2$-operator norm. In addition, we derive bounds on the Lipschitz constant for both models to establish a theoretical certificate for their robustness. The theoretical results enable us to propose a principled regularization scheme that we also evaluate experimentally and show that it improves the accuracy as well as the robustness of the models to adversarial perturbations. We showcase how this regularization can be combined with adversarial training, resulting in further improvements.

ICLR Conference 2022 Conference Paper

Escaping limit cycles: Global convergence for constrained nonconvex-nonconcave minimax problems

• Thomas Pethick
• Puya Latafat
• Panagiotis Patrinos
• Olivier Fercoq
• Volkan Cevher

This paper introduces a new extragradient-type algorithm for a class of nonconvex-nonconcave minimax problems. It is well-known that finding a local solution for general minimax problems is computationally intractable. This observation has recently motivated the study of structures sufficient for convergence of first order methods in the more general setting of variational inequalities when the so-called weak Minty variational inequality (MVI) holds. This problem class captures non-trivial structures as we demonstrate with examples, for which a large family of existing algorithms provably converge to limit cycles. Our results require a less restrictive parameter range in the weak MVI compared to what is previously known, thus extending the applicability of our scheme. The proposed algorithm is applicable to constrained and regularized problems, and involves an adaptive stepsize allowing for potentially larger stepsizes. Our scheme also converges globally even in settings where the underlying operator exhibits limit cycles.

NeurIPS Conference 2022 Conference Paper

Extra-Newton: A First Approach to Noise-Adaptive Accelerated Second-Order Methods

• Kimon Antonakopoulos
• Ali Kavis
• Volkan Cevher

In this work, we propose a universal and adaptive second-order method for minimization of second-order smooth, convex functions. Precisely, our algorithm achieves $O(\sigma / \sqrt{T})$ when the oracle feedback is stochastic with variance $\sigma$, and obtains the improved $O( 1 / T^3)$ convergence with deterministic oracles. Our method achieves this rate interpolation without knowing the nature of the oracle a priori, which was enabled by a parameter-free step-size that is oblivious to the knowledge of smoothness modulus, variance bounds and the diameter of the constrained set. To our knowledge, this is the first universal algorithm that achieves the aforementioned global guarantees within second-order convex optimization literature.

NeurIPS Conference 2022 Conference Paper

Extrapolation and Spectral Bias of Neural Nets with Hadamard Product: a Polynomial Net Study

• Yongtao Wu
• Zhenyu Zhu
• Fanghui Liu
• Grigorios Chrysos
• Volkan Cevher

Neural tangent kernel (NTK) is a powerful tool to analyze training dynamics of neural networks and their generalization bounds. The study on NTK has been devoted to typical neural network architectures, but it is incomplete for neural networks with Hadamard products (NNs-Hp), e. g. , StyleGAN and polynomial neural networks (PNNs). In this work, we derive the finite-width NTK formulation for a special class of NNs-Hp, i. e. , polynomial neural networks. We prove their equivalence to the kernel regression predictor with the associated NTK, which expands the application scope of NTK. Based on our results, we elucidate the separation of PNNs over standard neural networks with respect to extrapolation and spectral bias. Our two key insights are that when compared to standard neural networks, PNNs can fit more complicated functions in the extrapolation regime and admit a slower eigenvalue decay of the respective NTK, leading to a faster learning towards high-frequency functions. Besides, our theoretical results can be extended to other types of NNs-Hp, which expand the scope of our work. Our empirical results validate the separations in broader classes of NNs-Hp, which provide a good justification for a deeper understanding of neural architectures.

NeurIPS Conference 2022 Conference Paper

Generalization Properties of NAS under Activation and Skip Connection Search

• Zhenyu Zhu
• Fanghui Liu
• Grigorios Chrysos
• Volkan Cevher

Neural Architecture Search (NAS) has fostered the automatic discovery of state-of-the-art neural architectures. Despite the progress achieved with NAS, so far there is little attention to theoretical guarantees on NAS. In this work, we study the generalization properties of NAS under a unifying framework enabling (deep) layer skip connection search and activation function search. To this end, we derive the lower (and upper) bounds of the minimum eigenvalue of the Neural Tangent Kernel (NTK) under the (in)finite-width regime using a certain search space including mixed activation functions, fully connected, and residual neural networks. We use the minimum eigenvalue to establish generalization error bounds of NAS in the stochastic gradient descent training. Importantly, we theoretically and experimentally show how the derived results can guide NAS to select the top-performing architectures, even in the case without training, leading to a train-free algorithm based on our theory. Accordingly, our numerical validation shed light on the design of computationally efficient methods for NAS. Our analysis is non-trivial due to the coupling of various architectures and activation functions under the unifying framework and has its own interest in providing the lower bound of the minimum eigenvalue of NTK in deep learning theory.

ICLR Conference 2022 Conference Paper

High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize

• Ali Kavis
• Kfir Yehuda Levy
• Volkan Cevher

In this paper, we propose a new, simplified high probability analysis of AdaGrad for smooth, non-convex problems. More specifically, we focus on a particular accelerated gradient (AGD) template (Lan, 2020), through which we recover the original AdaGrad and its variant with averaging, and prove a convergence rate of $\mathcal O (1/ \sqrt{T})$ with high probability without the knowledge of smoothness and variance. We use a particular version of Freedman's concentration bound for martingale difference sequences (Kakade & Tewari, 2008) which enables us to achieve the best-known dependence of $\log (1 / \delta )$ on the probability margin $\delta$. We present our analysis in a modular way and obtain a complementary $\mathcal O (1 / T)$ convergence rate in the deterministic setting. To the best of our knowledge, this is the first high probability result for AdaGrad with a truly adaptive scheme, i.e., completely oblivious to the knowledge of smoothness and uniform variance bound, which simultaneously has best-known dependence of $\log( 1/ \delta)$. We further prove noise adaptation property of AdaGrad under additional noise assumptions.

NeurIPS Conference 2022 Conference Paper

Identifiability and generalizability from multiple experts in Inverse Reinforcement Learning

• Paul Rolland
• Luca Viano
• Norman Schürhoff
• Boris Nikolov
• Volkan Cevher

While Reinforcement Learning (RL) aims to train an agent from a reward function in a given environment, Inverse Reinforcement Learning (IRL) seeks to recover the reward function from observing an expert's behavior. It is well known that, in general, various reward functions can lead to the same optimal policy, and hence, IRL is ill-defined. However, \cite{cao2021identifiability} showed that, if we observe two or more experts with different discount factors or acting in different environments, the reward function can under certain conditions be identified up to a constant. This work starts by showing an equivalent identifiability statement from multiple experts in tabular MDPs based on a rank condition, which is easily verifiable and is shown to be also necessary. We then extend our result to various different scenarios, i. e. , we characterize reward identifiability in the case where the reward function can be represented as a linear combination of given features, making it more interpretable, or when we have access to approximate transition matrices. Even when the reward is not identifiable, we provide conditions characterizing when data on multiple experts in a given environment allows to generalize and train an optimal agent in a new environment. Our theoretical results on reward identifiability and generalizability are validated in various numerical experiments.

NeurIPS Conference 2022 Conference Paper

No-regret learning in games with noisy feedback: Faster rates and adaptivity via learning rate separation

• Yu-Guan Hsieh
• Kimon Antonakopoulos
• Volkan Cevher
• Panayotis Mertikopoulos

We examine the problem of regret minimization when the learner is involved in a continuous game with other optimizing agents: in this case, if all players follow a no-regret algorithm, it is possible to achieve significantly lower regret relative to fully adversarial environments. We study this problem in the context of variationally stable games (a class of continuous games which includes all convex-concave and monotone games), and when the players only have access to noisy estimates of their individual payoff gradients. If the noise is additive, the game-theoretic and purely adversarial settings enjoy similar regret guarantees; however, if the noise is \emph{multiplicative}, we show that the learners can, in fact, achieve \emph{constant} regret. We achieve this faster rate via an optimistic gradient scheme with \emph{learning rate separation} \textendash\ that is, the method's extrapolation and update steps are tuned to different schedules, depending on the noise profile. Subsequently, to eliminate the need for delicate hyperparameter tuning, we propose a fully adaptive method that smoothly interpolates between worst- and best-case regret guarantees.

