This paper addresses the problem of learning to sparsify stochastic linear bandits, where a decision-maker sequentially selects actions from a high-dimensional space subject to a sparsity constraint on the number of nonzero elements in the action vector. The key challenge lies in minimizing cumulative regret while tackling the potential NP-hardness of finding optimal sparse actions due to the inherent combinatorial structure of the problem. We propose an adaptively phased exploration and exploitation algorithmic framework, utilizing ordinary least squares for parameter learning and specialized subroutines for sparse action selection. When the action set is a Euclidean ball, optimal sparse actions can be efficiently computed, enabling us to establish a Õ(d √T) regret, where d is the dimension of the action vector and T is the time horizon length. For general convex and compact action sets where finding optimal sparse actions is intractable, we employ a greedy subroutine. For general strongly convex action sets, we derive a Õ(d √T) α-regret; for general compact sets lacking strong convexity, we establish a Õ(d T^(2/3)) α-regret, where α pertains to the approximation ratio of the greedy algorithm. Finally, we validate the performance of our algorithms using extensive experiments.
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Stochastic Decentralized Optimization of Non-Smooth Convex and Convex-Concave Problems over Time-Varying Networks
PDF ↗We study non-smooth stochastic decentralized optimization problems over time-varying networks, where objective functions are distributed across nodes and network connections may intermittently appear or break. Specifically, we consider two settings: (i) stochastic non-smooth (strongly) convex optimization, and (ii) stochastic non-smooth (strongly) convex–(strongly) concave saddle point optimization. Convex problems of this type commonly arise in deep neural network training, while saddle point problems are central to machine learning tasks such as the training of generative adversarial networks (GANs). Prior works have primarily focused on the smooth setting, or time-invariant network scenarios. We extend the existing theory to the more general non-smooth and stochastic setting over time-varying networks and saddle point problems. Our analysis establishes upper bounds on both the number of stochastic oracle calls and communication rounds, matching lower bounds for both convex and saddle point optimization problems.
S-D-RSM: Stochastic Distributed Regularized Splitting Method for Large-Scale Convex Optimization Problems
PDF ↗This paper investigates problems of large-scale distributed composite convex optimization, with motivations from a broad range of applications, including multi-agent systems, federated learning, smart grids, wireless sensor networks, compressed sensing, and so on. Stochastic gradient descent (SGD) and its variants are commonly employed to solve such problems. However, existing algorithms often rely on vanishing step sizes, strong convexity assumptions, or entail substantial computational overhead to ensure convergence or obtain favorable complexity. To bridge the gap between theory and practice, we integrate consensus optimization and operator splitting techniques (see Problem Reformulation) to develop a novel stochastic splitting algorithm, termed the stochastic distributed regularized splitting method (S-D-RSM). In practice, S-D-RSM performs parallel updates of proximal mappings and gradient information for only a randomly selected subset of agents at each iteration. By introducing regularization terms, it effectively mitigates consensus discrepancies among distributed nodes. In contrast to conventional stochastic methods, our theoretical analysis establishes that S-D-RSM achieves global convergence without requiring diminishing step sizes or strong convexity assumptions. Furthermore, it achieves an iteration complexity of 1/epsilon with respect to both the objective function value and the consensus error. Numerical experiments show that S-D-RSM achieves up to two to three times speedup compared with state-of-the-art baselines, while maintaining comparable or better accuracy. These results not only validate the algorithm's theoretical guarantees but also demonstrate its effectiveness in practical tasks such as compressed sensing and empirical risk minimization.
This paper examines a variety of classical optimization problems, including well-known minimization tasks and more general variational inequalities. We consider a stochastic formulation of these problems and, unlike most previous work, we take into account the complex Markov nature of the noise. We also consider the geometry of the problem in an arbitrary non-Euclidean setting and propose four methods based on the Mirror Descent iteration technique. The theoretical analysis is provided for smooth and convex minimization problems and variational inequalities with Lipschitz and monotone operators. The convergence guarantees obtained are optimal for first-order stochastic methods, as evidenced by the lower bound estimates provided in this paper. In order to validate the theoretical results, we present the relevant numerical experiments on various reinforcement learning tasks.
