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Yong Lin

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19 papers

2 author rows

TIST Journal 2026 Journal Article

An Improved Autoregressive Evaluation Paradigm for Large Language Models

Jipeng Zhang
Rui Pan
Yuzheng Hu
KaShun Shum
Guanyu Yao
Xiang Liu
Renjie Pi
Hanze Dong

The AI community has witnessed the emergence of various chat-style Large Language Models (LLMs) since the advent of ChatGPT. Despite significant progress in this area, evaluating these models remains a substantial challenge. The evaluations provided by humans or GPT-4 oracles are often taken as the gold standard, but they are neither automatic nor scalable. More recently, a series of (open source) LLM-based judge models have been introduced, yet they often exhibit model-specific biases, e.g., a LLaMA-family judge favors a LLaMA-family model. On the other hand, autoregressive evaluation metrics, which holds the potential to address the aforementioned issues, remains underexplored. Among them, likelihood-based metrics such as perplexity and Negative Log-Likelihood (NLL) are widely adopted and has proven effective in tracking the pre-training progress of LLMs. However, they struggle to evaluate the generation capabilities of fine-tuned models due to exposure bias, a phenomenon where the distribution of the model’s output gradually deviates from the ground-truth during inference. To address this key issue, in this article, we propose a novel autoregressive metric, Normalized Discounted Cumulative Gain (NDCG), to improve the evaluation of fine-tuned LLMs. Our experimental results demonstrate that NDCG significantly outperforms likelihood-based metrics: it shows over 45% improvement in both Spearman and Kendall’s tau correlation coefficients for commonsense QA tasks, and aligns more closely with GPT-4 Elo rankings for instruction-tuned models.

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TMLR Journal 2026 Journal Article

Rethinking Mixture-of-Agents: Is Mixing Different Large Language Models Beneficial?

Wenzhe Li
Yong Lin
Mengzhou Xia
Chi Jin

Ensembling outputs from diverse sources is a straightforward yet effective approach to boost performance. Mixture-of-Agents (MoA) is one such popular ensemble method that aggregates outputs from multiple \emph{different} Large Language Models (LLMs). This paper raises the question in the context of language models: is mixing different LLMs truly beneficial? We propose Self-MoA --- an ensemble method that aggregates outputs from only the \emph{single} top-performing LLM. Our extensive experiments reveal that, surprisingly, Self-MoA outperforms standard MoA that mixes different LLMs in a large number of scenarios: Self-MoA achieves $6.6\%$ improvement over MoA on the AlpacaEval 2.0 benchmark, and an average of $3.8\%$ improvement across various benchmarks, including MMLU, CRUX, and MATH. Applying Self-MoA to one of the top-ranking models in AlpacaEval 2.0 directly achieves the new state-of-the-art performance on the leaderboard. To understand the effectiveness of Self-MoA, we systematically investigate the trade-off between diversity and quality of outputs under various MoA settings. We confirm that the MoA performance is rather sensitive to the quality, and mixing different LLMs often lowers the average quality of the models. To complement the study, we identify the scenarios where mixing different LLMs could be helpful. This paper further introduces a sequential version of Self-MoA, that is capable of aggregating a large number of LLM outputs on-the-fly over multiple rounds, and is as effective as aggregating all outputs at once.

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JBHI Journal 2025 Journal Article

Effectiveness Evaluation for Clinical Depression Detection Using Deep Learning Based Synthetic House-Tree-Person Test

Zhuolong Chen
Xiaoqing Yin
Fan Yang
Xiaofan Li
Zixuan Zhao
Xueying Li
Jianghu Liu
Yubin Zhao

Depression is one of the most common mood disorders and the number of patients increases significantly in recent years. Due to the lack of biomarkers, conversation between patients and psychiatrists is still the main clinical diagnostic method which is easily influenced by subjectivity of both patients and psychiatrists. Synthetic House-tree-person test (S-HTP), a convenient and efficient mental assessment tool, minimizes subjective influences from patients, while its effectiveness is limited by the professional ability of analyst. Here we introduce a deep learning model DeHTP, a flexible and convenient depression detection method based on S-HTP without interaction between people. Experimental results demonstrate that DeHTP achieves 0. 963 AUC and 0. 9 accuracy, and outperforms the conventional manual analysis of S-HTP, which is conducted on the guideline of 50 conclusions from previous study related to depression. In addition, it reveals 22 depression-correlated drawing features aligned with conclusions above from the perspective of our proposed model. Leveraging the advantages of deep learning and S-HTP, this approach has the potential for widespread promotion and adoption as the available tool for daily self-mental monitoring, as well as the promising auxiliary diagnostic method in clinical.

