Zixiang Chen

Jingfeng Wu, Difan Zou, Zixiang Chen et al. · berkeley

This paper considers the problem of learning a single ReLU neuron with squared loss (a.k.a., ReLU regression) in the overparameterized regime, where the input dimension can exceed the number of samples. We analyze a Perceptron-type algorithm called GLM-tron (Kakade et al., 2011) and provide its dimension-free risk upper bounds for high-dimensional ReLU regression in both well-specified and misspecified settings. Our risk bounds recover several existing results as special cases. Moreover, in the well-specified setting, we provide an instance-wise matching risk lower bound for GLM-tron. Our upper and lower risk bounds provide a sharp characterization of the high-dimensional ReLU regression problems that can be learned via GLM-tron. On the other hand, we provide some negative results for stochastic gradient descent (SGD) for ReLU regression with symmetric Bernoulli data: if the model is well-specified, the excess risk of SGD is provably no better than that of GLM-tron ignoring constant factors, for each problem instance; and in the noiseless case, GLM-tron can achieve a small risk while SGD unavoidably suffers from a constant risk in expectation. These results together suggest that GLM-tron might be preferable to SGD for high-dimensional ReLU regression.

Jingfeng Wu, Difan Zou, Zixiang Chen et al. · berkeley

Transformers pretrained on diverse tasks exhibit remarkable in-context learning (ICL) capabilities, enabling them to solve unseen tasks solely based on input contexts without adjusting model parameters. In this paper, we study ICL in one of its simplest setups: pretraining a linearly parameterized single-layer linear attention model for linear regression with a Gaussian prior. We establish a statistical task complexity bound for the attention model pretraining, showing that effective pretraining only requires a small number of independent tasks. Furthermore, we prove that the pretrained model closely matches the Bayes optimal algorithm, i.e., optimally tuned ridge regression, by achieving nearly Bayes optimal risk on unseen tasks under a fixed context length. These theoretical findings complement prior experimental research and shed light on the statistical foundations of ICL.

Zixiang Chen, Yihe Deng, Yue Wu et al.

The Mixture-of-Experts (MoE) layer, a sparsely-activated model controlled by a router, has achieved great success in deep learning. However, the understanding of such architecture remains elusive. In this paper, we formally study how the MoE layer improves the performance of neural network learning and why the mixture model will not collapse into a single model. Our empirical results suggest that the cluster structure of the underlying problem and the non-linearity of the expert are pivotal to the success of MoE. To further understand this, we consider a challenging classification problem with intrinsic cluster structures, which is hard to learn using a single expert. Yet with the MoE layer, by choosing the experts as two-layer nonlinear convolutional neural networks (CNNs), we show that the problem can be learned successfully. Furthermore, our theory shows that the router can learn the cluster-center features, which helps divide the input complex problem into simpler linear classification sub-problems that individual experts can conquer. To our knowledge, this is the first result towards formally understanding the mechanism of the MoE layer for deep learning.

Yiwen Kou, Zixiang Chen, Yuanzhou Chen et al.

Modern deep learning models with great expressive power can be trained to overfit the training data but still generalize well. This phenomenon is referred to as \textit{benign overfitting}. Recently, a few studies have attempted to theoretically understand benign overfitting in neural networks. However, these works are either limited to neural networks with smooth activation functions or to the neural tangent kernel regime. How and when benign overfitting can occur in ReLU neural networks remains an open problem. In this work, we seek to answer this question by establishing algorithm-dependent risk bounds for learning two-layer ReLU convolutional neural networks with label-flipping noise. We show that, under mild conditions, the neural network trained by gradient descent can achieve near-zero training loss and Bayes optimal test risk. Our result also reveals a sharp transition between benign and harmful overfitting under different conditions on data distribution in terms of test risk. Experiments on synthetic data back up our theory.

Zixiang Chen, Chris Junchi Li, Angela Yuan et al.

With the increasing need for handling large state and action spaces, general function approximation has become a key technique in reinforcement learning (RL). In this paper, we propose a general framework that unifies model-based and model-free RL, and an Admissible Bellman Characterization (ABC) class that subsumes nearly all Markov Decision Process (MDP) models in the literature for tractable RL. We propose a novel estimation function with decomposable structural properties for optimization-based exploration and the functional eluder dimension as a complexity measure of the ABC class. Under our framework, a new sample-efficient algorithm namely OPtimization-based ExploRation with Approximation (OPERA) is proposed, achieving regret bounds that match or improve over the best-known results for a variety of MDP models. In particular, for MDPs with low Witness rank, under a slightly stronger assumption, OPERA improves the state-of-the-art sample complexity results by a factor of $dH$. Our framework provides a generic interface to design and analyze new RL models and algorithms.