Decentralized optimization is critical for solving large-scale machine learning problems over distributed networks, where multiple nodes collaborate through local communication. In practice, the variances of stochastic gradient estimators often differ across nodes, yet their impact on algorithm design and complexity remains unclear. To address this issue, we propose D-NSS, a decentralized algorithm with node-specific sampling, and establish its sample complexity depending on the arithmetic mean of local standard deviations, achieving tighter bounds than existing methods that rely on the worst-case or quadratic mean. We further derive a matching sample complexity lower bound under heterogeneous variance, thereby proving the optimality of this dependence. Moreover, we extend the framework with a variance reduction technique and develop D-NSS-VR, which under the mean-squared smoothness assumption attains an improved sample complexity bound while preserving the arithmetic-mean dependence. Finally, numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed algorithms.
Ensuring the safety of AI-enabled systems, particularly in high-stakes domains such as autonomous driving and healthcare, has become increasingly critical. Traditional formal verification tools fall short when faced with systems that embed both opaque, black-box AI components and complex stochastic dynamics. To address these challenges, we introduce LUCID (Learning-enabled Uncertainty-aware Certification of stochastIc Dynamical systems), a verification engine for certifying safety of black-box stochastic dynamical systems from a finite dataset of random state transitions. As such, LUCID is the first known tool capable of establishing quantified safety guarantees for such systems. Thanks to its modular architecture and extensive documentation, LUCID is designed for easy extensibility. LUCID employs a data-driven methodology rooted in control barrier certificates, which are learned directly from system transition data, to ensure formal safety guarantees. We use conditional mean embeddings to embed data into a Reproducing Kernel Hilbert Space (RKHS), where an RKHS ambiguity set is constructed that can be inflated to robustify the result to out-of-distribution behavior. A key innovation within LUCID is its use of a finite Fourier kernel expansion to reformulate a semi-infinite non-convex optimization problem into a tractable linear program. The resulting spectral barrier allows us to leverage the fast Fourier transform to generate the relaxed problem efficiently, offering a scalable yet distributionally robust framework for verifying safety. LUCID thus offers a robust and efficient verification framework, able to handle the complexities of modern black-box systems while providing formal guarantees of safety. These unique capabilities are demonstrated on challenging benchmarks.
Urban Network Security Games (UNSGs), which model the strategic allocation of limited security resources on city road networks, are critical for urban safety. However, finding a Nash Equilibrium (NE) in large-scale UNSGs is challenging due to their massive and combinatorial action spaces. One common approach to addressing these games is the Policy-Space Response Oracle (PSRO) framework, which requires computing best responses (BR) at each iteration. However, precisely computing exact BRs is impractical in large-scale games, and employing reinforcement learning to approximate BRs inevitably introduces errors that limit the overall effectiveness of the PSRO methods. Recent advancements in leveraging non-convex stochastic optimization to approximate an NE offer a promising alternative to the burdensome BR computation. However, utilizing existing stochastic optimization techniques with an unbiased loss function for UNSGs remains challenging because the action spaces are too vast to be effectively represented by neural networks. To address these issues, we introduce Tree-based Stochastic Optimization (TSO), a framework that bridges the gap between the stochastic optimization paradigm for NE-finding and the demands of UNSGs. Specifically, we employ the tree-based action representation that maps the whole action space onto a tree structure, addressing the challenge faced by neural networks in representing actions when the action space cannot be enumerated. We then incorporate this representation into the loss function and theoretically demonstrate its equivalence to the unbiased loss function. To further enhance the quality of the converged solution, we introduce a sample-and-prune mechanism that reduces the risk of being trapped in suboptimal local optima. Extensive experimental results indicate the superiority of TSO over other baseline algorithms in addressing the UNSGs.
Optimization · Stochastic
Sharpness-Aware Minimization (SAM) has established itself as a powerful and widely adopted optimizer for training machine learning models. By explicitly minimizing the sharpness of the loss landscape, SAM often improves generalization while delivering strong empirical performance. However, SAM and its variants, like most training algorithms, are sensitive to the choice of learning rate, which is typically tuned by trial and error or via schedulers. In this work, motivated by recent advances on the effectiveness of stochastic Polyak step sizes for Stochastic Gradient Descent (SGD), we derive Polyak schedulers tailored to SAM-style updates, yielding novel adaptive algorithms in both deterministic and stochastic settings. In smooth setting, for the proposed methods, we prove linear convergence for strongly convex objectives and an $O(1/T)$ rate (up to a neighborhood in the stochastic setting) for convex objectives. Numerical experiments demonstrate that the proposed Polyak schedulers match or surpass tuned SAM baselines while substantially reducing the need for learning-rate tuning.