NeurIPS Conference 2022 Conference Paper

On the Double Descent of Random Features Models Trained with SGD

• Fanghui Liu
• Johan Suykens
• Volkan Cevher

We study generalization properties of random features (RF) regression in high dimensions optimized by stochastic gradient descent (SGD) in under-/over-parameterized regime. In this work, we derive precise non-asymptotic error bounds of RF regression under both constant and polynomial-decay step-size SGD setting, and observe the double descent phenomenon both theoretically and empirically. Our analysis shows how to cope with multiple randomness sources of initialization, label noise, and data sampling (as well as stochastic gradients) with no closed-form solution, and also goes beyond the commonly-used Gaussian/spherical data assumption. Our theoretical results demonstrate that, with SGD training, RF regression still generalizes well for interpolation learning, and is able to characterize the double descent behavior by the unimodality of variance and monotonic decrease of bias. Besides, we also prove that the constant step-size SGD setting incurs no loss in convergence rate when compared to the exact minimum-norm interpolator, as a theoretical justification of using SGD in practice.

NeurIPS Conference 2022 Conference Paper

Proximal Point Imitation Learning

• Luca Viano
• Angeliki Kamoutsi
• Gergely Neu
• Igor Krawczuk
• Volkan Cevher

This work develops new algorithms with rigorous efficiency guarantees for infinite horizon imitation learning (IL) with linear function approximation without restrictive coherence assumptions. We begin with the minimax formulation of the problem and then outline how to leverage classical tools from optimization, in particular, the proximal-point method (PPM) and dual smoothing, for online and offline IL, respectively. Thanks to PPM, we avoid nested policy evaluation and cost updates for online IL appearing in the prior literature. In particular, we do away with the conventional alternating updates by the optimization of a single convex and smooth objective over both cost and $Q$-functions. When solved inexactly, we relate the optimization errors to the suboptimality of the recovered policy. As an added bonus, by re-interpreting PPM as dual smoothing with the expert policy as a center point, we also obtain an offline IL algorithm enjoying theoretical guarantees in terms of required expert trajectories. Finally, we achieve convincing empirical performance for both linear and neural network function approximation.

NeurIPS Conference 2022 Conference Paper

Robustness in deep learning: The good (width), the bad (depth), and the ugly (initialization)

• Zhenyu Zhu
• Fanghui Liu
• Grigorios Chrysos
• Volkan Cevher

We study the average robustness notion in deep neural networks in (selected) wide and narrow, deep and shallow, as well as lazy and non-lazy training settings. We prove that in the under-parameterized setting, width has a negative effect while it improves robustness in the over-parameterized setting. The effect of depth closely depends on the initialization and the training mode. In particular, when initialized with LeCun initialization, depth helps robustness with the lazy training regime. In contrast, when initialized with Neural Tangent Kernel (NTK) and He-initialization, depth hurts the robustness. Moreover, under the non-lazy training regime, we demonstrate how the width of a two-layer ReLU network benefits robustness. Our theoretical developments improve the results by [Huang et al. NeurIPS21; Wu et al. NeurIPS21] and are consistent with [Bubeck and Sellke NeurIPS21; Bubeck et al. COLT21].

ICML Conference 2022 Conference Paper

Score Matching Enables Causal Discovery of Nonlinear Additive Noise Models

• Paul Rolland
• Volkan Cevher
• Matthäus Kleindessner
• Chris Russell 0001
• Dominik Janzing
• Bernhard Schölkopf
• Francesco Locatello

This paper demonstrates how to recover causal graphs from the score of the data distribution in non-linear additive (Gaussian) noise models. Using score matching algorithms as a building block, we show how to design a new generation of scalable causal discovery methods. To showcase our approach, we also propose a new efficient method for approximating the score’s Jacobian, enabling to recover the causal graph. Empirically, we find that the new algorithm, called SCORE, is competitive with state-of-the-art causal discovery methods while being significantly faster.

NeurIPS Conference 2022 Conference Paper

Sound and Complete Verification of Polynomial Networks

• Elias Abad Rocamora
• Mehmet Fatih Sahin
• Fanghui Liu
• Grigorios Chrysos
• Volkan Cevher

Polynomial Networks (PNs) have demonstrated promising performance on face and image recognition recently. However, robustness of PNs is unclear and thus obtaining certificates becomes imperative for enabling their adoption in real-world applications. Existing verification algorithms on ReLU neural networks (NNs) based on classical branch and bound (BaB) techniques cannot be trivially applied to PN verification. In this work, we devise a new bounding method, equipped with BaB for global convergence guarantees, called Verification of Polynomial Networks or VPN for short. One key insight is that we obtain much tighter bounds than the interval bound propagation (IBP) and DeepT-Fast [Bonaert et al. , 2021] baselines. This enables sound and complete PN verification with empirical validation on MNIST, CIFAR10 and STL10 datasets. We believe our method has its own interest to NN verification. The source code is publicly available at https: //github. com/megaelius/PNVerification.

ICLR Conference 2022 Conference Paper

The Spectral Bias of Polynomial Neural Networks

• Moulik Choraria
• Leello Tadesse Dadi
• Grigorios Chrysos 0002
• Julien Mairal
• Volkan Cevher

Polynomial neural networks (PNNs) have been recently shown to be particularly effective at image generation and face recognition, where high-frequency information is critical. Previous studies have revealed that neural networks demonstrate a $\text{\it{spectral bias}}$ towards low-frequency functions, which yields faster learning of low-frequency components during training. Inspired by such studies, we conduct a spectral analysis of the Neural Tangent Kernel (NTK) of PNNs. We find that the $\Pi$-Net family, i.e., a recently proposed parametrization of PNNs, speeds up the learning of the higher frequencies. We verify the theoretical bias through extensive experiments. We expect our analysis to provide novel insights into designing architectures and learning frameworks by incorporating multiplicative interactions via polynomials.

ICML Conference 2022 Conference Paper

UnderGrad: A Universal Black-Box Optimization Method with Almost Dimension-Free Convergence Rate Guarantees

• Kimon Antonakopoulos
• Dong Quan Vu
• Volkan Cevher
• Kfir Yehuda Levy
• Panayotis Mertikopoulos

Universal methods achieve optimal convergence rate guarantees in convex optimization without any prior knowledge of the problem’s regularity parameters or the attributes of the gradient oracle employed by the method. In this regard, existing state-of-the-art algorithms achieve an $O(1/T^2)$ convergence rate in Lipschitz smooth problems with a perfect gradient oracle, and an $O(1/sqrt{T})$ convergence speed when the underlying problem is non-smooth and/or the gradient oracle is stochastic. On the downside, these methods do not take into account the dependence of these guarantees on the problem’s dimensionality, and this can have a catastrophic impact on a method’s convergence, in both theory and practice. Our paper aims to bridge this gap by providing a scalable universal method - dubbed UnDERGrad - which enjoys an almost dimension-free oracle complexity in problems with a favorable geometry (like the simplex, $\ell_1$-ball or trace-constraints), while retaining the order-optimal dependence on T described above. These "best of both worlds" guarantees are achieved via a primal-dual update scheme inspired by the dual exploration method for variational inequalities.

NeurIPS Conference 2022 Conference Paper

Understanding Deep Neural Function Approximation in Reinforcement Learning via $\epsilon$-Greedy Exploration

• Fanghui Liu
• Luca Viano
• Volkan Cevher

This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $\epsilon$-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e. g. , width and depth) beyond the ``linear'' regime. To be specific, we focus on the value based algorithm with the $\epsilon$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces, respectively, which aims at approximating an $\alpha$-smooth Q-function in a $d$-dimensional feature space. We prove that, with $T$ episodes, scaling the width $m = \widetilde{\mathcal{O}}(T^{\frac{d}{2\alpha + d}})$ and the depth $L=\mathcal{O}(\log T)$ of the neural network for deep RL is sufficient for learning with sublinear regret in Besov spaces. Moreover, for a two layer neural network endowed by the Barron space, scaling the width $\Omega(\sqrt{T})$ is sufficient. To achieve this, the key issue in our analysis is how to estimate the temporal difference error under deep neural function approximation as the $\epsilon$-greedy exploration is not enough to ensure "optimism". Our analysis reformulates the temporal difference error in an $L^2(\mathrm{d}\mu)$-integrable space over a certain averaged measure $\mu$, and transforms it to a generalization problem under the non-iid setting. This might have its own interest in RL theory for better understanding $\epsilon$-greedy exploration in deep RL.

