Stochastic Optimization

Both Dimensionality Reduction (DR) and Graph Drawing (GD) aim to visualize abstract, non-linear structures, yet rely on different optimization paradigms. This contrast is evident in Multidimensional Scaling (MDS), which typically depends on the SMACOF algorithm despite graph drawing results showing that simpler stochastic optimization schemes can be more effective for the same objective. We bridge these domains by adapting Stochastic Gradient Descent (SGD) techniques from graph drawing to vector data embedding. We present a scikit-learn compatible estimator that minimizes global stress through local pairwise updates, improving upon the existing implementation. Experiments on standard high-dimensional benchmarks show that our stochastic solver converges substantially faster than SMACOF while achieving comparable or lower stress.

We study high-probability (HP) convergence guarantees in decentralized stochastic optimization, where multiple agents collaborate to jointly train a model over a network. Existing HP results in decentralized settings almost exclusively focus on the Decentralized Stochastic Gradient Descent ($\mathtt{DSGD}$) algorithm, which requires strong assumptions, such as bounded data heterogeneity, or strong convexity of each agent's cost. This is contrary to the mean-squared error (MSE) results, where methods incorporating bias-correction techniques are known to converge under relaxed assumptions and achieve better practical performance. In this paper we provide the first step toward bridging the gap, by studying HP convergence of $\mathtt{DSGD}$ incorporating the gradient tracking technique, in the presence of noise satisfying a relaxed sub-Gaussian condition. We show that the resulting method, dubbed $\mathtt{GT-DSGD}$, achieves order-optimal HP convergence rates for both non-convex and Polyak-Łojasiewicz costs, of order $\mathcal{O}\Big(\frac{\log(1/δ)}{\sqrt{nT}}\Big)$ and $\mathcal{O}\Big(\frac{\log(1/δ)}{nT}\Big)$, respectively, where $n$ is the number of agents, $T$ is the time horizon and $δ\in (0,1)$ is the confidence parameter. Our results establish that $\mathtt{GT-DSGD}$ converges in the HP sense under the same conditions on the cost as in the MSE sense, while achieving comparable transient times. To the best of our knowledge, these are the first HP guarantees for decentralized optimization methods incorporating bias-correction. Numerical experiments on real and synthetic data verify our theoretical findings, underlining the superior performance of $\mathtt{GT-DSGD}$ and highlighting that the benefits of incorporating bias-correction are also maintained in the HP sense.

Information-theoretic generalization bounds analyze stochastic optimization by relating expected generalization error to the mutual information between learned parameters and training data. Virtual perturbation analyses of SGD add auxiliary Gaussian noise only in the proof, making mutual information tractable while leaving the actual SGD trajectory unchanged. Existing bounds, however, typically require perturbation covariances to be fixed independently of the optimization history, limiting their ability to represent geometries induced by moving gradient statistics, preconditioners, curvature proxies, and other pathwise information. We introduce predictable history-adaptive virtual perturbations, where the perturbation covariance at each iteration may depend on the past real SGD history but not on current or future randomness. This predictability enables a conditional Gaussian relative-entropy argument and yields generalization bounds for SGD with adaptive virtual-noise geometry. The bounds replace fixed sensitivity and gradient-deviation terms with conditional adaptive counterparts, include an output-sensitivity penalty from accumulated perturbation covariance, and reduce the deviation term to a conditional variance only under conditional unbiasedness. Since adaptive covariances may be data-dependent, we separate local Gaussian smoothing from global reference-kernel comparison. The resulting bound includes a covariance-comparison cost measuring the KL price of using an admissible reference geometry different from the actual adaptive covariance. Fixed-noise-style bounds are recovered under admissible synchronization, such as deterministic, public, or prefix-observable covariance rules. The framework recovers fixed isotropic and geometry-aware bounds as special cases while extending virtual perturbation analysis to history-dependent SGD without modifying the algorithm.

