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Theory · Learning Theory

Shaojie Li, Pengwei Tang, Bowei Zhu, Yong Liu

Stochastic gradient descent with momentum (SGDM) is widely used in machine learning, yet high-probability learning bounds for SGDM in non-convex settings remain scarce. In this paper, we provide high-probability convergence bounds and generalization bounds for SGDM. First, we establish such bounds for the gradient norm in the general non-convex case. The resulting convergence bounds are tighter than existing theoretical results, and the obtained generalization bounds seem to be the first for SGDM. Next, under the Polyak-{\L}ojasiewicz condition, we derive bounds for the function-value error instead of the gradient norm, and the corresponding learning rates are faster than in the general non-convex case. Finally, by additionally assuming a mild Bernstein condition on the gradient, we obtain even sharper generalization bounds whose learning rates can reach $\widetilde{\mathcal{O}}(1/n^2)$ in the low-noise regime, where $n$ is the sample size. Overall, we provide a systematic study of high-probability learning bounds for non-convex SGDM.

Optimization · Non-Convex

Pranav Vaidhyanathan, Lucas Schorling, Natalia Ares, Michael Osborne

We introduce Velocity-Regularized Adam (VRAdam), a physics-inspired optimizer for training deep neural networks that draws on ideas from quartic terms for kinetic energy with its stabilizing effects on various system dynamics. Previous algorithms, including the ubiquitous Adam, operate at the so-called adaptive edge of stability regime during training, leading to rapid oscillations and slowed convergence of loss. However, VRAdam adds a higher order penalty on the learning rate based on the velocity such that the algorithm automatically slows down whenever weight updates become large. In practice, we observe that the effective dynamic learning rate shrinks in high-velocity regimes, and damping oscillations. By combining this velocity‑based regularizer for global damping with Adam’s per‑parameter scaling, we create a powerful hybrid optimizer. For this optimizer, we provide rigorous theoretical analysis of operation at the edge of stability from a physical and control perspective for the momentum. Furthermore, we derive convergence bounds with the rate $\mathcal{O}(\ln(N)/\sqrt{N})$ for a stochastic non‑convex objective under mild assumptions. We demonstrate that VRAdam exceeds the performance against standard optimizers including AdamW. We benchmark various tasks such as image classification, language modeling, and generative modeling using diverse architectures and training methodologies including Convolutional Neural Networks (CNNs), Transformers, and GFlowNets.

Theory · Optimization

Savelii Chezhegov, Daniela Angela Parletta, Andrea Paudice, Eduard Gorbunov

We study the problem of minimizing a convex objective when only noisy gradient estimates are available. Assuming that stochastic gradients have finite $\alpha$-th moments for some $\alpha \in (1,2]$, we establish - for the first time - a high-probability convergence guarantee for the last iterate of clipped stochastic gradient descent (Clipped-SGD) on smooth objectives. In particular, we prove a rate of $1/K^{(2\alpha-2)/(3\alpha)}$ with only polylogarithmic dependence on the confidence parameter. In addition, we introduce a new technique for deriving in-expectation convergence guarantees from high-probability bounds for methods with almost surely bounded updates, and apply it to obtain expectation guarantees for Clipped-SGD. Finally, we complement our theoretical analysis with empirical results that support and illustrate our findings.

Optimization · Non-Convex

Fanfan Ji, Xiaotong Yuan

Online-to-non-convex conversion (O2NC) is an online updates learning framework for producing Goldstein $(\delta,\epsilon)$-stationary points of non-smooth non-convex functions with optimal oracle complexity $\mathcal{O}(\delta^{-1} \epsilon^{-3})$. Subject to auxiliary \emph{random interpolation or scaling}, O2NC recapitulates the stochastic gradient descent with momentum (SGDM) algorithm popularly used for training neural networks. Randomization, however, introduces deviations from practical SGDM. So a natural question arises: Can we derandomize O2NC to achieve the same optimal guarantees while resembling SGDM? On the negative side, the general answer is \emph{no} due to the impossibility results of~\citet{jordan23deterministic}, showing that no dimension-free rate can be achieved by deterministic algorithms. On the positive side, as the primary contribution of the present work, we show that O2NC can be naturally derandomized for \emph{weakly convex} functions. Remarkably, our deterministic algorithm converges at an optimal rate as long as the weak convexity parameter is no larger than $\mathcal{O}(\delta^{-1}\epsilon^{-1})$. In other words, the stronger stationarity is expected, the higher non-convexity can be tolerated by our optimizer. Additionally, we develop a periodically restarted variant of our method to enable more progressive updates when the iterates are far from stationarity. The resulting algorithm, which can be viewed as a momentum-restarted variant of SGDM, has been empirically demonstrated to be effective and efficient for training ResNet and ViT models.