NeurIPS Conference 2021 Conference Paper

A first-order primal-dual method with adaptivity to local smoothness

• Maria-Luiza Vladarean
• Yura Malitsky
• Volkan Cevher

We consider the problem of finding a saddle point for the convex-concave objective $\min_x \max_y f(x) + \langle Ax, y\rangle - g^*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vũ algorithm, which alternates between primal gradient steps and dual proximal steps. The method achieves stepsize adaptivity through a simple rule involving $\|A\|$ and the norm of recently computed gradients of $f$. Under standard assumptions, we prove an $\mathcal{O}(k^{-1})$ ergodic convergence rate. Furthermore, when $f$ is also locally strongly convex and $A$ has full row rank we show that our method converges with a linear rate. Numerical experiments are provided for illustrating the practical performance of the algorithm.

NeurIPS Conference 2021 Conference Paper

Convergence of adaptive algorithms for constrained weakly convex optimization

• Ahmet Alacaoglu
• Yura Malitsky
• Volkan Cevher

We analyze the adaptive first order algorithm AMSGrad, for solving a constrained stochastic optimization problem with a weakly convex objective. We prove the $\mathcal{\tilde O}(t^{-1/2})$ rate of convergence for the squared norm of the gradient of Moreau envelope, which is the standard stationarity measure for this class of problems. It matches the known rates that adaptive algorithms enjoy for the specific case of unconstrained smooth nonconvex stochastic optimization. Our analysis works with mini-batch size of $1$, constant first and second order moment parameters, and possibly unbounded optimization domains. Finally, we illustrate the applications and extensions of our results to specific problems and algorithms.

ICML Conference 2021 Conference Paper

Regret Minimization in Stochastic Non-Convex Learning via a Proximal-Gradient Approach

• Nadav Hallak
• Panayotis Mertikopoulos
• Volkan Cevher

This paper develops a methodology for regret minimization with stochastic first-order oracle feedback in online, constrained, non-smooth, non-convex problems. In this setting, the minimization of external regret is beyond reach for first-order methods, and there are no gradient-based algorithmic frameworks capable of providing a solution. On that account, we propose a conceptual approach that leverages non-convex optimality measures, leading to a suitable generalization of the learner’s local regret. We focus on a local regret measure defined via a proximal-gradient mapping, that also encompasses the original notion proposed by Hazan et al. (2017). To achieve no local regret in this setting, we develop a proximal-gradient method based on stochastic first-order feedback, and a simpler method for when access to a perfect first-order oracle is possible. Both methods are order-optimal (in the min-max sense), and we also establish a bound on the number of proximal-gradient queries these methods require. As an important application of our results, we also obtain a link between online and offline non-convex stochastic optimization manifested as a new proximal-gradient scheme with complexity guarantees matching those obtained via variance reduction techniques.

NeurIPS Conference 2021 Conference Paper

Robust Inverse Reinforcement Learning under Transition Dynamics Mismatch

• Luca Viano
• Yu-Ting Huang
• Parameswaran Kamalaruban
• Adrian Weller
• Volkan Cevher

We study the inverse reinforcement learning (IRL) problem under a transition dynamics mismatch between the expert and the learner. Specifically, we consider the Maximum Causal Entropy (MCE) IRL learner model and provide a tight upper bound on the learner's performance degradation based on the $\ell_1$-distance between the transition dynamics of the expert and the learner. Leveraging insights from the Robust RL literature, we propose a robust MCE IRL algorithm, which is a principled approach to help with this mismatch. Finally, we empirically demonstrate the stable performance of our algorithm compared to the standard MCE IRL algorithm under transition dynamics mismatches in both finite and continuous MDP problems.

NeurIPS Conference 2021 Conference Paper

Sifting through the noise: Universal first-order methods for stochastic variational inequalities

• Kimon Antonakopoulos
• Thomas Pethick
• Ali Kavis
• Panayotis Mertikopoulos
• Volkan Cevher

We examine a flexible algorithmic framework for solving monotone variational inequalities in the presence of randomness and uncertainty. The proposed template encompasses a wide range of popular first-order methods, including dual averaging, dual extrapolation and optimistic gradient algorithms – both adaptive and non-adaptive. Our first result is that the algorithm achieves the optimal rates of convergence for cocoercive problems when the profile of the randomness is known to the optimizer: $\mathcal{O}(1/\sqrt{T})$ for absolute noise profiles, and $\mathcal{O}(1/T)$ for relative ones. Subsequently, we drop all prior knowledge requirements (the absolute/relative variance of the randomness affecting the problem, the operator's cocoercivity constant, etc. ), and we analyze an adaptive instance of the method that gracefully interpolates between the above rates – i. e. it achieves $\mathcal{O}(1/\sqrt{T})$ and $\mathcal{O}(1/T)$ in the absolute and relative cases, respectively. To our knowledge, this is the first universality result of its kind in the literature and, somewhat surprisingly, it shows that an extra-gradient proxy step is not required to achieve optimal rates.

NeurIPS Conference 2021 Conference Paper

STORM+: Fully Adaptive SGD with Recursive Momentum for Nonconvex Optimization

• Kfir Levy
• Ali Kavis
• Volkan Cevher

In this work we investigate stochastic non-convex optimization problems where the objective is an expectation over smooth loss functions, and the goal is to find an approximate stationary point. The most popular approach to handling such problems is variance reduction techniques, which are also known to obtain tight convergence rates, matching the lower bounds in this case. Nevertheless, these techniques require a careful maintenance of anchor points in conjunction with appropriately selected ``mega-batchsizes". This leads to a challenging hyperparameter tuning problem, that weakens their practicality. Recently, [Cutkosky and Orabona, 2019] have shown that one can employ recursive momentum in order to avoid the use of anchor points and large batchsizes, and still obtain the optimal rate for this setting. Yet, their method called $\rm{STORM}$ crucially relies on the knowledge of the smoothness, as well a bound on the gradient norms. In this work we propose $\rm{STORM}^{+}$, a new method that is completely parameter-free, does not require large batch-sizes, and obtains the optimal $O(1/T^{1/3})$ rate for finding an approximate stationary point. Our work builds on the $\rm{STORM}$ algorithm, in conjunction with a novel approach to adaptively set the learning rate and momentum parameters.

NeurIPS Conference 2021 Conference Paper

Subquadratic Overparameterization for Shallow Neural Networks

• ChaeHwan Song
• Ali Ramezani-Kebrya
• Thomas Pethick
• Armin Eftekhari
• Volkan Cevher

Overparameterization refers to the important phenomenon where the width of a neural network is chosen such that learning algorithms can provably attain zero loss in nonconvex training. The existing theory establishes such global convergence using various initialization strategies, training modifications, and width scalings. In particular, the state-of-the-art results require the width to scale quadratically with the number of training data under standard initialization strategies used in practice for best generalization performance. In contrast, the most recent results obtain linear scaling either with requiring initializations that lead to the "lazy-training", or training only a single layer. In this work, we provide an analytical framework that allows us to adopt standard initialization strategies, possibly avoid lazy training, and train all layers simultaneously in basic shallow neural networks while attaining a desirable subquadratic scaling on the network width. We achieve the desiderata via Polyak-Lojasiewicz condition, smoothness, and standard assumptions on data, and use tools from random matrix theory.

NeurIPS Conference 2021 Conference Paper

The Effect of the Intrinsic Dimension on the Generalization of Quadratic Classifiers

• Fabian Latorre
• Leello Tadesse Dadi
• Paul Rolland
• Volkan Cevher

It has been recently observed that neural networks, unlike kernel methods, enjoy a reduced sample complexity when the distribution is isotropic (i. e. , when the covariance matrix is the identity). We find that this sensitivity to the data distribution is not exclusive to neural networks, and the same phenomenon can be observed on the class of quadratic classifiers (i. e. , the sign of a quadratic polynomial) with a nuclear-norm constraint. We demonstrate this by deriving an upper bound on the Rademacher Complexity that depends on two key quantities: (i) the intrinsic dimension, which is a measure of isotropy, and (ii) the largest eigenvalue of the second moment (covariance) matrix of the distribution. Our result improves the dependence on the dimension over the best previously known bound and precisely quantifies the relation between the sample complexity and the level of isotropy of the distribution.