Liangwei Yang, Jielin Qiu, Zixiang Chen et al.

Many decision-support settings require systems that adapt to individual users, but evaluation data for this problem remain limited. Existing benchmarks for user understanding often rely on simulated users or model-generated behavior, even though recent work cautions that model-based simulations can diverge systematically from human behavior. We introduce \textsc{BehaviorBench}, a benchmark for evaluating personalized decision modeling from real-world behavioral traces. \textsc{BehaviorBench} reconstructs wallet-level decision histories from observed public prediction-market and on-chain records, and organizes them into two complementary task layers: \emph{Belief prediction}, which predicts a user's final revealed stance and confidence in a market, and \emph{Trade prediction}, which predicts the direction and amount of individual transactions. Across 2,000 evaluation wallets, the benchmark contains 141,445 Belief instances and 1,485,972 Trade instances, with disjoint support pools for retrieval-based evaluation. We evaluate frontier and open-weight generative models under four history interfaces: no personalization, direct recent history, generated user profiles, and retrieved support-wallet evidence. Personalization improves Belief prediction more consistently than Trade prediction, model rankings change across task layers and metrics, and different history interfaces expose different failure modes. \textsc{BehaviorBench} provides an evaluation setting for studying whether personalized methods can use real-world behavioral evidence rather than simulated users alone.

Zixiang Chen, Yihe Deng, Huizhuo Yuan et al.

Harnessing the power of human-annotated data through Supervised Fine-Tuning (SFT) is pivotal for advancing Large Language Models (LLMs). In this paper, we delve into the prospect of growing a strong LLM out of a weak one without the need for acquiring additional human-annotated data. We propose a new fine-tuning method called Self-Play fIne-tuNing (SPIN), which starts from a supervised fine-tuned model. At the heart of SPIN lies a self-play mechanism, where the LLM refines its capability by playing against instances of itself. More specifically, the LLM generates its own training data from its previous iterations, refining its policy by discerning these self-generated responses from those obtained from human-annotated data. Our method progressively elevates the LLM from a nascent model to a formidable one, unlocking the full potential of human-annotated demonstration data for SFT. Theoretically, we prove that the global optimum to the training objective function of our method is achieved only when the LLM policy aligns with the target data distribution. Empirically, we evaluate our method on several benchmark datasets including the HuggingFace Open LLM Leaderboard, MT-Bench, and datasets from Big-Bench. Our results show that SPIN can significantly improve the LLM's performance across a variety of benchmarks and even outperform models trained through direct preference optimization (DPO) supplemented with extra GPT-4 preference data. This sheds light on the promise of self-play, enabling the achievement of human-level performance in LLMs without the need for expert opponents. Codes are available at https://github.com/uclaml/SPIN.

Zixiang Chen, Junkai Zhang, Yiwen Kou et al.

The challenge of overfitting, in which the model memorizes the training data and fails to generalize to test data, has become increasingly significant in the training of large neural networks. To tackle this challenge, Sharpness-Aware Minimization (SAM) has emerged as a promising training method, which can improve the generalization of neural networks even in the presence of label noise. However, a deep understanding of how SAM works, especially in the setting of nonlinear neural networks and classification tasks, remains largely missing. This paper fills this gap by demonstrating why SAM generalizes better than Stochastic Gradient Descent (SGD) for a certain data model and two-layer convolutional ReLU networks. The loss landscape of our studied problem is nonsmooth, thus current explanations for the success of SAM based on the Hessian information are insufficient. Our result explains the benefits of SAM, particularly its ability to prevent noise learning in the early stages, thereby facilitating more effective learning of features. Experiments on both synthetic and real data corroborate our theory.

Zhengyong Huang, Sijuan Zou, Guoshuai Wang et al.

Achieving accurate and automated tumor segmentation plays an important role in both clinical practice and radiomics research. Segmentation in medicine is now often performed manually by experts, which is a laborious, expensive and error-prone task. Manual annotation relies heavily on the experience and knowledge of these experts. In addition, there is much intra- and interobserver variation. Therefore, it is of great significance to develop a method that can automatically segment tumor target regions. In this paper, we propose a deep learning segmentation method based on multimodal positron emission tomography-computed tomography (PET-CT), which combines the high sensitivity of PET and the precise anatomical information of CT. We design an improved spatial attention network(ISA-Net) to increase the accuracy of PET or CT in detecting tumors, which uses multi-scale convolution operation to extract feature information and can highlight the tumor region location information and suppress the non-tumor region location information. In addition, our network uses dual-channel inputs in the coding stage and fuses them in the decoding stage, which can take advantage of the differences and complementarities between PET and CT. We validated the proposed ISA-Net method on two clinical datasets, a soft tissue sarcoma(STS) and a head and neck tumor(HECKTOR) dataset, and compared with other attention methods for tumor segmentation. The DSC score of 0.8378 on STS dataset and 0.8076 on HECKTOR dataset show that ISA-Net method achieves better segmentation performance and has better generalization. Conclusions: The method proposed in this paper is based on multi-modal medical image tumor segmentation, which can effectively utilize the difference and complementarity of different modes. The method can also be applied to other multi-modal data or single-modal data by proper adjustment.

