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Theory · Reinforcement Learning and Planning

Aditya Gangrade, Venkatesh Saligrama

We study safe linear bandits (SLBs), where an agent selects actions from a convex set to maximize an unknown linear objective subject to unknown linear constraints in each round. Existing methods for SLBs provide strong regret guarantees, but require solving expensive optimization problems (e.g., second-order cones, NP hard programs). To address this, we propose Constrained Linear Thompson Sampling (COLTS), a sampling-based framework that selects actions by solving perturbed linear programs, which significantly reduces computational costs while matching the regret and risk of prior methods. We develop two main variants: S-COLTS, which ensures zero risk and ${\tilde{O}(\sqrt{d^3 T})}$ regret given a safe action, and R-COLTS, which achieves ${\tilde{O}(\sqrt{d^3 T})}$ regret and risk with no instance information. In simulations, these methods match or outperform state of the art SLB approaches while substantially improving scalability. On the technical front, we introduce a novel coupled noise design that ensures frequent 'local optimism' about the true optimum, and a scaling-based analysis to handle the per-round variability of constraints.

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.

Optimization · Bayesian & Black-box Optimization

Guido Carnevale, Giuseppe Notarstefano

Model-free optimization methods typically rely on cost samples gathered by perturbing the current solution estimate along a finite and fixed set of directions. However, at each iteration, only the current cost samples are used, while potentially informative, previously collected samples are discarded. In this work, we challenge this conventional approach by introducing a simple yet effective memory mechanism that maintains an auxiliary vector of iteratively updated cost samples. By leveraging this stored information, our method estimates descent directions through an averaging of all perturbing directions weighted by the auxiliary vector components. This results in a faster convergence without increasing the number of function queries. By interpreting the resulting algorithm as a time-varying dynamical system, we are able to establish its convergence properties in the strongly convex case. In particular, by using tools from system theory based on timescale separation, we are able to guarantee a linear convergence rate toward an arbitrarily small neighborhood of the optimal solution. Numerical simulations on regression problems demonstrate that the proposed approach significantly outperforms existing model-free optimization methods.

Optimization · Stochastic & Convex Optimization

Liang Zhang, Bingcong Li, Kiran Thekumparampil, Sewoong Oh, Michael Muehlebach, Niao He

Zeroth-order methods are extensively used in machine learning applications where gradients are infeasible or expensive to compute, such as black-box attacks, reinforcement learning, and language model fine-tuning. Existing optimization theory focuses on convergence to an arbitrary stationary point, but less is known on the implicit regularization that provides a fine-grained characterization on which particular solutions are finally reached. We show that zeroth-order optimization with the standard two-point estimator favors solutions with small trace of Hessian, which is widely used in previous work to distinguish between sharp and flat minima. We further provide convergence rates of zeroth-order optimization to approximate flat minima for convex and sufficiently smooth functions, where flat minima are defined as the minimizers that achieve the smallest trace of Hessian among all optimal solutions. Experiments on binary classification tasks with convex losses and language model fine-tuning support our theoretical findings.

Optimization

Zhiwei Zhai, Wenjing Yan, Ying-Jun Zhang

Decentralized bilevel optimization has garnered significant attention due to its critical role in solving large-scale machine learning problems. However, existing methods often rely on prior knowledge of problem parameters—such as smoothness, convexity, or communication network topologies—to determine appropriate stepsizes. In practice, these problem parameters are typically unavailable, leading to substantial manual effort for hyperparameter tuning. In this paper, we propose \textbf{AdaSDBO}, a fully problem-parameter-free algorithm for decentralized bilevel optimization with a single-loop structure. AdaSDBO leverages adaptive stepsizes based on cumulative gradient norms to update all variables simultaneously, dynamically adjusting its progress and eliminating the need for problem-specific hyperparameter tuning. Through rigorous theoretical analysis, we establish that AdaSDBO achieves a convergence rate of $\widetilde{\mathcal{O}}\left(\frac{1}{T}\right)$, matching the performance of well-tuned state-of-the-art methods up to polylogarithmic factors. Extensive numerical experiments demonstrate that AdaSDBO delivers competitive performance compared to existing decentralized bilevel optimization methods while exhibiting remarkable robustness across diverse stepsize configurations.

