Generalization in Deep Neural Networks: Minimax Rates for Gradient Methods

Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory. While recent advances, particularly works under the Neural Tangent Kernel (NTK) regime, have shed light on the behavior of shallow architectures, the statistical generalization properties of deep neural networks (DNNs), especially in regression tasks, remain far less understood. In this paper, we make significant progress toward closing this gap by providing a comprehensive generalization analysis of DNNs trained using gradient-based methods. First, we establish, for the first time, a crucial connection between the learning dynamics of a DNN with smooth activation functions trained via gradient-based methods and those of kernel methods, showing that gradient-based methods on over-parameterized DNNs can fully inherit the favorable learning dynamics of their kernel counterparts. Building on this connection and the well-established optimality of kernel methods, we derive the first known minimax-optimal rates for the excess population risk of both gradient descent (GD) and stochastic gradient descent (SGD), under the assumption that network width scales polynomially with the sample size. Our results demonstrate that, with sufficient width, DNNs trained by GD or SGD can achieve generalization performance comparable to kernel-based methods.

Optimal Rates for Generalization of Gradient Descent Methods with Deep Neural Networks

Recent progress has been made in understanding the statistical generalization performance of gradient descent methods for overparameterized neural networks within the neural tangent kernel (NTK) regime. However, most of the existing work on regression problems is limited to shallow network architectures, leaving a notable gap in the theory of deep neural networks. This paper addresses this gap by presenting a comprehensive generalization analysis for deep ReLU networks trained using gradient descent (GD) and stochastic gradient descent (SGD). Specifically, we establish the first known minimax-optimal rates of excess population risk for both GD and SGD with deep ReLU networks, under the assumption that the network width scales polynomially with respect to the network depth and training sample size. Our results demonstrate that with sufficient width, gradient descent methods for deep ReLU networks can achieve optimal generalization rates on par with kernel methods.

Statistical Consistency and Generalization of Contrastive Representation Learning

Contrastive representation learning (CRL) underpins many modern foundation models. Despite recent theoretical progress, existing analyses suffer from several key limitations: (i) the statistical consistency of CRL remains poorly understood; (ii) available generalization bounds deteriorate as the number of negative samples increases, contradicting the empirical benefits of large negative sets; and (iii) the retrieval performance of CRL has received limited theoretical attention. In this paper, we develop a unified statistical learning theory for CRL. For downstream tasks, we evaluate retrieval quality using an AUC-type population criterion and show that the contrastive loss is \emph{statistically consistent} with optimal ranking. We further establish a \emph{calibration-style inequality} that quantitatively relates excess contrastive risk to excess retrieval suboptimality. For upstream training, we study both supervised and self-supervised contrastive objectives and derive generalization bounds of order $O(1/m + 1/\sqrt{n})$ and $O(1/\sqrt{m} + 1/\sqrt{n})$, respectively, where $m$ denotes the number of negative samples and $n$ the number of anchor points. These bounds not only explain the empirical advantages of large negative sets but also reveal an explicit trade-off between $m$ and $n$. Extensive experiments on large-scale vision--language models corroborate our theoretical predictions.

i-PhysGaussian: Implicit Physical Simulation for 3D Gaussian Splatting

Physical simulation predicts future states of objects based on material properties and external loads, enabling blueprints for both Industry and Engineering to conduct risk management. Current 3D reconstruction-based simulators typically rely on explicit, step-wise updates, which are sensitive to step time and suffer from rapid accuracy degradation under complicated scenarios, such as high-stiffness materials or quasi-static movement. To address this, we introduce i-PhysGaussian, a framework that couples 3D Gaussian Splatting (3DGS) with an implicit Material Point Method (MPM) integrator. Unlike explicit methods, our solution obtains an end-of-step state by minimizing a momentum-balance residual through implicit Newton-type optimization with a GMRES solver. This formulation significantly reduces time-step sensitivity and ensures physical consistency. Our results demonstrate that i-PhysGaussian maintains stability at up to 20x larger time steps than explicit baselines, preserving structural coherence and smooth motion even in complex dynamic transitions.