ICML Conference 2021 Conference Paper

The Limits of Min-Max Optimization Algorithms: Convergence to Spurious Non-Critical Sets

• Ya-Ping Hsieh
• Panayotis Mertikopoulos
• Volkan Cevher

Compared to minimization, the min-max optimization in machine learning applications is considerably more convoluted because of the existence of cycles and similar phenomena. Such oscillatory behaviors are well-understood in the convex-concave regime, and many algorithms are known to overcome them. In this paper, we go beyond this basic setting and characterize the convergence properties of many popular methods in solving non-convex/non-concave problems. In particular, we show that a wide class of state-of-the-art schemes and heuristics may converge with arbitrarily high probability to attractors that are in no way min-max optimal or even stationary. Our work thus points out a potential pitfall among many existing theoretical frameworks, and we corroborate our theoretical claims by explicitly showcasing spurious attractors in simple two-dimensional problems.

ICML Conference 2020 Conference Paper

A new regret analysis for Adam-type algorithms

• Ahmet Alacaoglu
• Yura Malitsky
• Panayotis Mertikopoulos
• Volkan Cevher

In this paper, we focus on a theory-practice gap for Adam and its variants (AMSGrad, AdamNC, etc.). In practice, these algorithms are used with a constant first-order moment parameter $\beta_{1}$ (typically between $0. 9$ and $0. 99$). In theory, regret guarantees for online convex optimization require a rapidly decaying $\beta_{1}\to0$ schedule. We show that this is an artifact of the standard analysis, and we propose a novel framework that allows us to derive optimal, data-dependent regret bounds with a constant $\beta_{1}$, without further assumptions. We also demonstrate the flexibility of our analysis on a wide range of different algorithms and settings.

ICML Conference 2020 Conference Paper

Conditional gradient methods for stochastically constrained convex minimization

• Maria-Luiza Vladarean
• Ahmet Alacaoglu
• Ya-Ping Hsieh
• Volkan Cevher

We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of combinatorial problems, which involve a number of constraints that is polynomial in the problem dimension. The most important feature of our framework is that only a subset of the constraints is processed at each iteration, thus gaining a computational advantage over prior works that require full passes. Our algorithms rely on variance reduction and smoothing used in conjunction with conditional gradient steps, and are accompanied by rigorous convergence guarantees. Preliminary numerical experiments are provided for illustrating the practical performance of the methods.

JMLR Journal 2020 Journal Article

Convergences of Regularized Algorithms and Stochastic Gradient Methods with Random Projections

• Junhong Lin
• Volkan Cevher

We study the least-squares regression problem over a Hilbert space, covering nonparametric regression over a reproducing kernel Hilbert space as a special case. We first investigate regularized algorithms adapted to a projection operator on a closed subspace of the Hilbert space. We prove convergence results with respect to variants of norms, under a capacity assumption on the hypothesis space and a regularity condition on the target function. As a result, we obtain optimal rates for regularized algorithms with randomized sketches, provided that the sketch dimension is proportional to the effective dimension up to a logarithmic factor. As a byproduct, we obtain similar results for Nystr\"{o}m regularized algorithms. Our results provide optimal, distribution-dependent rates that do not have any saturation effect for sketched/Nystr\"{o}m regularized algorithms, considering both the attainable and non-attainable cases, in the well-conditioned regimes. We then study stochastic gradient methods with projection over the subspace, allowing multi-pass over the data and minibatches, and we derive similar optimal statistical convergence results. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

ICML Conference 2020 Conference Paper

Double-Loop Unadjusted Langevin Algorithm

• Paul Rolland
• Armin Eftekhari
• Ali Kavis
• Volkan Cevher

A well-known first-order method for sampling from log-concave probability distributions is the Unadjusted Langevin Algorithm (ULA). This work proposes a new annealing step-size schedule for ULA, which allows to prove new convergence guarantees for sampling from a smooth log-concave distribution, which are not covered by existing state-of-the-art convergence guarantees. To establish this result, we derive a new theoretical bound that relates the Wasserstein distance to total variation distance between any two log-concave distributions that complements the reach of Talagrand $T_2$ inequality. Moreover, applying this new step size schedule to an existing constrained sampling algorithm, we show state-of-the-art convergence rates for sampling from a constrained log-concave distribution, as well as improved dimension dependence.

ICML Conference 2020 Conference Paper

Efficient Proximal Mapping of the 1-path-norm of Shallow Networks

• Fabian Latorre
• Paul Rolland
• Nadav Hallak
• Volkan Cevher

We demonstrate two new important properties of the 1-path-norm of shallow neural networks. First, despite its non-smoothness and non-convexity it allows a closed form proximal operator which can be efficiently computed, allowing the use of stochastic proximal-gradient-type methods for regularized empirical risk minimization. Second, when the activation functions is differentiable, it provides an upper bound on the Lipschitz constant of the network. Such bound is tighter than the trivial layer-wise product of Lipschitz constants, motivating its use for training networks robust to adversarial perturbations. In practical experiments we illustrate the advantages of using the proximal mapping and we compare the robustness-accuracy trade-off induced by the 1-path-norm, L1-norm and layer-wise constraints on the Lipschitz constant (Parseval networks).

ICLR Conference 2020 Conference Paper

Lipschitz constant estimation of Neural Networks via sparse polynomial optimization

• Fabian Latorre
• Paul Rolland
• Volkan Cevher

We introduce LiPopt, a polynomial optimization framework for computing increasingly tighter upper bound on the Lipschitz constant of neural networks. The underlying optimization problems boil down to either linear (LP) or semidefinite (SDP) programming. We show how to use the sparse connectivity of a network, to significantly reduce the complexity of computation. This is specially useful for convolutional as well as pruned neural networks. We conduct experiments on networks with random weights as well as networks trained on MNIST, showing that in the particular case of the $\ell_\infty$-Lipschitz constant, our approach yields superior estimates as compared to other baselines available in the literature.

NeurIPS Conference 2020 Conference Paper

On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems

• Panayotis Mertikopoulos
• Nadav Hallak
• Ali Kavis
• Volkan Cevher

In this paper, we analyze the trajectories of stochastic gradient descent (SGD) with the aim of understanding their convergence properties in non-convex problems. We first show that the sequence of iterates generated by SGD remains bounded and converges with probability $1$ under a very broad range of step-size schedules. Subsequently, we prove that the algorithm's rate of convergence to local minimizers with a positive-definite Hessian is $O(1/n^p)$ if the method is run with a $Θ(1/n^p)$ step-size. This provides an important guideline for tuning the algorithm's step-size as it suggests that a cool-down phase with a vanishing step-size could lead to significant performance gains; we demonstrate this heuristic using ResNet architectures on CIFAR. Finally, going beyond existing positive probability guarantees, we show that SGD avoids strict saddle points/manifolds with probability $1$ for the entire spectrum of step-size policies considered.

JMLR Journal 2020 Journal Article

Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithms

• Junhong Lin
• Volkan Cevher

We study generalization properties of distributed algorithms in the setting of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We first investigate distributed stochastic gradient methods (SGM), with mini-batches and multi-passes over the data. We show that optimal generalization error bounds (up to logarithmic factor) can be retained for distributed SGM provided that the partition level is not too large. We then extend our results to spectral algorithms (SA), including kernel ridge regression (KRR), kernel principal component regression, and gradient methods. Our results show that distributed SGM has a smaller theoretical computational complexity, compared with distributed KRR and classic SGM. Moreover, even for a general non-distributed SA, they provide optimal, capacity-dependent convergence rates, for the case that the regression function may not be in the RKHS in the well-conditioned regimes. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

ICML Conference 2020 Conference Paper

Random extrapolation for primal-dual coordinate descent

• Ahmet Alacaoglu
• Olivier Fercoq
• Volkan Cevher

We introduce a randomly extrapolated primal-dual coordinate descent method that adapts to sparsity of the data matrix and the favorable structures of the objective function. Our method updates only a subset of primal and dual variables with sparse data, and it uses large step sizes with dense data, retaining the benefits of the specific methods designed for each case. In addition to adapting to sparsity, our method attains fast convergence guarantees in favorable cases \emph{without any modifications}. In particular, we prove linear convergence under metric subregularity, which applies to strongly convex-strongly concave problems and piecewise linear quadratic functions. We show almost sure convergence of the sequence and optimal sublinear convergence rates for the primal-dual gap and objective values, in the general convex-concave case. Numerical evidence demonstrates the state-of-the-art empirical performance of our method in sparse and dense settings, matching and improving the existing methods.