Jiawei Huang, Minming Li, Hu Ding

Contrastive self-supervised learning has emerged as a powerful paradigm for extracting meaningful representations without labels. While effective at capturing broad categorical distinctions, current methods often struggle to preserve the fine-grained and hierarchical relationships inherent in real-world data. From the perspective of semantic alignment, conventional contrastive learning aligns representations to semantic structure at a global level, treating the entire embedding space uniformly and frequently overlooking rich local structural information. In this paper, we propose \emph{Adaptive Multi-scale Affinity alignment (AMA-alignment)}, a framework that introduces localized contrastive objectives and a dynamic multi-scale optimization strategy to adaptively identify and refine poorly aligned regions within the embedding space. Although our model is inherently more complex due to its \emph{multi-scale} and \emph{adaptive} design, we provide the theoretical guarantees indicating that its convergence rate remains comparable to that of standard smooth non-convex optimization. We conduct a set of experiments on diverse benchmarks to show that AMA-alignment can effectively preserve hierarchical structure; moreover, AMA-alignment also outperforms existing contrastive methods on a range of downstream tasks.

Optimization · Stochastic & Convex Optimization

Da Chang, Ganzhao Yuan

Efficient optimization is essential for training large language models. Although intra-layer selective updates have been explored, a general mechanism that enables fine-grained control while ensuring convergence guarantees is still lacking. To bridge this gap, we propose \textbf{MGUP}, a novel mechanism for selective updates. \textbf{MGUP} augments standard momentum-based optimizers by applying larger step-sizes to a selected fixed proportion of parameters in each iteration, while applying smaller, non-zero step-sizes to the rest. As a nearly {plug-and-play} module, \textbf{MGUP} seamlessly integrates with optimizers such as AdamW, Lion, and Muon. This yields powerful variants such as \textbf{MGUP-AdamW}, \textbf{MGUP-Lion}, and \textbf{MGUP-Muon}. Under standard assumptions, we provide theoretical convergence guarantees for \textbf{MGUP-AdamW} (without weight decay) in stochastic optimization. Extensive experiments across diverse tasks, including MAE pretraining, LLM pretraining, and downstream fine-tuning, demonstrate that our \textbf{MGUP}-enhanced optimizers achieve superior or more stable performance compared to their original base optimizers. We offer a principled, versatile, and theoretically grounded strategy for efficient intra-layer selective updates, accelerating and stabilizing the training of large-scale models. The code is publicly available at https://github.com/MaeChd/MGUP.

Optimization

Elfarouk Harb, Yousef Yassin, Chandra Chekuri

We consider the following question: given a submodular or supermodular set function $f:2^V \to \mathbb{R}$, how should one minimize or maximize its average value $f(S)/|S|$ over non-empty subsets $S\subseteq V$? This problem generalizes several well-known objectives including Densest Subgraph (DSG), Densest Supermodular Set (DSS), and Submodular Function Minimization (SFM). Motivated by recent applications [39, 31], we formalize two new broad problems: the Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS) which allow negative and non-monotone functions. Using classical results we observe that DSS, SFM, USSS, UDSS, and MNP are all equivalent under strongly polynomial-time reductions. This equivalence enables algorithmic cross-over: methods designed for one problem can be repurposed to solve others efficiently. In particular we use the perspective of the minimum norm point in the base polyhedron of a sub/supermodular function which, via Fujishige's results, yields the dense decomposition as a byproduct. Via this perspective we show that a recent converging heuristic for DSS, \textsc{SuperGreedy++} [15, 29], and Wolfe’s minimum norm point algorithm are both universal solvers for all of these problems. On the theoretical front, we explain the observation made in recent work [39, 31] that \textsc{SuperGreedy++} appears to work well even in settings beyond DSS. Surprisingly, we also show that this simple algorithm can be used for Submodular Function Minimization, including for example that it can act as an effective minimum $st$ cut algorithm. On the empirical front, we explore the utility of several different algorithms including Fujishige-Wolfe min-norm point algorithm for recent problems. We conduct over 400 experiments across seven problem types and large-scale synthetic and real-world datasets (up to $\approx 100$ million edges). Our results reveal that methods historically considered inefficient, such as convex-programming methods, flow-based solvers, and Fujishige-Wolfe’s algorithm, outperform state-of-the-art task-specific baselines by orders of magnitude on concrete problems like HNSN [39]. These findings challenge prevailing assumptions and demonstrate that with the right framing, general optimization algorithms can be both scalable and state-of-the-art for supermodular and submodular ratio problems.

Optimization · Stochastic & Convex Optimization

Francesco Orabona, Ryan D'Orazio

The Polyak stepsize has been proven to be a fundamental stepsize in convex optimization, giving near optimal gradient descent rates across a wide range of assumptions. The universality of the Polyak stepsize has also inspired many stochastic variants, with theoretical guarantees and strong empirical performance. Despite the many theoretical results, our understanding of the convergence properties and shortcomings of the Polyak stepsize or its variants is both incomplete and fractured across different analyses. We propose a new, unified, and simple perspective for the Polyak stepsize and its variants as gradient descent on a surrogate loss. We show that each variant is equivalent to minimize a surrogate function with stepsizes that adapt to a guaranteed local curvature. Our general surrogate loss perspective is then used to provide a unified analysis of existing variants across different assumptions. Moreover, we show a number of negative results proving that the non-convergence results in some of the upper bounds is indeed real.