Data-driven decision-making under uncertainty typically presumes the collection of historical data from an unknown target probability distribution. However, one may have no access to any data from the target distribution prior to decision-making. To address this challenge, we propose robust out-of-distribution stochastic optimization, a novel data-driven framework that effectively utilizes relevant data distributions for robust decision-making under unseen distributions. A key feature of our framework is that all data distributions are assumed to be randomly generated from a meta-distribution over distributions. To describe uncertainty in distribution generation, we propose to learn a data-driven uncertainty set in a reproducing kernel Hilbert space (RKHS) from relevant data distributions, with adjustable conservatism. We then incorporate this set into a min-max stochastic program to derive robust decisions. Notably, under randomness of distribution generation, we establish rigorous out-of-distribution generalization guarantees for the uncertainty set as well as the solution. To ease problem-solving in RKHS, an approximate parametrization with a provably bounded suboptimality and a row generation strategy are presented. Extensive numerical experiments on multi-item newsvendor and portfolio optimization demonstrate the superior out-of-distribution performance of our decision-making framework under unseen data distribution, even when only a small or moderate number of relevant sources are available.

Maintaining long-term accuracy of stereo camera calibration parameters is important for autonomous systems' perception. This work proposes Online Tracking of Essential Matrix by Stochastic Optimization (TESO). The core mechanisms of TESO are: 1) a robust loss function based on kernel correlation over tentative correspondences, 2) an adaptive online stochastic optimization on the essential manifold. TESO has low CPU and memory requirements, relies on a few hyperparameters, and eliminates the need for data-driven training, enabling the usage in resource-constrained online perception systems. We evaluated the influence of TESO on geometric precision, rectification quality, and stereo depth consistency. On the large-scale MAN TruckScenes dataset, TESO tracks rotational calibration drift with 0.12 deg precision in the Y-axis (critical for stereo accuracy) while the X- and Z-axes are five times more precise. Tracking applied to sequences with simulated drift shows similar precision with respect to the reference as tracking applied to no-drift sequences, indicating the tracker is unbiased. On the KITTI dataset, TESO revealed systematic inconsistencies in extrinsic parameters across stereo pairs, confirming previous published findings. We verified that intrinsic decalibration affected these errors, as evidenced by the conflicting behavior of the rectification and depth metrics. After correcting the reference calibration, TESO improved its rotation precision around the Y-axis 20 times to 0.025 deg and its depth accuracy 50 times. Despite its lightweight design, direct optimization of the proposed TESO loss function alone achieves accuracy comparable to that of neural network-based single-frame methods.

Parameter-free stochastic optimization aims to design algorithms that are agnostic to the underlying problem parameters while still achieving convergence rates competitive with optimally tuned methods. While some parameter-free methods do not require the specific values of the problem parameters, they still rely on prior knowledge, such as the lower or upper bounds of them. We refer to such methods as partially parameter-free''. In this work, we target achieving fully parameter-free'' methods, i.e., the algorithmic inputs do not need to satisfy any unverifiable condition related to the true problem parameters. We propose a powerful and general grid search framework, named \textsc{Grasp}, with a novel self-bounding analysis technique that effectively determines the search ranges of parameters, in contrast to previous work. Our method demonstrates generality in: (i) the non-convex case, where we propose a fully parameter-free method that achieves near-optimal convergence rate, up to logarithmic factors; (ii) the convex case, where our parameter-free methods are competitive with strong performance in terms of acceleration and universality. Finally, we contribute a sharper guarantee for the model ensemble, a final step of the grid search framework, under interpolated variance characterization.