NeurIPS Conference 2020 Conference Paper

Robust Reinforcement Learning via Adversarial training with Langevin Dynamics

• Parameswaran Kamalaruban
• Yu-Ting Huang
• Ya-Ping Hsieh
• Paul Rolland
• Cheng Shi
• Volkan Cevher

We introduce a \emph{sampling} perspective to tackle the challenging task of training robust Reinforcement Learning (RL) agents. Leveraging the powerful Stochastic Gradient Langevin Dynamics, we present a novel, scalable two-player RL algorithm, which is a sampling variant of the two-player policy gradient method. Our algorithm consistently outperforms existing baselines, in terms of generalization across different training and testing conditions, on several MuJoCo environments. Our experiments also show that, even for objective functions that entirely ignore potential environmental shifts, our sampling approach remains highly robust in comparison to standard RL algorithms.

ICML Conference 2019 Conference Paper

A Conditional-Gradient-Based Augmented Lagrangian Framework

• Alp Yurtsever
• Olivier Fercoq
• Volkan Cevher

This paper considers a generic convex minimization template with affine constraints over a compact domain, which covers key semidefinite programming applications. The existing conditional gradient methods either do not apply to our template or are too slow in practice. To this end, we propose a new conditional gradient method, based on a unified treatment of smoothing and augmented Lagrangian frameworks. The proposed method maintains favorable properties of the classical conditional gradient method, such as cheap linear minimization oracle calls and sparse representation of the decision variable. We prove $O(1/\sqrt{k})$ convergence rate for our method in the objective residual and the feasibility gap. This rate is essentially the same as the state of the art CG-type methods for our problem template, but the proposed method is arguably superior in practice compared to existing methods in various applications.

ICML Conference 2019 Conference Paper

Almost surely constrained convex optimization

• Olivier Fercoq
• Ahmet Alacaoglu
• Ion Necoara
• Volkan Cevher

We propose a stochastic gradient framework for solving stochastic composite convex optimization problems with (possibly) infinite number of linear inclusion constraints that need to be satisfied almost surely. We use smoothing and homotopy techniques to handle constraints without the need for matrix-valued projections. We show for our stochastic gradient algorithm $\mathcal{O}(\log(k)/\sqrt{k})$ convergence rate for general convex objectives and $\mathcal{O}(\log(k)/k)$ convergence rate for restricted strongly convex objectives. These rates are known to be optimal up to logarithmic factor, even without constraints. We conduct numerical experiments on basis pursuit, hard margin support vector machines and portfolio optimization problems and show that our algorithm achieves state-of-the-art practical performance.

NeurIPS Conference 2019 Conference Paper

An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints

• Mehmet Fatih Sahin
• Armin Eftekhari
• Ahmet Alacaoglu
• Fabian Latorre
• Volkan Cevher

We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. We characterize the total computational complexity of our method subject to a verifiable geometric condition, which is closely related to the Polyak-Lojasiewicz and Mangasarian-Fromowitz conditions. In particular, when a first-order solver is used for the inner iterates, we prove that iALM finds a first-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^3)$ calls to the first-order oracle. {If, in addition, the problem is smooth and} a second-order solver is used for the inner iterates, iALM finds a second-order stationary point with $\tilde{\mathcal{O}}(1/\epsilon^5)$ calls to the second-order oracle. These complexity results match the known theoretical results in the literature. We also provide strong numerical evidence on large-scale machine learning problems, including the Burer-Monteiro factorization of semidefinite programs, and a novel nonconvex relaxation of the standard basis pursuit template. For these examples, we also show how to verify our geometric condition.

ICML Conference 2019 Conference Paper

Conditional Gradient Methods via Stochastic Path-Integrated Differential Estimator

• Alp Yurtsever
• Suvrit Sra
• Volkan Cevher

We propose a class of variance-reduced stochastic conditional gradient methods. By adopting the recent stochastic path-integrated differential estimator technique (SPIDER) of Fang et. al. (2018) for the classical Frank-Wolfe (FW) method, we introduce SPIDER-FW for finite-sum minimization as well as the more general expectation minimization problems. SPIDER-FW enjoys superior complexity guarantees in the non-convex setting, while matching the best known FW variants in the convex case. We also extend our framework a la conditional gradient sliding (CGS) of Lan & Zhou. (2016), and propose SPIDER-CGS.

ICML Conference 2019 Conference Paper

Efficient learning of smooth probability functions from Bernoulli tests with guarantees

• Paul Rolland
• Ali Kavis
• Alexander Immer
• Adish Singla
• Volkan Cevher

We study the fundamental problem of learning an unknown, smooth probability function via point-wise Bernoulli tests. We provide a scalable algorithm for efficiently solving this problem with rigorous guarantees. In particular, we prove the convergence rate of our posterior update rule to the true probability function in L2-norm. Moreover, we allow the Bernoulli tests to depend on contextual features, and provide a modified inference engine with provable guarantees for this novel setting. Numerical results show that the empirical convergence rates match the theory, and illustrate the superiority of our approach in handling contextual features over the state-of-the-art.

NeurIPS Conference 2019 Conference Paper

Fast and Provable ADMM for Learning with Generative Priors

• Fabian Latorre
• Armin Eftekhari
• Volkan Cevher

In this work, we propose a (linearized) Alternating Direction Method-of-Multipliers (ADMM) algorithm for minimizing a convex function subject to a nonconvex constraint. We focus on the special case where such constraint arises from the specification that a variable should lie in the range of a neural network. This is motivated by recent successful applications of Generative Adversarial Networks (GANs) in tasks like compressive sensing, denoising and robustness against adversarial examples. The derived rates for our algorithm are characterized in terms of certain geometric properties of the generator network, which we show hold for feedforward architectures, under mild assumptions. Unlike gradient descent (GD), it can efficiently handle non-smooth objectives as well as exploit efficient partial minimization procedures, thus being faster in many practical scenarios.

ICML Conference 2019 Conference Paper

Finding Mixed Nash Equilibria of Generative Adversarial Networks

• Ya-Ping Hsieh
• Chen Liu 0027
• Volkan Cevher

Generative adversarial networks (GANs) are known to achieve the state-of-the-art performance on various generative tasks, but these results come at the expense of a notoriously difficult training phase. Current training strategies typically draw a connection to optimization theory, whose scope is restricted to local convergence due to the presence of non-convexity. In this work, we tackle the training of GANs by rethinking the problem formulation from the mixed Nash Equilibria (NE) perspective. Via a classical lifting trick, we show that essentially all existing GAN objectives can be relaxed into their mixed strategy forms, whose global optima can be solved via sampling, in contrast to the exclusive use of optimization framework in previous work. We further propose a mean-approximation sampling scheme, which allows to systematically exploit methods for bi-affine games to delineate novel, practical training algorithms of GANs. Finally, we provide experimental evidence that our approach yields comparable or superior results to contemporary training algorithms, and outperforms classical methods such as SGD, Adam, and RMSProp.

IJCAI Conference 2019 Conference Paper

Interactive Teaching Algorithms for Inverse Reinforcement Learning

• Parameswaran Kamalaruban
• Rati Devidze
• Volkan Cevher
• Adish Singla

We study the problem of inverse reinforcement learning (IRL) with the added twist that the learner is assisted by a helpful teacher. More formally, we tackle the following algorithmic question: How could a teacher provide an informative sequence of demonstrations to an IRL learner to speed up the learning process? We present an interactive teaching framework where a teacher adaptively chooses the next demonstration based on learner's current policy. In particular, we design teaching algorithms for two concrete settings: an omniscient setting where a teacher has full knowledge about the learner's dynamics and a blackbox setting where the teacher has minimal knowledge. Then, we study a sequential variant of the popular MCE-IRL learner and prove convergence guarantees of our teaching algorithm in the omniscient setting. Extensive experiments with a car driving simulator environment show that the learning progress can be speeded up drastically as compared to an uninformative teacher.

AAAI Conference 2019 Conference Paper

Iterative Classroom Teaching

• Teresa Yeo
• Parameswaran Kamalaruban
• Adish Singla
• Arpit Merchant
• Thibault Asselborn
• Louis Faucon
• Pierre Dillenbourg
• Volkan Cevher

We consider the machine teaching problem in a classroom-like setting wherein the teacher has to deliver the same examples to a diverse group of students. Their diversity stems from differences in their initial internal states as well as their learning rates. We prove that a teacher with full knowledge about the learning dynamics of the students can teach a target concept to the entire classroom using O min {d, N} log 1 examples, where d is the ambient dimension of the problem, N is the number of learners, and is the accuracy parameter. We show the robustness of our teaching strategy when the teacher has limited knowledge of the learners’ internal dynamics as provided by a noisy oracle. Further, we study the trade-off between the learners’ workload and the teacher’s cost in teaching the target concept. Our experiments validate our theoretical results and suggest that appropriately partitioning the classroom into homogenous groups provides a balance between these two objectives.