General Machine Learning · Transfer, Multitask and Meta-learning

Wenju Sun, Qingyong Li, Wen Wang, Yang Liu, Yangliao Geng, Boyang Li

Multi-task model merging aims to consolidate knowledge from multiple fine-tuned task-specific experts into a unified model while minimizing performance degradation. Existing methods primarily approach this by minimizing differences between task-specific experts and the unified model, either from a parameter-level or a task-loss perspective. However, parameter-level methods exhibit a significant performance gap compared to the upper bound, while task-loss approaches entail costly secondary training procedures. In contrast, we observe that performance degradation closely correlates with feature drift, i.e., differences in feature representations of the same sample caused by model merging. Motivated by this observation, we propose Layer-wise Optimal Task Vector Merging (LOT Merging), a technique that explicitly minimizes feature drift between task-specific experts and the unified model in a layer-by-layer manner. LOT Merging can be formulated as a convex quadratic optimization problem, enabling us to analytically derive closed-form solutions for the parameters of linear and normalization layers. Consequently, LOT Merging achieves efficient model consolidation through basic matrix operations. Extensive experiments across vision and vision-language benchmarks demonstrate that LOT Merging significantly outperforms baseline methods, achieving improvements of up to 4.4% (ViT-B/32) over state-of-the-art approaches. The source code is available at https://github.com/SunWenJu123/model-merging.

Optimization · Discrete Optimization

Marie-Charlotte Brandenburg, Katharina Jochemko

We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and a log-likelihood loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the log-likelihood loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

Optimization · Stochastic & Convex Optimization

Yipeng Li, Xinchen Lyu, Zhenyu Liu

We aim to provide a unified convergence analysis for permutation-based Stochastic Gradient Descent (SGD), where data examples are permuted before each epoch. By examining the relations among permutations, we categorize existing permutation-based SGD algorithms into three categories: Arbitrary Permutations, Independent Permutations (including Random Reshuffling and FlipFlop Rajput et al., 2022), Dependent Permutations (including GraBs Lu et al., 2022a; Cooper et al., 2023). Existing unified analyses failed to encompass the Dependent Permutations category due to the inter-epoch permutation dependency. In this work, we propose a generalized assumption that explicitly characterizes the dependence of permutations across epochs. Building upon this assumption, we develop a unified framework for permutation-based SGD with arbitrary permutations of examples, incorporating all the existing permutation-based SGD algorithms. Furthermore, we adapt our framework for Federated Learning (FL), developing a unified framework for regularized client participation FL with arbitrary permutations of clients.

General Machine Learning · Representation Learning

Hoang Son Nguyen, Xiao Fu

Latent component identification from unknown *nonlinear* mixtures is a foundational challenge in machine learning, with applications in tasks such as self-supervised learning and causal representation learning. Prior work in *nonlinear independent component analysis* (nICA) has shown that auxiliary signals---such as weak supervision---can support *identifiability* of conditionally independent latent components. More recent approaches explore structural assumptions, like sparsity in the Jacobian of the mixing function, to relax such requirements. In this work, we introduce *Diverse Influence Component Analysis* (DICA), a framework that exploits the convex geometry of the mixing function’s Jacobian. We propose a *Jacobian Volume Maximization* (J-VolMax) criterion, which enables latent component identification by encouraging diversity in their influence on the observed variables. Under suitable conditions, this approach achieves identifiability without relying on auxiliary information, latent component independence, or Jacobian sparsity assumptions. These results extend the scope of identifiability analysis and offer a complementary perspective to existing methods.