We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite $p$-th order moments, $1 < p \leq 2$. This scheme---which we call the Lévy mirror flow (LMF)---arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when $p < 2$---the heavy noise regime---the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains $\epsilon$-optimality within $\mathcal{O}(\epsilon^{-p/(p-1)})$ time in the convex case, and within $\tilde{\mathcal{O}}(\epsilon^{-1/(p-1)})$ time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.
Optimization · Zero-order and Black-box Optimization
This paper deals with stochastic optimization problems involving Markovian noise with a zero-order oracle. We present and analyze a novel derivative-free method for solving such problems in strongly convex smooth and non-smooth settings with both one-point and two-point feedback oracles. Using a randomized batching scheme, we show that when mixing time $\tau$ of the underlying noise sequence is less than the dimension of the problem $d$, the convergence estimates of our method do not depend on $\tau$. This observation provides an efficient way to interact with Markovian stochasticity: instead of invoking the expensive first-order oracle, one should use the zero-order oracle. Finally, we complement our upper bounds with the corresponding lower bounds. This confirms the optimality of our results.
Optimization · Stochastic
We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size becomes too large. For convex problems, we show that this quantity directly impacts the suboptimality bound of the method. Most importantly, our analysis provides direct *theoretical* evidence that adaptive step size methods, such as SPS or NGN, are more robust than SGD. This allows us to quantify the advantage of these adaptive methods beyond empirical evaluation. Finally, we show through experiments that our theoretical bound qualitatively mirrors the actual performance as a function of the step size, even for nonconvex problems.
Deep Learning · Generative Models and Autoencoders
Discrete flow matching, a recent framework for modeling categorical data, has shown competitive performance with autoregressive models. However, unlike continuous flow matching, the rectification strategy cannot be applied due to the stochasticity of discrete paths, necessitating alternative methods to minimize state transitions. We propose a dynamic-optimal-transport-like minimization objective and derive its Kantorovich formulation for discrete flows with convex interpolants, where transport cost depends solely on inter-state similarity and can be optimized via minibatch strategies. We show that such methods can reduce the number of transitions up to 8 times (1024 to 128) to reach the same generative perplexity without compromising diversity. Additionally, path nondeterminism in discrete flows precludes an instantaneous change-of-variables analogue, preventing precise probability estimation available to continuous flows. We therefore propose two upper bounds on perplexity, enabling principled training, evaluation and model comparison. Finally, we introduce Multimask Flows which outperform masked flows in generative perplexity without compromising diversity, particularly when utilizing minibatch Optimal Transport.
Optimization · Convex
In distributed machine learning, efficiently training across multiple agents with heterogeneous data distributions remains a central challenge. We address the problem of stochastic, strongly convex distributed optimization by applying accelerated gradient ascent to the dual variables and multi-step stochastic gradient descent (SGD) to the primal variables in the Lagrangian formulation. This approach naturally enables local computation, as the inner SGD loops require no inter-agent communication. We prove that the method converges for any number of local updates, attaining the optimal communication complexity when local computation is sufficient. Our analysis builds on an inexact accelerated gradient framework, where the partial gradient of the Lagrangian with respect to the dual variables is treated as an inexact gradient of the dual function. A notable byproduct of this framework is an algorithm that achieves optimal reproducibility guarantees under biased gradient estimates.
Social Aspects · Privacy
Differentially private (DP) stochastic optimization algorithms are widely used in privacy-preserving deep learning, where per-sample gradient clipping and noise injection protect sensitive information. However, these operations limit existing DP algorithms to converge within a constant-radius neighborhood of the first-order stationary point, leading to solution bias and the well-known privacy-utility trade-off. To enhance model utility, we propose a novel algorithmic framework called DP-C4, which is designed to be error-Consistently-vanishing, Coupledly-clipped, solution-Calibrated, and Convergence-guaranteed. Specifically, it incorporates a carefully designed coupled clipping scheme with shifted threshold strategy, ensuring that both clipping bias and noise variance asymptotically vanish, thereby eliminating the DP-induced solution bias. Moreover, we extend existing sensitivity analysis techniques and develop a tailored privacy budget allocation to guarantee the privacy of DP-C4. Compared with the well-recognized DP-SGD, our framework injects significantly less noise under the same privacy level. In addition, we prove that our framework converges to the optimum in strongly-convex case and to a diminishing neighborhood of the first-order stationary point in non-convex case. Experiments show that DP-C4 achieves superior privacy-utility trade-off over existing baselines across various tasks and datasets.
Optimization · Convex
Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessians in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish sharp convergence for stochastic mirror descent, matching state-of-the-art under classic smoothness. Our theory also extends to non-convex and composite optimization, which may shed light on practical usages of mirror descent, including pre-training and post-training of LLMs.