ICML Conference 2019 Conference Paper

On Certifying Non-Uniform Bounds against Adversarial Attacks

• Chen Liu 0027
• Ryota Tomioka
• Volkan Cevher

This work studies the robustness certification problem of neural network models, which aims to find certified adversary-free regions as large as possible around data points. In contrast to the existing approaches that seek regions bounded uniformly along all input features, we consider non-uniform bounds and use it to study the decision boundary of neural network models. We formulate our target as an optimization problem with nonlinear constraints. Then, a framework applicable for general feedforward neural networks is proposed to bound the output logits so that the relaxed problem can be solved by the augmented Lagrangian method. Our experiments show the non-uniform bounds have larger volumes than uniform ones. Compared with normal models, the robust models have even larger non-uniform bounds and better interpretability. Further, the geometric similarity of the non-uniform bounds gives a quantitative, data-agnostic metric of input features’ robustness.

NeurIPS Conference 2019 Conference Paper

Stochastic Frank-Wolfe for Composite Convex Minimization

• Francesco Locatello
• Alp Yurtsever
• Olivier Fercoq
• Volkan Cevher

A broad class of convex optimization problems can be formulated as a semidefinite program (SDP), minimization of a convex function over the positive-semidefinite cone subject to some affine constraints. The majority of classical SDP solvers are designed for the deterministic setting where problem data is readily available. In this setting, generalized conditional gradient methods (aka Frank-Wolfe-type methods) provide scalable solutions by leveraging the so-called linear minimization oracle instead of the projection onto the semidefinite cone. Most problems in machine learning and modern engineering applications, however, contain some degree of stochasticity. In this work, we propose the first conditional-gradient-type method for solving stochastic optimization problems under affine constraints. Our method guarantees O(k^{-1/3}) convergence rate in expectation on the objective residual and O(k^{-5/12}) on the feasibility gap.

NeurIPS Conference 2019 Conference Paper

UniXGrad: A Universal, Adaptive Algorithm with Optimal Guarantees for Constrained Optimization

• Ali Kavis
• Kfir Y. Levy
• Francis Bach
• Volkan Cevher

We propose a novel adaptive, accelerated algorithm for the stochastic constrained convex optimization setting. Our method, which is inspired by the Mirror-Prox method, \emph{simultaneously} achieves the optimal rates for smooth/non-smooth problems with either deterministic/stochastic first-order oracles. This is done without any prior knowledge of the smoothness nor the noise properties of the problem. To the best of our knowledge, this is the first adaptive, unified algorithm that achieves the optimal rates in the constrained setting. We demonstrate the practical performance of our framework through extensive numerical experiments.

ICML Conference 2018 Conference Paper

A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming

• Alp Yurtsever
• Olivier Fercoq
• Francesco Locatello
• Volkan Cevher

We propose a conditional gradient framework for a composite convex minimization template with broad applications. Our approach combines smoothing and homotopy techniques under the CGM framework, and provably achieves the optimal convergence rate. We demonstrate that the same rate holds if the linear subproblems are solved approximately with additive or multiplicative error. In contrast with the relevant work, we are able to characterize the convergence when the non-smooth term is an indicator function. Specific applications of our framework include the non-smooth minimization, semidefinite programming, and minimization with linear inclusion constraints over a compact domain. Numerical evidence demonstrates the benefits of our framework.

NeurIPS Conference 2018 Conference Paper

Adversarially Robust Optimization with Gaussian Processes

• Ilija Bogunovic
• Jonathan Scarlett
• Stefanie Jegelka
• Volkan Cevher

In this paper, we consider the problem of Gaussian process (GP) optimization with an added robustness requirement: The returned point may be perturbed by an adversary, and we require the function value to remain as high as possible even after this perturbation. This problem is motivated by settings in which the underlying functions during optimization and implementation stages are different, or when one is interested in finding an entire region of good inputs rather than only a single point. We show that standard GP optimization algorithms do not exhibit the desired robustness properties, and provide a novel confidence-bound based algorithm StableOpt for this purpose. We rigorously establish the required number of samples for StableOpt to find a near-optimal point, and we complement this guarantee with an algorithm-independent lower bound. We experimentally demonstrate several potential applications of interest using real-world data sets, and we show that StableOpt consistently succeeds in finding a stable maximizer where several baseline methods fail.

ICML Conference 2018 Conference Paper

Let's be Honest: An Optimal No-Regret Framework for Zero-Sum Games

• Ehsan Asadi Kangarshahi
• Ya-Ping Hsieh
• Mehmet Fatih Sahin
• Volkan Cevher

We revisit the problem of solving two-player zero-sum games in the decentralized setting. We propose a simple algorithmic framework that simultaneously achieves the best rates for honest regret as well as adversarial regret, and in addition resolves the open problem of removing the logarithmic terms in convergence to the value of the game. We achieve this goal in three steps. First, we provide a novel analysis of the optimistic mirror descent (OMD), showing that it can be modified to guarantee fast convergence for both honest regret and value of the game, when the players are playing collaboratively. Second, we propose a new algorithm, dubbed as robust optimistic mirror descent (ROMD), which attains optimal adversarial regret without knowing the time horizon beforehand. Finally, we propose a simple signaling scheme, which enables us to bridge OMD and ROMD to achieve the best of both worlds. Numerical examples are presented to support our theoretical claims and show that our non-adaptive ROMD algorithm can be competitive to OMD with adaptive step-size selection.

NeurIPS Conference 2018 Conference Paper

Mirrored Langevin Dynamics

• Ya-Ping Hsieh
• Ali Kavis
• Paul Rolland
• Volkan Cevher

We consider the problem of sampling from constrained distributions, which has posed significant challenges to both non-asymptotic analysis and algorithmic design. We propose a unified framework, which is inspired by the classical mirror descent, to derive novel first-order sampling schemes. We prove that, for a general target distribution with strongly convex potential, our framework implies the existence of a first-order algorithm achieving O~(\epsilon^{-2}d) convergence, suggesting that the state-of-the-art O~(\epsilon^{-6}d^5) can be vastly improved. With the important Latent Dirichlet Allocation (LDA) application in mind, we specialize our algorithm to sample from Dirichlet posteriors, and derive the first non-asymptotic O~(\epsilon^{-2}d^2) rate for first-order sampling. We further extend our framework to the mini-batch setting and prove convergence rates when only stochastic gradients are available. Finally, we report promising experimental results for LDA on real datasets.

NeurIPS Conference 2018 Conference Paper

Online Adaptive Methods, Universality and Acceleration

• Kfir Y. Levy
• Alp Yurtsever
• Volkan Cevher

We present a novel method for convex unconstrained optimization that, without any modifications ensures: (1) accelerated convergence rate for smooth objectives, (2) standard convergence rate in the general (non-smooth) setting, and (3) standard convergence rate in the stochastic optimization setting. To the best of our knowledge, this is the first method that simultaneously applies to all of the above settings. At the heart of our method is an adaptive learning rate rule that employs importance weights, in the spirit of adaptive online learning algorithms [duchi2011adaptive, levy2017online], combined with an update that linearly couples two sequences, in the spirit of [AllenOrecchia2017]. An empirical examination of our method demonstrates its applicability to the above mentioned scenarios and corroborates our theoretical findings.

ICML Conference 2018 Conference Paper

Optimal Distributed Learning with Multi-pass Stochastic Gradient Methods

• Junhong Lin
• Volkan Cevher

We study generalization properties of distributed algorithms in the setting of nonparametric regression over a reproducing kernel Hilbert space (RKHS). We investigate distributed stochastic gradient methods (SGM), with mini-batches and multi-passes over the data. We show that optimal generalization error bounds can be retained for distributed SGM provided that the partition level is not too large. Our results are superior to the state-of-the-art theory, covering the cases that the regression function may not be in the hypothesis spaces. Particularly, our results show that distributed SGM has a smaller theoretical computational complexity, compared with distributed kernel ridge regression (KRR) and classic SGM.