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TMLR Journal 2025 Journal Article

Entropy-Regularized Process Reward Model

Hanning Zhang
Pengcheng Wang
Shizhe Diao
Yong Lin
Rui Pan
Hanze Dong
Dylan Zhang
Pavlo Molchanov

Large language models (LLMs) have shown promise in performing complex multi-step reasoning, yet they continue to struggle with mathematical reasoning, often making systematic errors. A promising solution is reinforcement learning (RL) guided by reward models, particularly those focusing on process rewards, which score each intermediate step rather than solely evaluating the final outcome. This approach is more effective at guiding policy models towards correct reasoning trajectories. In this work, we propose an entropy-regularized process reward model (ER-PRM) that integrates KL-regularized Markov Decision Processes (MDP) to balance policy optimization with the need to prevent the policy from shifting too far from its initial distribution. We derive a novel reward construction method based on the theoretical results. Our theoretical analysis shows that we could derive the optimal reward model from the initial policy sampling. Our empirical experiments on the MATH and GSM8K benchmarks demonstrate that ER-PRM consistently outperforms existing process reward models, achieving 1% improvement on GSM8K and 2-3% improvement on MATH under best-of-N evaluation, and more than 1% improvement under RLHF. These results highlight the efficacy of entropy-regularization in enhancing LLMs' reasoning capabilities.

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NeurIPS Conference 2025 Conference Paper

Ineq-Comp: Benchmarking Human-Intuitive Compositional Reasoning in Automated Theorem Proving of Inequalities

Haoyu Zhao
Yihan Geng
Shange Tang
Yong Lin
Bohan Lyu
Hongzhou Lin
Chi Jin
Sanjeev Arora

LLM-based formal proof assistants (e. g. , in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question in context of mathematical inequalities---specifically the prover's ability to recognize that the given problem simplifies by applying a known inequality such as AM/GM. Specifically, we are interested in their ability to do this in a {\em compositional setting} where multiple inequalities must be applied as part of a solution. We introduce \ineqcomp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers---including Goedel, STP, and Kimina-7B---struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness, but still suffers a 20\% performance drop (pass@32). Even for DeepSeek-Prover-V2-671B model, the gap between compositional variants and seed problems exists, implying that simply scaling up the model size alone does not fully solve the compositional weakness. Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition. All data and evaluation code can be found at \url{https: //github. com/haoyuzhao123/LeanIneqComp}.

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TMLR Journal 2025 Journal Article

Localize-and-Stitch: Efficient Model Merging via Sparse Task Arithmetic

Yifei He
Yuzheng Hu
Yong Lin
Tong Zhang
Han Zhao

Model merging offers an effective strategy to combine the strengths of multiple finetuned models into a unified model that preserves the specialized capabilities of each. Existing methods merge models in a global manner, performing arithmetic operations across all model parameters. However, such global merging often leads to task interference, degrading the performance of the merged model. In this work, we introduce Localize-and-Stitch, a novel approach that merges models in a localized way. Our algorithm works in two steps: i) Localization: identify tiny ($1\%$ of the total parameters) localized regions in the finetuned models containing essential skills for the downstream tasks, and ii) Stitching: reintegrate only these essential regions back into the pretrained model for task synergy. We demonstrate that our approach effectively locates sparse regions responsible for finetuned performance, and the localized regions could be treated as compact and interpretable representations of the finetuned models (tasks). Empirically, we evaluate our method on various vision and language benchmarks, showing that it outperforms existing model merging methods under different data availability scenarios. Beyond strong empirical performance, our algorithm also facilitates model compression and preserves pretrained knowledge, enabling flexible and continual skill composition from multiple finetuned models with minimal storage and computational overhead.