Optimization · Non-Convex

Jinghui Yuan, Xie Fangyuan, Feiping Nie, Xuelong Li

The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and compared with other existing relaxations, it can flexibly incorporate class information. We prove that this relaxation forms a manifold, which we call the Relaxed Indicator Matrix Manifold (RIM manifold). Based on Riemannian geometry, we develop a Riemannian toolbox for optimization on the RIM manifold. Specifically, we provide several methods of Retraction, including a fast Retraction method to obtain geodesics. We point out that the RIM manifold is a generalization of the double stochastic manifold, and it is much faster than existing methods on the double stochastic manifold, which has a complexity of \( \mathcal{O}(n^3) \), while RIM manifold optimization is \( \mathcal{O}(n) \) and often yields better results. We conducted extensive experiments, including image denoising, with millions of variables to support our conclusion, and applied the RIM manifold to Ratio Cut, we provide a rigorous convergence proof and achieve clustering results that outperform the state-of-the-art methods. Our Code is presented in Appendix H.

Optimization · Non-Convex

Xuan Tang, Jichu Li, Difan Zou

The rapid scaling of large language models (LLMs) has made low-precision training essential for reducing memory, improving efficiency, and enabling larger models and datasets. Existing convergence theories for adaptive optimizers, however, assume all components are exact and neglect hardware-aware quantization, leaving open the question of why low-precision training remains effective. We introduce the first theoretical framework for analyzing the convergence of adaptive optimizers, including Adam and Muon, under floating-point quantization of gradients, weights, and optimizer states (e.g., moment estimates). Within this framework, we derive convergence rates on smooth non-convex objectives under standard stochastic gradient assumptions, explicitly characterizing how quantization errors from different components affect convergence. We show that both algorithms retain rates close to their full-precision counterparts provided mantissa length scales only logarithmically with the number of iterations. Our analysis further reveals that Adam is highly sensitive to weights and second-moment quantization due to its reliance on $\beta_2 \to 1$, while Muon requires weaker error control and is thus potentially more robust. These results narrow the gap between empirical success and theoretical understanding of low-precision training methods. Numerical experiments on synthetic and real-world data corroborate our theory.

Wenhan Xian, Feihu Huang, Heng Huang

Hierarchical optimization is attracting significant attentions as it can be applied to a broad range of machine learning tasks. Recently, many algorithms are proposed to improve the theoretical results of minimax and bilevel optimizations. Among these works, a core issue that has not been well studies is to escape saddle point and find local minimum. In this paper, thus, we investigate the methods to achieve second-order optimality for nonconvex minimax and bilevel optimization. Specifically, we propose a new algorithm named PRGDA without the computation of second order derivative of the primal function. In nonconvex-strongly-concave minimax optimization, we prove that our algorithm can find a second-order stationary point with the gradient complexity that matches state-of-the-art result to find first-order stationary point. To our best knowledge, PRGDA is the first stochastic algorithm that is guaranteed to obtain the second-order stationary point for nonconvex minimax problems. In nonconvex-strongly-convex bilevel optimization, our method also achieves better gradient complexity to find local minimum. Finally, we conduct two numerical experiments to validate the performance of our new method.

Xiaokang Pan, Jin Liu, Hulin Kuang, Youqi Li, Lixing Chen, Zhe Qu

The use of estimators instead of stochastic gradients for updates has been shown to improve algorithm convergence rates of, but their impact on generalization remains under-explored. In this paper, we investigate how estimators influence generalization. Our focus is on two widely studied problems: stochastic optimization (SO) and stochastic compositional optimization (SCO), both under convex and non-convex settings. For SO problems, we first analyze the generalization error of the STORM algorithm as a foundational step. We then extend our analysis to SCO problems by introducing an algorithmic framework that encompasses several popular algorithmic approaches. Through this framework, we conduct a generalization analysis, uncovering new insights into the impact of estimators on generalization. Subsequently, we provide a detailed analysis of three specific algorithms within this framework: SCGD, SCSC, and COVER, to explore the effects of different estimator strategies. Furthermore, in the context of SCO, we propose a novel definition of stability and a new decomposition of excess risk in the non-convex setting. Our analysis indicates two key findings: (1) In SCO problems, eliminating the estimator for the gradient of the inner function does not impact generalization performance while significantly reducing computational and storage overhead. (2) Faster convergence rates are consistently associated with better generalization performance.