ICML Conference 2018 Conference Paper

Optimal Rates of Sketched-regularized Algorithms for Least-Squares Regression over Hilbert Spaces

• Junhong Lin
• Volkan Cevher

We investigate regularized algorithms combining with projection for least-squares regression problem over a Hilbert space, covering nonparametric regression over a reproducing kernel Hilbert space. We prove convergence results with respect to variants of norms, under a capacity assumption on the hypothesis space and a regularity condition on the target function. As a result, we obtain optimal rates for regularized algorithms with randomized sketches, provided that the sketch dimension is proportional to the effective dimension up to a logarithmic factor. As a byproduct, we obtain similar results for Nyström regularized algorithms. Our results provide optimal, distribution-dependent rates for sketched/Nyström regularized algorithms, considering both the attainable and non-attainable cases.

STOC Conference 2017 Conference Paper

An adaptive sublinear-time block sparse fourier transform

• Volkan Cevher
• Michael Kapralov
• Jonathan Scarlett
• Amir Zandieh

NeurIPS Conference 2017 Conference Paper

Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data

• Joel Tropp
• Alp Yurtsever
• Madeleine Udell
• Volkan Cevher

Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a sequence of linear updates. Because of storage limitations, it may only be possible to retain a sketch of the psd matrix. This paper develops a new algorithm for fixed-rank psd approximation from a sketch. The approach combines the Nyström approximation with a novel mechanism for rank truncation. Theoretical analysis establishes that the proposed method can achieve any prescribed relative error in the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that the proposed method dominates alternative techniques for fixed-rank psd matrix approximation across a wide range of examples.

NeurIPS Conference 2017 Conference Paper

Phase Transitions in the Pooled Data Problem

• Jonathan Scarlett
• Volkan Cevher

In this paper, we study the {\em pooled data} problem of identifying the labels associated with a large collection of items, based on a sequence of pooled tests revealing the counts of each label within the pool. In the noiseless setting, we identify an exact asymptotic threshold on the required number of tests with optimal decoding, and prove a {\em phase transition} between complete success and complete failure. In addition, we present a novel {\em noisy} variation of the problem, and provide an information-theoretic framework for characterizing the required number of tests for general random noise models. Our results reveal that noise can make the problem considerably more difficult, with strict increases in the scaling laws even at low noise levels. Finally, we demonstrate similar behavior in an {\em approximate recovery} setting, where a given number of errors is allowed in the decoded labels.

ICML Conference 2017 Conference Paper

Robust Submodular Maximization: A Non-Uniform Partitioning Approach

• Ilija Bogunovic
• Slobodan Mitrovic
• Jonathan Scarlett
• Volkan Cevher

We study the problem of maximizing a monotone submodular function subject to a cardinality constraint $k$, with the added twist that a number of items $\tau$ from the returned set may be removed. We focus on the worst-case setting considered by Orlin et al. \ (2016), in which a constant-factor approximation guarantee was given for $\tau = o(\sqrt{k})$. In this paper, we solve a key open problem raised therein, presenting a new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $\tau = o(k)$. Our algorithm constructs partitions consisting of buckets with exponentially increasing sizes, and applies standard submodular optimization subroutines on the buckets in order to construct the robust solution. We numerically demonstrate the performance of PRo in data summarization and influence maximization, demonstrating gains over both the greedy algorithm and the algorithm of Orlin et al. \ (2016).

NeurIPS Conference 2017 Conference Paper

Smooth Primal-Dual Coordinate Descent Algorithms for Nonsmooth Convex Optimization

• Ahmet Alacaoglu
• Quoc Tran Dinh
• Olivier Fercoq
• Volkan Cevher

We propose a new randomized coordinate descent method for a convex optimization template with broad applications. Our analysis relies on a novel combination of four ideas applied to the primal-dual gap function: smoothing, acceleration, homotopy, and coordinate descent with non-uniform sampling. As a result, our method features the first convergence rate guarantees among the coordinate descent methods, that are the best-known under a variety of common structure assumptions on the template. We provide numerical evidence to support the theoretical results with a comparison to state-of-the-art algorithms.

NeurIPS Conference 2017 Conference Paper

Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach

• Slobodan Mitrovic
• Ilija Bogunovic
• Ashkan Norouzi-Fard
• Jakub Tarnawski
• Volkan Cevher

We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm’s memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance.

NeurIPS Conference 2016 Conference Paper

An Efficient Streaming Algorithm for the Submodular Cover Problem

• Ashkan Norouzi-Fard
• Abbas Bazzi
• Ilija Bogunovic
• Marwa El Halabi
• Ya-Ping Hsieh
• Volkan Cevher

We initiate the study of the classical Submodular Cover (SC) problem in the data streaming model which we refer to as the Streaming Submodular Cover (SSC). We show that any single pass streaming algorithm using sublinear memory in the size of the stream will fail to provide any non-trivial approximation guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only seek to find a partial cover. We design the first Efficient bicriteria Submodular Cover Streaming (ESC-Streaming) algorithm for this problem, and provide theoretical guarantees for its performance supported by numerical evidence. Our algorithm finds solutions that are competitive with the near-optimal offline greedy algorithm despite requiring only a single pass over the data stream. In our numerical experiments, we evaluate the performance of ESC-Streaming on active set selection and large-scale graph cover problems.

SODA Conference 2016 Conference Paper

Phase Transitions in Group Testing

• Jonathan Scarlett
• Volkan Cevher

NeurIPS Conference 2016 Conference Paper

Stochastic Three-Composite Convex Minimization

• Alp Yurtsever
• Bang Cong Vu
• Volkan Cevher

We propose a stochastic optimization method for the minimization of the sum of three convex functions, one of which has Lipschitz continuous gradient as well as restricted strong convexity. Our approach is most suitable in the setting where it is computationally advantageous to process smooth term in the decomposition with its stochastic gradient estimate and the other two functions separately with their proximal operators, such as doubly regularized empirical risk minimization problems. We prove the convergence characterization of the proposed algorithm in expectation under the standard assumptions for the stochastic gradient estimate of the smooth term. Our method operates in the primal space and can be considered as a stochastic extension of the three-operator splitting method. Finally, numerical evidence supports the effectiveness of our method in real-world problems.

NeurIPS Conference 2016 Conference Paper

Truncated Variance Reduction: A Unified Approach to Bayesian Optimization and Level-Set Estimation

• Ilija Bogunovic
• Jonathan Scarlett
• Andreas Krause
• Volkan Cevher

We present a new algorithm, truncated variance reduction (TruVaR), that treats Bayesian optimization (BO) and level-set estimation (LSE) with Gaussian processes in a unified fashion. The algorithm greedily shrinks a sum of truncated variances within a set of potential maximizers (BO) or unclassified points (LSE), which is updated based on confidence bounds. TruVaR is effective in several important settings that are typically non-trivial to incorporate into myopic algorithms, including pointwise costs and heteroscedastic noise. We provide a general theoretical guarantee for TruVaR covering these aspects, and use it to recover and strengthen existing results on BO and LSE. Moreover, we provide a new result for a setting where one can select from a number of noise levels having associated costs. We demonstrate the effectiveness of the algorithm on both synthetic and real-world data sets.

NeurIPS Conference 2015 Conference Paper

A Universal Primal-Dual Convex Optimization Framework

• Alp Yurtsever
• Quoc Tran Dinh
• Volkan Cevher

We propose a new primal-dual algorithmic framework for a prototypical constrained convex optimization template. The algorithmic instances of our framework are universal since they can automatically adapt to the unknown Holder continuity degree and constant within the dual formulation. They are also guaranteed to have optimal convergence rates in the objective residual and the feasibility gap for each Holder smoothness degree. In contrast to existing primal-dual algorithms, our framework avoids the proximity operator of the objective function. We instead leverage computationally cheaper, Fenchel-type operators, which are the main workhorses of the generalized conditional gradient (GCG)-type methods. In contrast to the GCG-type methods, our framework does not require the objective function to be differentiable, and can also process additional general linear inclusion constraints, while guarantees the convergence rate on the primal problem.

JMLR Journal 2015 Journal Article

Composite Self-Concordant Minimization

• Quoc Tran-Dinh
• Anastasios Kyrillidis
• Volkan Cevher

We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data. [abs] [ pdf ][ bib ] &copy JMLR 2015. ( edit, beta )

NeurIPS Conference 2015 Conference Paper

Preconditioned Spectral Descent for Deep Learning

• David Carlson
• Edo Collins
• Ya-Ping Hsieh
• Lawrence Carin
• Volkan Cevher

Deep learning presents notorious computational challenges. These challenges include, but are not limited to, the non-convexity of learning objectives and estimating the quantities needed for optimization algorithms, such as gradients. While we do not address the non-convexity, we present an optimization solution that ex- ploits the so far unused “geometry” in the objective function in order to best make use of the estimated gradients. Previous work attempted similar goals with preconditioned methods in the Euclidean space, such as L-BFGS, RMSprop, and ADA-grad. In stark contrast, our approach combines a non-Euclidean gradient method with preconditioning. We provide evidence that this combination more accurately captures the geometry of the objective function compared to prior work. We theoretically formalize our arguments and derive novel preconditioned non-Euclidean algorithms. The results are promising in both computational time and quality when applied to Restricted Boltzmann Machines, Feedforward Neural Nets, and Convolutional Neural Nets.