Optimization · Stochastic
The LogSumExp function, dual to the Kullback-Leibler (KL) divergence, plays a central role in many important optimization problems, including entropy-regularized optimal transport (OT) and distributionally robust optimization (DRO). In practice, when the number of exponential terms inside the logarithm is large or infinite, optimization becomes challenging since computing the gradient requires differentiating every term. We propose a novel convexity- and smoothness-preserving approximation to LogSumExp that can be efficiently optimized using stochastic gradient methods. This approximation is rooted in a sound modification of the KL divergence in the dual, resulting in a new $f$-divergence called the *safe KL divergence*. Our experiments and theoretical analysis of the LogSumExp-based stochastic optimization, arising in DRO and continuous OT, demonstrate the advantages of our approach over existing baselines.
Theory · Optimization
We propose a mathematically principled PDE gradient flow framework for distributionally robust optimization (DRO). Exploiting the recent advances in the intersection of Monte Carlo sampling and statistical optimal transport, we show that our theoretical framework can be implemented as practical algorithms for sampling from worst-case distributions and, consequently, DRO. While numerous previous works have relied on dual reformulation techniques, we contribute a sound and complete gradient flow view based on SDEs or PDEs that can be used to construct new algorithms for general, potentially non-convex, losses. Without loss of generality, we solve a class of Wasserstein and entropy-regularized DRO problems using the recently-discovered Wasserstein Fisher-Rao and Stein variational gradient flows. Notably, we also show some simple reductions of our framework recover exactly previously proposed popular DRO methods, and provide new insights into their theoretical limits and optimization dynamics of DRO. Numerical studies based on stochastic gradient descent on machine learning tasks provide empirical backing for our theoretical findings.
Optimization · Stochastic
Sparse tangent portfolio optimization aims to learn an interpretable, low-cardinality portfolio in the tangency direction of the mean–variance frontier, yet the associated cardinality-constrained formulation is NP-hard and standard predict-then-optimize pipelines often misalign forecasting accuracy with downstream portfolio quality. We propose an end-to-end decision-focused learning framework that reformulates Sharpe-ratio maximization as a Disciplined Parametrized Programming (DPP)-compliant convex programming layer and replaces discrete selection with a smooth top-k operator enforcing an exact sum-to-k sparsity budget. This enables gradient flow through prediction, asset selection, and re-optimization, allowing the predictive model to directly optimize the portfolio performance. Across five major equity markets, our method consistently delivers higher out-of-sample Sharpe ratios than historical and prediction-focused baselines while producing meaningful sparse selections.
Theory · Optimization
In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness. Our finite-time bounds reveal a sharp decomposition of tracking error into transient, noise-induced, and drift-induced components. This decomposition exposes a fundamental trade-off: while momentum is often used as a gradient-smoothing heuristic, under distribution shift it incurs an explicit drift-amplification penalty that diverges as the momentum parameter $\beta$ approaches 1, yielding systematic tracking lag. We complement these upper bounds with minimax lower bounds under gradient-variation constraints, proving this momentum-induced tracking penalty is not an analytical artifact but an information-theoretic barrier: in drift-dominated regimes, momentum is unavoidably worse because stale-gradient averaging forces systematic lag. Our results provide theoretical grounding for the empirical instability of momentum in dynamic settings and precisely delineate regime boundaries where vanilla SGD provably outperforms its accelerated counterparts.
Theory · Learning Theory
The choice of loss function in classification involves a fundamental trade-off: smooth losses (like Cross-Entropy) enable fast optimization rates but yield slow square-root consistency bounds, while piecewise-linear losses (like Hinge) offer fast linear consistency rates but suffer from non-differentiability. We propose *Linear-Core (LC) Surrogates*, a new family of convex loss functions that resolve this tension by stitching a linear core to a smooth tail. We prove that these surrogates are differentiable everywhere while retaining strict linear $H$-consistency bounds, effectively combining the optimization benefits of smoothness with the statistical efficiency of margin-based losses. In the structured prediction setting, we show that this smoothness unlocks a massive computational and energy advantage: it allows for an unbiased stochastic gradient estimator that bypasses the quadratic complexity $O(|\mathcal{Y}|^2)$ of exact inference (e.g., Viterbi). Empirically, our method achieves a *23$\times$ speedup* over Structured SVMs on large-vocabulary sequence tagging tasks and demonstrates superior robustness to instance-dependent label noise, outperforming Cross-Entropy by *2.6\%* on corrupted CIFAR-10.