Abdoulatif Cissé, Xenophon Evangelopoulos, Vladimir V. Gusev, Andrew I. Cooper

Many important scientific problems involve multivariate optimization coupled with slow and laborious experimental measurements. These high-dimensional searches can be defined by complex, non-convex optimization landscapes that resemble needle-in-a-haystack surfaces, leading to entrapment in local minima. Contextualizing optimizers with human domain knowledge is a powerful approach to guide searches to localized fruitful regions. However, this approach is susceptible to human confirmation bias. It is also challenging for domain experts to keep track of the rapidly expanding scientific literature. Here, we propose the use of Large Language Models (LLMs) for contextualizing Bayesian optimization (BO) via a hybrid optimization framework that intelligently and economically blends stochastic inference with domain knowledge-based insights from the LLM, which is used to suggest new, better-performing areas of the search space for exploration. Our method fosters user engagement by offering real-time commentary on the optimization progress, explaining the reasoning behind the search strategies. We validate the effectiveness of our approach on synthetic benchmarks with up to 15 variables and demonstrate the ability of LLMs to reason in four real-world experimental tasks where context-aware suggestions boost optimization performance substantially.

Haeun Jeon, Hyunglip Bae, Minsu Park, Chanyeong Kim, Woo Chang Kim

In decision-making problems under uncertainty, predicting unknown parameters is often considered independent of the optimization part. Decision-focused learning (DFL) is a task-oriented framework that integrates prediction and optimization by adapting the predictive model to give better decisions for the corresponding task. Here, an inevitable challenge arises when computing the gradients of the optimal decision with respect to the parameters. Existing research copes with this issue by smoothly reforming surrogate optimization or constructing surrogate loss functions that mimic task loss. However, they are applied to restricted optimization domains. In this paper, we propose Locally Convex Global Loss Network (LCGLN), a global surrogate loss model that can be implemented in a general DFL paradigm. LCGLN learns task loss via a partial input convex neural network which is guaranteed to be convex for chosen inputs while keeping the non-convex global structure for the other inputs. This enables LCGLN to admit general DFL through only a single surrogate loss without any sense for choosing appropriate parametric forms. We confirm the effectiveness and flexibility of LCGLN by evaluating our proposed model with three stochastic decision-making problems.

Puning Zhao, Jiafei Wu, Zhe Liu, Chong Wang, Rongfei Fan, Qingming Li

We study convex optimization problems under differential privacy (DP). With heavy-tailed gradients, existing works achieve suboptimal rates. The main obstacle is that existing gradient estimators have suboptimal tail property, resulting in a superfluous factor of d in the union bound. In this paper, we explore algorithms achieving optimal rates of DP optimization with heavy-tailed gradients. Our first method is a simple clipping approach. Under bounded p-th order moments of gradients, with n samples, it achieves minimax optimal population risk with epsilon less than 1/d. We then propose an iterative updating method, which is more complex but achieves this rate for all epsilon smaller than 1. The results significantly improve over existing methods. Such improvement relies on a careful treatment of the tail behavior of gradient estimators. Our results match the minimax lower bound, indicating that the theoretical limit of stochastic convex optimization under DP is achievable.

Ruijia Zhang, Mingxi Lei, Meng Ding, Zihang Xiang, Jinhui Xu, Di Wang

In this paper, we study the problem of (finite sum) minimax optimization in the Differential Privacy (DP) model. Unlike most of the previous studies on the (strongly) convex-concave settings or loss functions satisfying the Polyak-Lojasiewicz condition, here we mainly focus on the nonconvex-strongly-concave one, which encapsulates many models in deep learning such as deep AUC maximization. Specifically, we first analyze a DP version of Stochastic Gradient Descent Ascent (SGDA) and show the utility bound in terms of the Euclidean norm of the gradient for the empirical risk function. We then propose a new method with less gradient noise variance and improve the upper bound to the best-known result for DP Empirical Risk Minimization with non-convex loss. We also discussed several lower bounds of private minimax optimization. Finally, experiments on AUC maximization, generative adversarial networks, and temporal difference learning with real-world data support our theoretical analysis.