UAI Conference 2014 Conference Paper

A variational approach to stable principal component pursuit

• Aleksandr Y. Aravkin
• Stephen Becker
• Volkan Cevher
• Peder A. Olsen

We introduce a new convex formulation for stable principal component pursuit (SPCP) to decompose noisy signals into low-rank and sparse representations. For numerical solutions of our SPCP formulation, we first develop a convex variational framework and then accelerate it with quasi-Newton methods. We show, via synthetic and real data experiments, that our approach offers advantages over the classical SPCP formulations in scalability and practical parameter selection.

NeurIPS Conference 2014 Conference Paper

Constrained convex minimization via model-based excessive gap

• Quoc Tran-Dinh
• Volkan Cevher

We introduce a model-based excessive gap technique to analyze first-order primal- dual methods for constrained convex minimization. As a result, we construct first- order primal-dual methods with optimal convergence rates on the primal objec- tive residual and the primal feasibility gap of their iterates separately. Through a dual smoothing and prox-center selection strategy, our framework subsumes the augmented Lagrangian, alternating direction, and dual fast-gradient methods as special cases, where our rates apply.

SODA Conference 2014 Conference Paper

Model-based Sketching and Recovery with Expanders

• Bubacarr Bah
• Luca Baldassarre
• Volkan Cevher

AAAI Conference 2014 Conference Paper

Scalable Sparse Covariance Estimation via Self-Concordance

• Anastasios Kyrillidis
• Rabeeh Karimi Mahabadi
• Quoc Tran Dinh
• Volkan Cevher

We consider the class of convex minimization problems, composed of a self-concordant function, such as the log det metric, a convex data fidelity term h(·) and, a regularizing – possibly non-smooth – function g(·). This type of problems have recently attracted a great deal of interest, mainly due to their omnipresence in top-notch applications. Under this locally Lipschitz continuous gradient setting, we analyze the convergence behavior of proximal Newton schemes with the added twist of a probable presence of inexact evaluations. We prove attractive convergence rate guarantees and enhance state-of-the-art optimization schemes to accommodate such developments. Experimental results on sparse covariance estimation show the merits of our algorithm, both in terms of recovery efficiency and complexity.

NeurIPS Conference 2014 Conference Paper

Time--Data Tradeoffs by Aggressive Smoothing

• John Bruer
• Joel Tropp
• Volkan Cevher
• Stephen Becker

This paper proposes a tradeoff between sample complexity and computation time that applies to statistical estimators based on convex optimization. As the amount of data increases, we can smooth optimization problems more and more aggressively to achieve accurate estimates more quickly. This work provides theoretical and experimental evidence of this tradeoff for a class of regularized linear inverse problems.

ICML Conference 2013 Conference Paper

A proximal Newton framework for composite minimization: Graph learning without Cholesky decompositions and matrix inversions

• Quoc Tran-Dinh
• Anastasios Kyrillidis
• Volkan Cevher

We propose an algorithmic framework for convex minimization problems of composite functions with two terms: a self-concordant part and a possibly nonsmooth regularization part. Our method is a new proximal Newton algorithm with local quadratic convergence rate. As a specific problem instance, we consider sparse precision matrix estimation problems in graph learning. Via a careful dual formulation and a novel analytic step-size selection, we instantiate an algorithm within our framework for graph learning that avoids Cholesky decompositions and matrix inversions, making it attractive for parallel and distributed implementations.

NeurIPS Conference 2013 Conference Paper

High-Dimensional Gaussian Process Bandits

• Josip Djolonga
• Andreas Krause
• Volkan Cevher

Many applications in machine learning require optimizing unknown functions defined over a high-dimensional space from noisy samples that are expensive to obtain. We address this notoriously hard challenge, under the assumptions that the function varies only along some low-dimensional subspace and is smooth (i. e. , it has a low norm in a Reproducible Kernel Hilbert Space). In particular, we present the SI-BO algorithm, which leverages recent low-rank matrix recovery techniques to learn the underlying subspace of the unknown function and applies Gaussian Process Upper Confidence sampling for optimization of the function. We carefully calibrate the exploration–exploitation tradeoff by allocating sampling budget to subspace estimation and function optimization, and obtain the first subexponential cumulative regret bounds and convergence rates for Bayesian optimization in high-dimensions under noisy observations. Numerical results demonstrate the effectiveness of our approach in difficult scenarios.

ICML Conference 2013 Conference Paper

Sparse projections onto the simplex

• Anastasios Kyrillidis
• Stephen Becker
• Volkan Cevher
• Christoph Koch 0001

Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the \ell_1-norm. However, several important learning applications cannot benefit from this approach as they feature these convex norms as constraints in addition to the non-convex rank and sparsity constraints. In this setting, we derive efficient sparse projections onto the simplex and its extension, and illustrate how to use them to solve high-dimensional learning problems in quantum tomography, sparse density estimation and portfolio selection with non-convex constraints.

NeurIPS Conference 2012 Conference Paper

Active Learning of Multi-Index Function Models

• Tyagi Hemant
• Volkan Cevher

We consider the problem of actively learning \textit{multi-index} functions of the form $f(\vecx) = g(\matA\vecx)= \sum_{i=1}^k g_i(\veca_i^T\vecx)$ from point evaluations of $f$. We assume that the function $f$ is defined on an $\ell_2$-ball in $\Real^d$, $g$ is twice continuously differentiable almost everywhere, and $\matA \in \mathbb{R}^{k \times d}$ is a rank $k$ matrix, where $k \ll d$. We propose a randomized, active sampling scheme for estimating such functions with uniform approximation guarantees. Our theoretical developments leverage recent techniques from low rank matrix recovery, which enables us to derive an estimator of the function $f$ along with sample complexity bounds. We also characterize the noise robustness of the scheme, and provide empirical evidence that the high-dimensional scaling of our sample complexity bounds are quite accurate.

ICML Conference 2010 Conference Paper

Submodular Dictionary Selection for Sparse Representation

• Andreas Krause 0001
• Volkan Cevher

NeurIPS Conference 2009 Conference Paper

Learning with Compressible Priors

• Volkan Cevher

We describe probability distributions, dubbed compressible priors, whose independent and identically distributed (iid) realizations result in compressible signals. A signal is compressible when sorted magnitudes of its coefficients exhibit a power-law decay so that the signal can be well-approximated by a sparse signal. Since compressible signals live close to sparse signals, their intrinsic information can be stably embedded via simple non-adaptive linear projections into a much lower dimensional space whose dimension grows logarithmically with the ambient signal dimension. By using order statistics, we show that N-sample iid realizations of generalized Pareto, Student’s t, log-normal, Frechet, and log-logistic distributions are compressible, i. e. , they have a constant expected decay rate, which is independent of N. In contrast, we show that generalized Gaussian distribution with shape parameter q is compressible only in restricted cases since the expected decay rate of its N-sample iid realizations decreases with N as 1/[q log(N/q)]. We use compressible priors as a scaffold to build new iterative sparse signal recovery algorithms based on Bayesian inference arguments. We show how tuning of these algorithms explicitly depends on the parameters of the compressible prior of the signal, and how to learn the parameters of the signal’s compressible prior on the fly during recovery.

NeurIPS Conference 2008 Conference Paper

Sparse Signal Recovery Using Markov Random Fields

• Volkan Cevher
• Marco Duarte
• Chinmay Hegde
• Richard Baraniuk

Compressive Sensing (CS) combines sampling and compression into a single sub-Nyquist linear measurement process for sparse and compressible signals. In this paper, we extend the theory of CS to include signals that are concisely represented in terms of a graphical model. In particular, we use Markov Random Fields (MRFs) to represent sparse signals whose nonzero coefficients are clustered. Our new model-based reconstruction algorithm, dubbed Lattice Matching Pursuit (LaMP), stably recovers MRF-modeled signals using many fewer measurements and computations than the current state-of-the-art algorithms.