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JMLR Journal 2025 Journal Article

Optimal Sample Selection Through Uncertainty Estimation and Its Application in Deep Learning

Yong Lin
Chen Liu
Chenlu Ye
Qing Lian
Yuan Yao
Tong Zhang

Modern deep learning heavily relies on large labeled datasets, which often comse with high costs in terms of both manual labeling and computational resources. To mitigate these challenges, researchers have explored the use of informative subset selection techniques. In this study, we present a theoretically optimal solution for addressing both sampling with and without labels within the context of linear softmax regression. Our proposed method, COPS (unCertainty based OPtimal Sub-sampling), is designed to minimize the expected loss of a model trained on subsampled data. Unlike existing approaches that rely on explicit calculations of the inverse covariance matrix, which are not easily applicable to deep learning scenarios, COPS leverages the model's logits to estimate the sampling ratio. This sampling ratio is closely associated with model uncertainty and can be effectively applied to deep learning tasks. Furthermore, we address the challenge of model sensitivity to misspecification by incorporating a down-weighting approach for low-density samples, drawing inspiration from previous works. To assess the effectiveness of our proposed method, we conducted extensive empirical experiments using deep neural networks on benchmark datasets. The results consistently showcase the superior performance of COPS compared to baseline methods, reaffirming its efficacy. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

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NeurIPS Conference 2024 Conference Paper

A Sober Look at the Robustness of CLIPs to Spurious Features

Qizhou Wang
Yong Lin
Yongqiang Chen
Ludwig Schmidt
Bo Han
Tong Zhang

Large vision language models, such as CLIP, demonstrate impressive robustness to spurious features than single-modal models trained on ImageNet. However, existing test datasets are typically curated based on ImageNet-trained models, which aim to capture the spurious features inherited in ImageNet. Benchmarking CLIP models based on the ImageNet-oriented spurious features may not be sufficient to reflect the extent to which CLIP models are robust to spurious correlations within CLIP training data, e. g. , LAION. To this end, we craft a new challenging dataset named CounterAnimal designed to reveal the reliance of CLIP models on realistic spurious features. Specifically, we split animal photos into groups according to the backgrounds, and then identify a pair of groups for each class where a CLIP model shows high-performance drops across the two groups. Our evaluations show that the spurious features captured by CounterAnimal are generically learned by CLIP models with different backbones and pre-train data, yet have limited influence for ImageNet models. We provide theoretical insights that the CLIP objective cannot offer additional robustness. Furthermore, we also re-evaluate strategies such as scaling up parameters and high-quality pre-trained data. We find that they still help mitigate the spurious features, providing a promising path for future developments.

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NeurIPS Conference 2024 Conference Paper

Regularizing Hidden States Enables Learning Generalizable Reward Model for LLMs

Rui Yang
Ruomeng Ding
Yong Lin
Huan ZHang
Tong Zhang

Reward models trained on human preference data have been proven to effectively align Large Language Models (LLMs) with human intent within the framework of reinforcement learning from human feedback (RLHF). However, current reward models have limited generalization capabilities to unseen prompts and responses, which can lead to an unexpected phenomenon known as reward over-optimization, resulting in a decline in actual performance due to excessive optimization of rewards. While previous research has advocated for constraining policy optimization, our study introduces a novel approach to enhance the reward model's generalization ability against distribution shifts by regularizing the hidden states. Specifically, we retain the base model's language model head and incorporate a suite of text-generation losses to preserve the hidden states' text-generation capabilities, while concurrently learning a reward head behind the same hidden states. Our experimental results demonstrate that the introduced regularization technique markedly improves the accuracy of learned reward models across a variety of out-of-distribution (OOD) tasks and effectively alleviates the over-optimization issue in RLHF, offering a more reliable and robust preference learning paradigm.

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ICLR Conference 2024 Conference Paper

Spurious Feature Diversification Improves Out-of-distribution Generalization

Yong Lin
Lu Tan
Yifan Hao 0002
Honam Wong
Hanze Dong
Weizhong Zhang
Yujiu Yang 0001
Tong Zhang 0001