Mingjing Xu, Peizhong Ju, Jia Liu, Haibo Yang

Multi-objective optimization (MOO) lies at the core of many machine learning (ML) applications that involve multiple, potentially conflicting objectives (e.g., multi-task learning, multi-objective reinforcement learning, among many others). Despite the long history of MOO, recent years have witnessed a surge in interest within the ML community in the development of gradient manipulation algorithms for MOO, thanks to the availability of gradient information in many ML problems. However, existing gradient manipulation methods for MOO often suffer from long training times, primarily due to the need for computing dynamic weights by solving an additional optimization problem to determine a common descent direction that can decrease all objectives simultaneously. To address this challenge, we propose a new and efficient algorithm called Periodic Stochastic Multi-Gradient Descent (PSMGD) to accelerate MOO. PSMGD is motivated by the key observation that dynamic weights across objectives exhibit small changes under minor updates over short intervals during the optimization process. Consequently, our PSMGD algorithm is designed to periodically compute these dynamic weights and utilizes them repeatedly, thereby effectively reducing the computational overload. Theoretically, we prove that PSMGD can achieve state-of-the-art convergence rates for strongly-convex, general convex, and non-convex functions. Additionally, we introduce a new computational complexity measure, termed backpropagation complexity, and demonstrate that PSMGD could achieve an objective-independent backpropagation complexity. Through extensive experiments, we verify that PSMGD can provide comparable or superior performance to state-of-the-art MOO algorithms while significantly reducing training time.

Xiaolin Hu, Zixuan Gong, Gengze Xu, Wei Liu, Jian Luan, Bin Wang, Yong Liu

Zeroth-order (ZO) optimization as the gradient-free method has become a powerful tool when the first-order gradient is unavailable or expensive to obtain, especially in decentralized learning scenarios where data and computational resources are distributed across multiple clients. There have been many efforts to analyze the optimization convergence rate of zeroth-order decentralized stochastic gradient descent (ZO-DSGD) algorithms. However, the generalization of these methods has not been well studied. In this paper, we provide a generalization analysis of ZO-DSGD with changing topology, where the clients run zeroth-order SGD with local data and communicate with each other according to time-varying topology. We systematically analyze the generalization error in convex, strongly convex, and non-convex cases. The obtained results in the convex and strongly convex cases with zeroth-order oracles recover the results of SGD. Moreover, the generalization bounds derived in non-convex cases align with that of DSGD. To capture the influence of communication topology on the generalization performance, we analyze local generalization bounds concerning local models held at different clients. The obtained results reflect the influence of the number of clients, local sample size, and topology on the generalization error. To the best of our knowledge, this is the first work that provides a generalization analysis of zeroth-order decentralized stochastic gradient descent methods and recovers the results of SGD.

Jun Chen, Hong Chen, Bin Gu, Guodong Liu, Yingjie Wang, Weifu Li

In recent years, there have been a growing number of works studying the generalization properties of stochastic gradient descent (SGD) from the perspective of algorithmic stability. However, few of them devote to simultaneously studying the generalization and optimization for the non-convex setting, especially pairwise SGD with heavy-tailed gradient noise. This paper considers the impact of the heavy-tailed gradient noise obeying sub-Weibull distribution on the stability-based learning guarantees for non-convex pairwise SGD by investigating its generalization and optimization jointly. Specifically, based on two novel pairwise uniform model stability tools, we firstly bound the generalization error of pairwise SGD in the general non-convex setting after bridging the quantitative relationships between stability and generalization error. Then, we further consider the practical heavy-tailed sub-Weibull gradient noise condition to establish a refined generalization bound without the bounded gradient condition. Finally, sharper error bounds for generalization and optimization are built by introducing the gradient dominance condition. Comparing these results reveals that sub-Weibull gradient noise brings some positive dependencies on the heavy-tailed strength for generalization and optimization. Furthermore, we extend our analysis to the corresponding pairwise minibatch SGD and derive the first stability-based near-optimal generalization and optimization bounds which are consistent with many empirical observations.