Yiwen Kou, Zixiang Chen, Quanquan Gu

The implicit bias towards solutions with favorable properties is believed to be a key reason why neural networks trained by gradient-based optimization can generalize well. While the implicit bias of gradient flow has been widely studied for homogeneous neural networks (including ReLU and leaky ReLU networks), the implicit bias of gradient descent is currently only understood for smooth neural networks. Therefore, implicit bias in non-smooth neural networks trained by gradient descent remains an open question. In this paper, we aim to answer this question by studying the implicit bias of gradient descent for training two-layer fully connected (leaky) ReLU neural networks. We showed that when the training data are nearly-orthogonal, for leaky ReLU activation function, gradient descent will find a network with a stable rank that converges to $1$, whereas for ReLU activation function, gradient descent will find a neural network with a stable rank that is upper bounded by a constant. Additionally, we show that gradient descent will find a neural network such that all the training data points have the same normalized margin asymptotically. Experiments on both synthetic and real data backup our theoretical findings.

Zixiang Chen, Yihe Deng, Yuanzhi Li et al.

Multi-modal learning has become increasingly popular due to its ability to leverage information from different data sources (e.g., text and images) to improve the model performance. Recently, CLIP has emerged as an effective approach that employs vision-language contrastive pretraining to learn joint image and text representations and exhibits remarkable performance in zero-shot learning and text-guided natural image generation. Despite the huge practical success of CLIP, its theoretical understanding remains elusive. In this paper, we formally study transferrable representation learning underlying CLIP and demonstrate how features from different modalities get aligned. We also analyze its zero-shot transfer performance on the downstream tasks. Inspired by our analysis, we propose a new CLIP-type approach, which achieves better performance than CLIP and other state-of-the-art methods on benchmark datasets.

Huizhuo Yuan, Zixiang Chen, Kaixuan Ji et al.

Fine-tuning Diffusion Models remains an underexplored frontier in generative artificial intelligence (GenAI), especially when compared with the remarkable progress made in fine-tuning Large Language Models (LLMs). While cutting-edge diffusion models such as Stable Diffusion (SD) and SDXL rely on supervised fine-tuning, their performance inevitably plateaus after seeing a certain volume of data. Recently, reinforcement learning (RL) has been employed to fine-tune diffusion models with human preference data, but it requires at least two images ("winner" and "loser" images) for each text prompt. In this paper, we introduce an innovative technique called self-play fine-tuning for diffusion models (SPIN-Diffusion), where the diffusion model engages in competition with its earlier versions, facilitating an iterative self-improvement process. Our approach offers an alternative to conventional supervised fine-tuning and RL strategies, significantly improving both model performance and alignment. Our experiments on the Pick-a-Pic dataset reveal that SPIN-Diffusion outperforms the existing supervised fine-tuning method in aspects of human preference alignment and visual appeal right from its first iteration. By the second iteration, it exceeds the performance of RLHF-based methods across all metrics, achieving these results with less data.

Zixiang Chen, Huizhuo Yuan, Yongqian Li et al.

Discrete diffusion models have emerged as powerful tools for high-quality data generation. Despite their success in discrete spaces, such as text generation tasks, the acceleration of discrete diffusion models remains under-explored. In this paper, we propose discrete non-Markov diffusion models (DNDM), which naturally induce the predetermined transition time set. This enables a training-free sampling algorithm that significantly reduces the number of function evaluations (i.e., calls to the neural network), making the sampling process much faster. Furthermore, we study the transition from finite to infinite step sampling, offering new insights into bridging the gap between discrete and continuous-time processes for discrete diffusion models. Extensive experiments on natural language generation and machine translation tasks demonstrate the superior performance of our method in terms of both generation speed and sample quality compared to existing methods for discrete diffusion models.

Jun Han, Zixiang Chen, Yongqian Li et al.