Optimization · Stochastic & Convex Optimization

Xingyu Chen, Bokun Wang, Ming Yang, Qihang Lin, Tianbao Yang

Finite-sum Coupled Compositional Optimization (FCCO), characterized by its coupled compositional objective structure, emerges as an important optimization paradigm for addressing a wide range of machine learning problems. In this paper, we focus on a challenging class of non-convex non-smooth FCCO, where the outer functions are non-smooth weakly convex or convex and the inner functions are smooth or weakly convex. Existing state-of-the-art result face two key limitations: (1) a high iteration complexity of $O(1/\epsilon^6)$ under the assumption that the stochastic inner functions are Lipschitz continuous in expectation; (2) reliance on vanilla SGD-type updates, which are not suitable for deep learning applications. Our main contributions are two fold: (i) We propose stochastic momentum methods tailored for non-smooth FCCO that come with provable convergence guarantees; (ii) We establish a **new state-of-the-art** iteration complexity of $O(1/\epsilon^5)$. Moreover, we apply our algorithms to multiple inequality constrained non-convex optimization problems involving smooth or weakly convex functional inequality constraints. By optimizing a smoothed hinge penalty based formulation, we achieve a **new state-of-the-art** complexity of $O(1/\epsilon^5)$ for finding an (nearly) $\epsilon$-level KKT solution. Experiments on three tasks demonstrate the effectiveness of the proposed algorithms.

Optimization · Stochastic & Convex Optimization

Junyu Huang, Qilong Feng, Zhen Zhang, Beirong Cui, Jianxin Wang

Local search is a powerful clustering technique that provides high-quality solutions with theoretical guarantees. With distance-based sampling strategies, local search methods can achieve constant approximations for clustering with linear running time in data size. Despite their effectiveness, existing algorithms still face scalability issues as they require scanning the entire dataset for iterative center swaps. This typically leads to an O(ndk) running time, where n is the data size, d is the dimension, k is the number of clusters. To further improve the efficiency of local search algorithms, we propose new methods based on adaptive sampling and bandit strategies. Specifically, adaptive sampling can well approximate the distance-based sampling distribution without maintaining pairwise distances between data points and the centers, enabling fast and accurate sampling in sublinear time after an $\tilde{O}(nd)$ time preprocessing step. The bandit strategy models the best swap pair selection as a bandit problem, where a grouping strategy is proposed for fast identification of the optimal swap pair. With these techniques, our proposed algorithm can achieve constant approximation in expected running time $\tilde{O}(nd + k^4)$ under mild assumptions on optimal clusters and swap pair distributions. Our approach also extends naturally to the k-median objective, achieving constant approximation in expected running time $\tilde{O}(nd + \sqrt{n}k^3)$ without distributional assumptions. Empirical results demonstrate that our algorithm achieves up to 1000× speedup over existing local search methods on datasets with 100 million points, while delivering comparable clustering quality. Compared to coreset-based approaches, it also provides up to 80× speedup and consistently yields better clustering results.

Theory · Learning Theory

Qinglun Li, Yingqi Liu, Miao Zhang, Xiaochun Cao, Quanjun Yin, Li Shen

Decentralized training removes the centralized server, making it a communication-efficient approach that can significantly improve training efficiency, but it often suffers from degraded performance compared to centralized training. Multi-Gossip Steps (MGS) serve as a simple yet effective bridge between decentralized and centralized training, significantly reducing experiment performance gaps. However, the theoretical reasons for its effectiveness and whether this gap can be fully eliminated by MGS remain open questions. In this paper, we derive upper bounds on the generalization error and excess error of MGS using stability analysis, systematically answering these two key questions. 1). Optimization Error Reduction: MGS reduces the optimization error bound at an exponential rate, thereby exponentially tightening the generalization error bound and enabling convergence to better solutions. 2). Gap to Centralization: Even as MGS approaches infinity, a non-negligible gap in generalization error remains compared to centralized mini-batch SGD ($\mathcal{O}(T^{\frac{c\beta}{c\beta +1}}/{n m})$ in centralized and $\mathcal{O}(T^{\frac{2c\beta}{2c\beta +2}}/{n m^{\frac{1}{2c\beta +2}}})$ in decentralized). Furthermore, we provide the first unified analysis of how factors like learning rate, data heterogeneity, node count, per-node sample size, and communication topology impact the generalization of MGS under non-convex settings without the bounded gradients assumption, filling a critical theoretical gap in decentralized training. Finally, promising experiments on CIFAR datasets support our theoretical findings.