Rayna Andreeva, James Ward, Primoz Skraba, Jie Gao, Rik Sarkar

Metric magnitude of a point cloud is a measure of its ``size." It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization algorithms. But its usability is limited due to the computational cost when the dataset is large or when the computation must be carried out repeatedly (e.g. in model training). In this paper, we study the magnitude computation problem, and show efficient ways of approximating it. We show that it can be cast as a convex optimization problem, but not as a submodular optimization. The paper describes two new algorithms -- an iterative approximation algorithm that converges fast and is accurate in practice, and a subset selection method that makes the computation even faster. It has previously been proposed that the magnitude of model sequences generated during stochastic gradient descent is correlated to the generalization gap. Extension of this result using our more scalable algorithms shows that longer sequences bear higher correlations. We also describe new applications of magnitude in machine learning -- as an effective regularizer for neural network training, and as a novel clustering criterion.

Antonios Alexos, Ian Domingo, Pierre Baldi

We present theory of synaptic neural balance and we show experimentally that synaptic neural balance can improve deep learning speed, and accuracy, even in data-scarce environments. Given an additive cost function (regularizer) of the synaptic weights, a neuron is said to be in balance if the total cost of its incoming weights is equal to the total cost of its outgoing weights. For large classes of networks, activation functions, and regularizers, neurons can be balanced fully or partially using scaling operations that do not change their functionality. Furthermore, these balancing operations are associated with a strictly convex optimization problem with a single optimum and can be carried out in any order. In our simulations, we systematically observe that: (1) Fully balancing before training results in better performance as compared to several other training approaches; (2) Interleaving partial (layer-wise) balancing and stochastic gradient descent steps during training results in faster learning convergence and better overall accuracy (with L1 balancing converging faster than L2 balancing); and (3) When given limited training data, neural balanced models outperform plain or regularized models; and this is observed in both feedforward and recurrent networks. In short, the evidence supports that neural balancing operations could be added to the arsenal of methods used to regularize and train neural networks. Furthermore, balancing operations are entirely local and can be carried out asynchronously, making them plausible for biological or neuromorphic systems.

Bingchen Wang, Zhaoxuan Wu, Fusheng Liu, Bryan Kian Hsiang Low

Collaborative machine learning (CML) provides a promising paradigm for democratizing advanced technologies by enabling cost-sharing among participants. However, the potential for rent-seeking behaviors among parties can undermine such collaborations. Contract theory presents a viable solution by rewarding participants with models of varying accuracy based on their contributions. However, unlike monetary compensation, using models as rewards introduces unique challenges, particularly due to the stochastic nature of these rewards when contribution costs are privately held information. This paper formalizes the optimal contracting problem within CML and proposes a transformation that simplifies the non-convex optimization problem into one that can be solved through convex optimization algorithms. We conduct a detailed analysis of the properties that an optimal contract must satisfy when models serve as the rewards, and we explore the potential benefits and welfare implications of these contract-driven CML schemes through numerical experiments.

Kangkang Deng, Jiang Hu

This paper studies decentralized optimization over a compact submanifold within a communication network of n nodes, where each node possesses a smooth non-convex local cost function, and the goal is to jointly minimize the sum of these local costs. We focus particularly on the online setting, where local data is processed in real-time as it streams in, without the need for full data storage. We propose a decentralized projected Riemannian stochastic recursive momentum (DPRSRM) method that employs local hybrid stochastic gradient estimators and uses the network to track the global gradient. DPRSRM achieves an oracle complexity of O(epsilon^(-3/2)), outperforming existing methods that have at most O(epsilon^(-2)) complexity. Our method requires only O(1) gradient evaluations per iteration for each local node and does not require restarting with a large batch gradient. Furthermore, we demonstrate the effectiveness of our proposed methods compared to state-of-the-art ones through numerical experiments on principal component analysis problems and low-rank matrix completion.

Optimization · Stochastic & Convex Optimization

Yi He, Xingyu Zhou

Policy optimization (PO) is a cornerstone of modern reinforcement learning (RL), with diverse applications spanning robotics, healthcare, and large language model training. The increasing deployment of PO in sensitive domains, however, raises significant privacy concerns. In this paper, we initiate a theoretical study of differentially private policy optimization, focusing explicitly on its sample complexity. We first formalize an appropriate definition of differential privacy (DP) tailored to PO, addressing the inherent challenges arising from on-policy learning dynamics and the subtlety involved in defining the unit of privacy. We then systematically analyze the sample complexity of widely-used PO algorithms, including policy gradient (PG), natural policy gradient (NPG) and more, under DP constraints and various settings, via a unified framework. Our theoretical results demonstrate that privacy costs can often manifest as lower-order terms in the sample complexity, while also highlighting subtle yet important observations in private PO settings. These offer valuable practical insights for privacy-preserving PO algorithms.