Generalization to out-of-distribution (OOD) data is a critical challenge in machine learning. Ensemble-based methods, like weight space ensembles that interpolate model parameters, have been shown to achieve superior OOD performance. However, the underlying mechanism for their effectiveness remains unclear. In this study, we closely examine WiSE-FT, a popular weight space ensemble method that interpolates between a pre-trained and a fine-tuned model. We observe an unexpected ``FalseFalseTrue" phenomenon, in which WiSE-FT successfully corrects many cases where each individual model makes incorrect predictions, which contributes significantly to its OOD effectiveness. To gain further insights, we conduct theoretical analysis in a multi-class setting with a large number of spurious features. Our analysis predicts the above phenomenon and it further shows that ensemble-based models reduce prediction errors in the OOD settings by utilizing a more diverse set of spurious features. Contrary to the conventional wisdom that focuses on learning invariant features for better OOD performance, our findings suggest that incorporating a large number of diverse spurious features weakens their individual contributions, leading to improved overall OOD generalization performance. Additionally, our findings provide the first explanation for the mysterious phenomenon of weight space ensembles outperforming output space ensembles in OOD. Empirically we demonstrate the effectiveness of utilizing diverse spurious features on a MultiColorMNIST dataset, and our experimental results are consistent with the theoretical analysis. Building upon the new theoretical insights into the efficacy of ensemble methods, we further identify an issue of WiSE-FT caused by the overconfidence of fine-tuned models in OOD situations. This overconfidence magnifies the fine-tuned model's incorrect prediction, leading to deteriorated OOD ensemble performance. To remedy this problem, we propose a novel method called BAlaNced averaGing (BANG) to mitigate the overconfidence problem, which significantly enhances the OOD performance of WiSE-FT.

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ICLR Conference 2023 Conference Paper

A Holistic View of Label Noise Transition Matrix in Deep Learning and Beyond

Yong Lin
Renjie Pi
Weizhong Zhang
Xiaobo Xia
Jiahui Gao
Xiao Zhou
Tongliang Liu
Bo Han 0003

In this paper, we explore learning statistically consistent classifiers under label noise by estimating the noise transition matrix T. We first provide a holistic view of existing T-estimation methods including those with or without anchor point assumptions. We unified them into the Minimum Geometric Envelope Operator (MGEO) framework, which tries to find the smallest T (in terms of a certain metric) that elicits a convex hull to enclose the posteriors of all the training data. Although MGEO methods show appealing theoretical properties and empirical results, we find them prone to failing when the noisy posterior estimation is imperfect, which is inevitable in practice. Specifically, we show that MGEO methods are in-consistent even with infinite samples if the noisy posterior is not estimated accurately. In view of this, we make the first effort to address this issue by proposing a novel T-estimation framework via the lens of bilevel optimization, and term it RObust Bilevel OpTimzation (ROBOT). ROBOT paves a new road beyond MGEO framework, which enjoys strong theoretical properties: identifibility, consistency and finite-sample generalization guarantees. Notably, ROBOT neither requires the perfect posterior estimation nor assumes the existence of anchor points. We further theoretically demonstrate that ROBOT is more robust in the case where MGEO methods fail. Experimentally, our framework also shows superior performance across multiple benchmarks.

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NeurIPS Conference 2023 Conference Paper

ID and OOD Performance Are Sometimes Inversely Correlated on Real-world Datasets

Damien Teney
Yong Lin
Seong Joon Oh
Ehsan Abbasnejad

Several studies have compared the in-distribution (ID) and out-of-distribution (OOD) performance of models in computer vision and NLP. They report a frequent positive correlation and some surprisingly never even observe an inverse correlation indicative of a necessary trade-off. The possibility of inverse patterns is important to determine whether ID performance can serve as a proxy for OOD generalization capabilities. This paper shows that inverse correlations between ID and OOD performance do happen with multiple real-world datasets, not only in artificial worst-case settings. We explain theoretically how these cases arise and how past studies missed them because of improper methodologies that examined a biased selection of models. Our observations lead to recommendations that contradict those found in much of the current literature. - High OOD performance sometimes requires trading off ID performance. - Focusing on ID performance alone may not lead to optimal OOD performance. It may produce diminishing (eventually negative) returns in OOD performance. - In these cases, studies on OOD generalization that use ID performance for model selection (a common recommended practice) will necessarily miss the best-performing models, making these studies blind to a whole range of phenomena.