On Discriminative Probabilistic Modeling for Self-Supervised Representation Learning

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

Differentially Private Non-convex Learning for Multi-layer Neural Networks

This paper focuses on the problem of Differentially Private Stochastic Optimization for (multi-layer) fully connected neural networks with a single output node. In the first part, we examine cases with no hidden nodes, specifically focusing on Generalized Linear Models (GLMs). We investigate the well-specific model where the random noise possesses a zero mean, and the link function is both bounded and Lipschitz continuous. We propose several algorithms and our analysis demonstrates the feasibility of achieving an excess population risk that remains invariant to the data dimension. We also delve into the scenario involving the ReLU link function, and our findings mirror those of the bounded link function. We conclude this section by contrasting well-specified and misspecified models, using ReLU regression as a representative example. In the second part of the paper, we extend our ideas to two-layer neural networks with sigmoid or ReLU activation functions in the well-specified model. In the third part, we study the theoretical guarantees of DP-SGD in Abadi et al. (2016) for fully connected multi-layer neural networks. By utilizing recent advances in Neural Tangent Kernel theory, we provide the first excess population risk when both the sample size and the width of the network are sufficiently large. Additionally, we discuss the role of some parameters in DP-SGD regarding their utility, both theoretically and empirically.

Outlier Robust Adversarial Training

Supervised learning models are challenged by the intrinsic complexities of training data such as outliers and minority subpopulations and intentional attacks at inference time with adversarial samples. While traditional robust learning methods and the recent adversarial training approaches are designed to handle each of the two challenges, to date, no work has been done to develop models that are robust with regard to the low-quality training data and the potential adversarial attack at inference time simultaneously. It is for this reason that we introduce Outlier Robust Adversarial Training (ORAT) in this work. ORAT is based on a bi-level optimization formulation of adversarial training with a robust rank-based loss function. Theoretically, we show that the learning objective of ORAT satisfies the $\mathcal{H}$-consistency in binary classification, which establishes it as a proper surrogate to adversarial 0/1 loss. Furthermore, we analyze its generalization ability and provide uniform convergence rates in high probability. ORAT can be optimized with a simple algorithm. Experimental evaluations on three benchmark datasets demonstrate the effectiveness and robustness of ORAT in handling outliers and adversarial attacks. Our code is available at https://github.com/discovershu/ORAT.

Stability and Generalization of Stochastic Compositional Gradient Descent Algorithms

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

Three-Way Trade-Off in Multi-Objective Learning: Optimization, Generalization and Conflict-Avoidance

Multi-objective learning (MOL) problems often arise in emerging machine learning problems when there are multiple learning criteria or multiple learning tasks. Recent works have developed various dynamic weighting algorithms for MOL such as MGDA and its variants, where the central idea is to find an update direction that avoids conflicts among objectives. Albeit its appealing intuition, empirical studies show that dynamic weighting methods may not always outperform static ones. To understand this theory-practical gap, we focus on a new stochastic variant of MGDA - the Multi-objective gradient with Double sampling (MoDo) algorithm, and study the generalization performance of the dynamic weighting-based MoDo and its interplay with optimization through the lens of algorithm stability. Perhaps surprisingly, we find that the key rationale behind MGDA -- updating along conflict-avoidant direction - may hinder dynamic weighting algorithms from achieving the optimal ${\cal O}(1/\sqrt{n})$ population risk, where $n$ is the number of training samples. We further demonstrate the variability of dynamic weights on the three-way trade-off among optimization, generalization, and conflict avoidance that is unique in MOL.

Generalization Guarantees of Gradient Descent for Multi-Layer Neural Networks

Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling parameters. In this paper, we greatly extend the previous work \cite{lei2022stability,richards2021stability} by conducting a comprehensive stability and generalization analysis of GD for multi-layer NNs. For two-layer NNs, our results are established under general network scaling parameters, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of over-parameterization. As a direct application of our general findings, we derive the excess risk rate of $O(1/\sqrt{n})$ for GD algorithms in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for under-parameterized and over-parameterized NNs trained by GD to attain the desired risk rate of $O(1/\sqrt{n})$. Moreover, we demonstrate that as the scaling parameter increases or the network complexity decreases, less over-parameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of $O(1/n)$ for GD in both two-layer and three-layer NNs.