Electronic health records (EHRs) are a pivotal data source that enables numerous applications in computational medicine, e.g., disease progression prediction, clinical trial design, and health economics and outcomes research. Despite wide usability, their sensitive nature raises privacy and confidentially concerns, which limit potential use cases. To tackle these challenges, we explore the use of generative models to synthesize artificial, yet realistic EHRs. While diffusion-based methods have recently demonstrated state-of-the-art performance in generating other data modalities and overcome the training instability and mode collapse issues that plague previous GAN-based approaches, their applications in EHR generation remain underexplored. The discrete nature of tabular medical code data in EHRs poses challenges for high-quality data generation, especially for continuous diffusion models. To this end, we introduce a novel tabular EHR generation method, EHR-D3PM, which enables both unconditional and conditional generation using the discrete diffusion model. Our experiments demonstrate that EHR-D3PM significantly outperforms existing generative baselines on comprehensive fidelity and utility metrics while maintaining less attribute and membership vulnerability risks. Furthermore, we show EHR-D3PM is effective as a data augmentation method and enhances performance on downstream tasks when combined with real data.

Yiwen Kou, Zixiang Chen, Quanquan Gu et al.

The $k$-sparse parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-sparse parity problem with sign stochastic gradient descent, a variant of stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that this approach can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\leq O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{Θ(k)}$ neurons, matching the established $Ω(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with sign SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. To the best of our knowledge, this is the first result that matches the SQ lower bound for solving $k$-sparse parity problem using gradient-based methods.

Zixiang Chen, Greg Yang, Qingyue Zhao et al.

Despite deep neural networks' powerful representation learning capabilities, theoretical understanding of how networks can simultaneously achieve meaningful feature learning and global convergence remains elusive. Existing approaches like the neural tangent kernel (NTK) are limited because features stay close to their initialization in this parametrization, leaving open questions about feature properties during substantial evolution. In this paper, we investigate the training dynamics of infinitely wide, $L$-layer neural networks using the tensor program (TP) framework. Specifically, we show that, when trained with stochastic gradient descent (SGD) under the Maximal Update parametrization ($μ$P) and mild conditions on the activation function, SGD enables these networks to learn linearly independent features that substantially deviate from their initial values. This rich feature space captures relevant data information and ensures that any convergent point of the training process is a global minimum. Our analysis leverages both the interactions among features across layers and the properties of Gaussian random variables, providing new insights into deep representation learning. We further validate our theoretical findings through experiments on real-world datasets.

Yuan Cao, Zixiang Chen, Mikhail Belkin et al.

Modern neural networks often have great expressive power and can be trained to overfit the training data, while still achieving a good test performance. This phenomenon is referred to as "benign overfitting". Recently, there emerges a line of works studying "benign overfitting" from the theoretical perspective. However, they are limited to linear models or kernel/random feature models, and there is still a lack of theoretical understanding about when and how benign overfitting occurs in neural networks. In this paper, we study the benign overfitting phenomenon in training a two-layer convolutional neural network (CNN). We show that when the signal-to-noise ratio satisfies a certain condition, a two-layer CNN trained by gradient descent can achieve arbitrarily small training and test loss. On the other hand, when this condition does not hold, overfitting becomes harmful and the obtained CNN can only achieve a constant level test loss. These together demonstrate a sharp phase transition between benign overfitting and harmful overfitting, driven by the signal-to-noise ratio. To the best of our knowledge, this is the first work that precisely characterizes the conditions under which benign overfitting can occur in training convolutional neural networks.

Zixiang Chen, Dongruo Zhou, Quanquan Gu

Escaping from saddle points and finding local minimum is a central problem in nonconvex optimization. Perturbed gradient methods are perhaps the simplest approach for this problem. However, to find $(ε, \sqrtε)$-approximate local minima, the existing best stochastic gradient complexity for this type of algorithms is $\tilde O(ε^{-3.5})$, which is not optimal. In this paper, we propose LENA (Last stEp shriNkAge), a faster perturbed stochastic gradient framework for finding local minima. We show that LENA with stochastic gradient estimators such as SARAH/SPIDER and STORM can find $(ε, ε_{H})$-approximate local minima within $\tilde O(ε^{-3} + ε_{H}^{-6})$ stochastic gradient evaluations (or $\tilde O(ε^{-3})$ when $ε_H = \sqrtε$). The core idea of our framework is a step-size shrinkage scheme to control the average movement of the iterates, which leads to faster convergence to the local minima.

Spencer Frei, Difan Zou, Zixiang Chen et al.