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ICLR Conference 2023 Conference Paper

Particle-based Variational Inference with Preconditioned Functional Gradient Flow

Hanze Dong
Xi Wang
Yong Lin
Tong Zhang 0001

Particle-based variational inference (VI) minimizes the KL divergence between model samples and the target posterior with gradient flow estimates. With the popularity of Stein variational gradient descent (SVGD), the focus of particle-based VI algorithms has been on the properties of functions in Reproducing Kernel Hilbert Space (RKHS) to approximate the gradient flow. However, the requirement of RKHS restricts the function class and algorithmic flexibility. This paper offers a general solution to this problem by introducing a functional regularization term that encompasses the RKHS norm as a special case. This allows us to propose a new particle-based VI algorithm called preconditioned functional gradient flow (PFG). Compared to SVGD, PFG has several advantages. It has a larger function class, improved scalability in large particle-size scenarios, better adaptation to ill-conditioned distributions, and provable continuous-time convergence in KL divergence. Additionally, non-linear function classes such as neural networks can be incorporated to estimate the gradient flow. Our theory and experiments demonstrate the effectiveness of the proposed framework.

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ICLR Conference 2023 Conference Paper

Self-Guided Noise-Free Data Generation for Efficient Zero-Shot Learning

Jiahui Gao
Renjie Pi
Yong Lin
Hang Xu 0004
Jiacheng Ye
Zhiyong Wu 0003
Weizhong Zhang
Xiaodan Liang

There is a rising interest in further exploring the zero-shot learning potential of large pre-trained language models (PLMs). A new paradigm called data-generation-based zero-shot learning has achieved impressive success. In this paradigm, the synthesized data from the PLM acts as the carrier of knowledge, which is used to train a task-specific model with orders of magnitude fewer parameters than the PLM, achieving both higher performance and efficiency than prompt-based zero-shot learning methods on PLMs. The main hurdle of this approach is that the synthesized data from PLM usually contains a significant portion of low-quality samples. Fitting on such data will greatly hamper the performance of the task-specific model, making it unreliable for deployment. Previous methods remedy this issue mainly by filtering synthetic data using heuristic metrics(e.g., output confidence), or refining the data with the help of a human expert, which comes with excessive manual tuning or expensive costs. In this paper, we propose a novel noise-robust re-weighting framework SunGen to automatically construct high-quality data for zero-shot classification problems. Our framework features the ability to learn the sample weights indicating data quality without requiring any human annotation. We theoretically and empirically verify the ability of our method to help construct good-quality synthetic datasets. Notably, SunGen-LSTM yields a 9.8% relative improvement than the baseline on average accuracy across eight different established text classification tasks.

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AAAI Conference 2023 Conference Paper

Stable Learning via Sparse Variable Independence

Han Yu
Peng Cui
Yue He
Zheyan Shen
Yong Lin
Renzhe Xu
Xingxuan Zhang

The problem of covariate-shift generalization has attracted intensive research attention. Previous stable learning algorithms employ sample reweighting schemes to decorrelate the covariates when there is no explicit domain information about training data. However, with finite samples, it is difficult to achieve the desirable weights that ensure perfect independence to get rid of the unstable variables. Besides, decorrelating within stable variables may bring about high variance of learned models because of the over-reduced effective sample size. A tremendous sample size is required for these algorithms to work. In this paper, with theoretical justification, we propose SVI (Sparse Variable Independence) for the covariate-shift generalization problem. We introduce sparsity constraint to compensate for the imperfectness of sample reweighting under the finite-sample setting in previous methods. Furthermore, we organically combine independence-based sample reweighting and sparsity-based variable selection in an iterative way to avoid decorrelating within stable variables, increasing the effective sample size to alleviate variance inflation. Experiments on both synthetic and real-world datasets demonstrate the improvement of covariate-shift generalization performance brought by SVI.

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ICML Conference 2022 Conference Paper

Model Agnostic Sample Reweighting for Out-of-Distribution Learning

Xiao Zhou
Yong Lin
Renjie Pi
Weizhong Zhang
Renzhe Xu
Peng Cui 0001
Tong Zhang 0001

Distributionally robust optimization (DRO) and invariant risk minimization (IRM) are two popular methods proposed to improve out-of-distribution (OOD) generalization performance of machine learning models. While effective for small models, it has been observed that these methods can be vulnerable to overfitting with large overparameterized models. This work proposes a principled method, Model Agnostic samPLe rEweighting (MAPLE), to effectively address OOD problem, especially in overparameterized scenarios. Our key idea is to find an effective reweighting of the training samples so that the standard empirical risk minimization training of a large model on the weighted training data leads to superior OOD generalization performance. The overfitting issue is addressed by considering a bilevel formulation to search for the sample reweighting, in which the generalization complexity depends on the search space of sample weights instead of the model size. We present theoretical analysis in linear case to prove the insensitivity of MAPLE to model size, and empirically verify its superiority in surpassing state-of-the-art methods by a large margin.