Theory · Learning Theory

Shivani Agarwal

A central question in the theory of machine learning concerns the identification of classes of data distributions for which one can provide computationally efficient learning algorithms with provable statistical learning guarantees. Indeed, in the context of probably approximately correct (PAC) learning, there has been much interest in exploring intermediate PAC learning models that, unlike the realizable PAC learning setting, allow for some stochasticity in the labels, and unlike the fully agnostic PAC learning setting, also admit computationally efficient learning algorithms with finite sample complexity bounds. Some examples of such models include random classification noise (RCN), probabilistic concepts, Massart noise, and generalized linear models (GLMs); in general, most of this work has focused on binary classification problems. In this paper, we study what we call realizable-statistic models (RSMs), wherein we allow stochastic labels but assume that some vector-valued statistic of the conditional label distribution comes from some known function class. RSMs are a flexible class of models that interpolate between the realizable and fully agnostic settings, and that also recover several previously studied models as special cases. We show that for a broad range of RSM learning problems, where the statistic of interest can be accurately estimated via a convex ‘strongly proper composite’ surrogate loss, minimizing this convex surrogate loss yields a computationally efficient learning algorithm with finite sample complexity bounds. We then apply this result to show that various commonly used (and in some cases, not so commonly used) convex surrogate risk minimization algorithms yield computationally efficient learning algorithms with finite sample complexity bounds for a variety of RSM learning problems including binary classification, multiclass classification, multi-label prediction, and subset ranking. For the special case of binary classification with sigmoid-of-linear class probabilities (also a special case of GLMs), our results show that minimizing the standard binary logistic loss has a similar sample complexity as the GLM-tron algorithm of Kakade et al. (2011), but is computationally more efficient. In terms of the distribution over the domain/instance space, our results are all distribution-independent. To our knowledge, these are the first such results for PAC learning with stochastic labels for such a broad range of learning problems.

Theory · Learning Theory

Yuzhou Cao, Han Bao, Lei Feng, Bo An

Surrogate regret bounds, also known as excess risk bounds, bridge the gap between the convergence rates of surrogate and target losses. The regret transfer is lossless if the surrogate regret bound is linear. While convex smooth surrogate losses are appealing in particular due to the efficient estimation and optimization, the existence of a trade-off between the loss smoothness and linear regret bound has been believed in the community. Under this scenario, the better optimization and estimation properties of convex smooth surrogate losses may inevitably deteriorate after undergoing the regret transfer onto a target loss. We overcome this dilemma for arbitrary discrete target losses by constructing a convex smooth surrogate loss, which entails a linear surrogate regret bound composed with a tailored prediction link. The construction is based on Fenchel--Young losses generated by the *convolutional negentropy*, which are equivalent to the infimal convolution of a generalized negentropy and the target Bayes risk. Consequently, the infimal convolution enables us to derive a smooth loss while maintaining the surrogate regret bound linear. We additionally benefit from the infimal convolution to have a consistent estimator of the underlying class probability. Our results are overall a novel demonstration of how convex analysis penetrates into optimization and statistical efficiency in risk minimization.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Haoyuan Liang, Shilei Cao, Li, Zhiyu Ye, Haohuan Fu, Juepeng Zheng

Federated learning (FL) has recently emerged as the primary approach to overcoming data silos, enabling collaborative model training without sharing sensitive or proprietary data. Parallel federated learning (PFL) aggregates models trained independently on each client’s local data, which can lead to suboptimal convergence due to limited data exposure. In contrast, Sequential Federated Learning (SFL) allows models to traverse client datasets sequentially, enhancing data utilization. However, SFL effectiveness is limited in real-world non-IID scenarios characterized by category shift (inconsistent class distributions) and domain shift (distribution discrepancies). These shifts cause two critical issues: update order sensitivity, where model performance varies significantly with the sequence of client updates, and catastrophic forgetting, where the model forgets previously learned features when trained on new client data. We propose SPFL, a novel updating method that can be integrated into existing FL methods, integrating sequential updates with parallel aggregation to enhance data utilization and ease update order sensitivity. At the same time, we give the convergence analysis of SPFL under strong convex, general convex, and non-convex conditions, proving that this update scheme is significantly better than PFL and SFL. Additionally, we introduce the Global-Local Alignment Module to mitigate catastrophic forgetting by aligning the predictions of the global model with those of the local and previous models during training. Our extensive experiments demonstrate that integrating SPFL into existing PFL methods significantly improves performance under category and domain shifts.

Optimization

Shuchang Zhang, Yaoyun Zeng, Kangkang Deng, Hongxia Wang

Learning pseudo-contractive denoisers is a fundamental challenge in the theoretical analysis of Plug-and-Play (PnP) methods and the Regularization by Denoising (RED) framework. While spectral methods attempt to address this challenge using the power iteration method, they fail to guarantee the truly pseudo-contractive property and suffer from high computational complexity. In this work, we rethink gradient step (GS) denoisers and establish a theoretical connection between GS denoisers and pseudo-contractive operators. We show that GS denoisers, with the gradients of convex potential functions parameterized by input convex neural networks (ICNNs), can achieve truly pseudo-contractive properties. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO (RED via fixed-point projection) model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments confirm that the learned GS denoiser satisfies the truly pseudo-contractive property and, when integrated into RED-PRO, provides a favorable trade-off between interpretability and empirical performance on inverse problems.