We consider a binary classification problem when the data comes from a mixture of two rotationally symmetric distributions satisfying concentration and anti-concentration properties enjoyed by log-concave distributions among others. We show that there exists a universal constant $C_{\mathrm{err}}>0$ such that if a pseudolabeler $\boldsymbolβ_{\mathrm{pl}}$ can achieve classification error at most $C_{\mathrm{err}}$, then for any $\varepsilon>0$, an iterative self-training algorithm initialized at $\boldsymbolβ_0 := \boldsymbolβ_{\mathrm{pl}}$ using pseudolabels $\hat y = \mathrm{sgn}(\langle \boldsymbolβ_t, \mathbf{x}\rangle)$ and using at most $\tilde O(d/\varepsilon^2)$ unlabeled examples suffices to learn the Bayes-optimal classifier up to $\varepsilon$ error, where $d$ is the ambient dimension. That is, self-training converts weak learners to strong learners using only unlabeled examples. We additionally show that by running gradient descent on the logistic loss one can obtain a pseudolabeler $\boldsymbolβ_{\mathrm{pl}}$ with classification error $C_{\mathrm{err}}$ using only $O(d)$ labeled examples (i.e., independent of $\varepsilon$). Together our results imply that mixture models can be learned to within $\varepsilon$ of the Bayes-optimal accuracy using at most $O(d)$ labeled examples and $\tilde O(d/\varepsilon^2)$ unlabeled examples by way of a semi-supervised self-training algorithm.

Zixiang Chen, Dongruo Zhou, Quanquan Gu

We study reinforcement learning for two-player zero-sum Markov games with simultaneous moves in the finite-horizon setting, where the transition kernel of the underlying Markov games can be parameterized by a linear function over the current state, both players' actions and the next state. In particular, we assume that we can control both players and aim to find the Nash Equilibrium by minimizing the duality gap. We propose an algorithm Nash-UCRL based on the principle "Optimism-in-Face-of-Uncertainty". Our algorithm only needs to find a Coarse Correlated Equilibrium (CCE), which is computationally efficient. Specifically, we show that Nash-UCRL can provably achieve an $\tilde{O}(dH\sqrt{T})$ regret, where $d$ is the linear function dimension, $H$ is the length of the game and $T$ is the total number of steps in the game. To assess the optimality of our algorithm, we also prove an $\tildeΩ( dH\sqrt{T})$ lower bound on the regret. Our upper bound matches the lower bound up to logarithmic factors, which suggests the optimality of our algorithm.

Zixiang Chen, Yuan Cao, Quanquan Gu et al.

A recent breakthrough in deep learning theory shows that the training of over-parameterized deep neural networks can be characterized by a kernel function called \textit{neural tangent kernel} (NTK). However, it is known that this type of results does not perfectly match the practice, as NTK-based analysis requires the network weights to stay very close to their initialization throughout training, and cannot handle regularizers or gradient noises. In this paper, we provide a generalized neural tangent kernel analysis and show that noisy gradient descent with weight decay can still exhibit a "kernel-like" behavior. This implies that the training loss converges linearly up to a certain accuracy. We also establish a novel generalization error bound for two-layer neural networks trained by noisy gradient descent with weight decay.

Zixiang Chen, Yuan Cao, Difan Zou et al.

A recent line of research on deep learning focuses on the extremely over-parameterized setting, and shows that when the network width is larger than a high degree polynomial of the training sample size $n$ and the inverse of the target error $ε^{-1}$, deep neural networks learned by (stochastic) gradient descent enjoy nice optimization and generalization guarantees. Very recently, it is shown that under certain margin assumptions on the training data, a polylogarithmic width condition suffices for two-layer ReLU networks to converge and generalize (Ji and Telgarsky, 2019). However, whether deep neural networks can be learned with such a mild over-parameterization is still an open question. In this work, we answer this question affirmatively and establish sharper learning guarantees for deep ReLU networks trained by (stochastic) gradient descent. In specific, under certain assumptions made in previous work, our optimization and generalization guarantees hold with network width polylogarithmic in $n$ and $ε^{-1}$. Our results push the study of over-parameterized deep neural networks towards more practical settings.

Tianyang Hu, Zixiang Chen, Hanxi Sun et al.

We propose two novel samplers to generate high-quality samples from a given (un-normalized) probability density. Motivated by the success of generative adversarial networks, we construct our samplers using deep neural networks that transform a reference distribution to the target distribution. Training schemes are developed to minimize two variations of the Stein discrepancy, which is designed to work with un-normalized densities. Once trained, our samplers are able to generate samples instantaneously. We show that the proposed methods are theoretically sound and experience fewer convergence issues compared with traditional sampling approaches according to our empirical studies.