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ICML Conference 2022 Conference Paper

Probabilistic Bilevel Coreset Selection

Xiao Zhou
Renjie Pi
Weizhong Zhang
Yong Lin
Zonghao Chen
Tong Zhang 0001

The goal of coreset selection in supervised learning is to produce a weighted subset of data, so that training only on the subset achieves similar performance as training on the entire dataset. Existing methods achieved promising results in resource-constrained scenarios such as continual learning and streaming. However, most of the existing algorithms are limited to traditional machine learning models. A few algorithms that can handle large models adopt greedy search approaches due to the difficulty in solving the discrete subset selection problem, which is computationally costly when coreset becomes larger and often produces suboptimal results. In this work, for the first time we propose a continuous probabilistic bilevel formulation of coreset selection by learning a probablistic weight for each training sample. The overall objective is posed as a bilevel optimization problem, where 1) the inner loop samples coresets and train the model to convergence and 2) the outer loop updates the sample probability progressively according to the model’s performance. Importantly, we develop an efficient solver to the bilevel optimization problem via unbiased policy gradient without trouble of implicit differentiation. We theoretically prove the convergence of this training procedure and demonstrate the superiority of our algorithm against various coreset selection methods in various tasks, especially in more challenging label-noise and class-imbalance scenarios.

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ICML Conference 2022 Conference Paper

Sparse Invariant Risk Minimization

Xiao Zhou
Yong Lin
Weizhong Zhang
Tong Zhang 0001

Invariant Risk Minimization (IRM) is an emerging invariant feature extracting technique to help generalization with distributional shift. However, we find that there exists a basic and intractable contradiction between the model trainability and generalization ability in IRM. On one hand, recent studies on deep learning theory indicate the importance of large-sized or even overparameterized neural networks to make the model easy to train. On the other hand, unlike empirical risk minimization that can be benefited from overparameterization, our empirical and theoretical analyses show that the generalization ability of IRM is much easier to be demolished by overfitting caused by overparameterization. In this paper, we propose a simple yet effective paradigm named Sparse Invariant Risk Minimization (SparseIRM) to address this contradiction. Our key idea is to employ a global sparsity constraint as a defense to prevent spurious features from leaking in during the whole IRM process. Compared with sparisfy-after-training prototype by prior work which can discard invariant features, the global sparsity constraint limits the budget for feature selection and enforces SparseIRM to select the invariant features. We illustrate the benefit of SparseIRM through a theoretical analysis on a simple linear case. Empirically we demonstrate the power of SparseIRM through various datasets and models and surpass state-of-the-art methods with a gap up to 29%.

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NeurIPS Conference 2022 Conference Paper

ZIN: When and How to Learn Invariance Without Environment Partition?

Yong Lin
Shengyu Zhu
Lu Tan
Peng Cui

It is commonplace to encounter heterogeneous data, of which some aspects of the data distribution may vary but the underlying causal mechanisms remain constant. When data are divided into distinct environments according to the heterogeneity, recent invariant learning methods have proposed to learn robust and invariant models using this environment partition. It is hence tempting to utilize the inherent heterogeneity even when environment partition is not provided. Unfortunately, in this work, we show that learning invariant features under this circumstance is fundamentally impossible without further inductive biases or additional information. Then, we propose a framework to jointly learn environment partition and invariant representation, assisted by additional auxiliary information. We derive sufficient and necessary conditions for our framework to provably identify invariant features under a fairly general setting. Experimental results on both synthetic and real world datasets validate our analysis and demonstrate an improved performance of the proposed framework. Our findings also raise the need of making the role of inductive biases more explicit when learning invariant models without environment partition in future works. Codes are available at https: //github. com/linyongver/ZIN_official.

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