Combinatorial black-box optimization in high-dimensional settings demands a careful trade-off between exploiting promising regions of the search space and preserving sufficient exploration to identify multiple optima. Although Estimation-of-Distribution Algorithms (EDAs) provide a powerful model-based framework, they often concentrate on a single region of interest, which may result in premature convergence when facing complex or multimodal objective landscapes. In this work, we incorporate the Stein operator to introduce a repulsive mechanism among particles in the parameter space, thereby encouraging the population to disperse and jointly explore several modes of the fitness landscape. Empirical evaluations across diverse benchmark problems show that the proposed method achieves performance competitive with, and in several cases superior to, leading state-of-the-art approaches, particularly on large-scale instances. These findings highlight the potential of Stein variational gradient descent as a promising direction for addressing large, computationally expensive, discrete black-box optimization problems.

Reward fine-tuning of diffusion and flow models and sampling from tilted or Boltzmann distributions can both be formulated as stochastic optimal control (SOC) problems, where learning an optimal generative dynamics corresponds to optimizing a control under SDE constraints. In this work, we revisit and generalize Adjoint Matching, a recently proposed SOC-based method for learning optimal controls, and place it on a rigorous footing by deriving it from the Stochastic Maximum Principle (SMP). We formulate a general Hamiltonian adjoint matching objective for SOC problems with control-dependent drift and diffusion and convex running costs, and show that its expected value has the same first variation as the original SOC objective. As a consequence, critical points satisfy the Hamilton--Jacobi--Bellman (HJB) stationarity conditions. In the important practical case of state- and control-independent diffusion, we recover the lean adjoint matching loss previously introduced, which avoids second-order terms and whose critical points coincide with the optimal control under mild uniqueness assumptions. Numerical experiments confirm that the extra terms it discards become necessary once the diffusion is state-dependent. Finally, we show that adjoint matching can be precisely interpreted as a continuous-time method of successive approximations induced by the SMP, yielding a practical and implementable alternative to classical SMP-based algorithms, which are obstructed by intractable martingale terms in the stochastic setting. These results are also of independent interest to the stochastic control community, providing new implementable objectives and a viable pathway for SMP-based iterations in stochastic problems.

We consider a stochastic optimization problem involving two random variables: a context variable $X$ and a dependent variable $Y$. The objective is to minimize the expected value of a nonlinear loss functional applied to the conditional expectation $\mathbb{E}[f(X, Y,β) \mid X]$, where $f$ is a nonlinear function and $β$ represents the decision variables. We focus on the practically important setting in which direct sampling from the conditional distribution of $Y \mid X$ is infeasible, and only a stream of i.i.d. observation pairs $\{(X^k, Y^k)\}_{k=0,1,2,\ldots}$ is available. In our approach, the conditional expectation is approximated within a prespecified parametric function class. We analyze a simultaneous learning-and-optimization algorithm that jointly estimates the conditional expectation and optimizes the outer objective. Using a specially designed measure of non-optimality, combining the squared norm of the objective function's gradient and the mean square error of the auxiliary parametric model, we establish that the method achieves a convergence rate of order $\mathcal{O}\big(1/\sqrt{N}\big)$, where $N$ denotes the number of observed pairs.

This paper addresses the distributed stochastic minimax optimization problem subject to stochastic constraints. We propose a novel first-order Softmax-Weighted Switching Gradient method tailored for federated learning. Under full client participation, our algorithm achieves the standard $\tilde{\mathcal{O}}(ε^{-4})$ oracle complexity to satisfy a unified bound $ε$ for both the optimality gap and feasibility tolerance. We extend our theoretical analysis to the practical partial participation regime by quantifying client sampling noise through a stochastic superiority assumption. Furthermore, by relaxing standard boundedness assumptions on the objective functions, we establish a strictly tighter lower bound for the softmax hyperparameter. We provide a unified error decomposition and establish a sharp $\mathcal{O}(\log\frac{1}δ)$ high-probability convergence guarantee. Ultimately, our framework demonstrates that a single-loop primal-only switching mechanism provides a stable alternative for optimizing worst-case client performance, effectively bypassing the hyperparameter sensitivity and convergence oscillations often encountered in traditional primal-dual or penalty-based approaches. We verify the efficacy of our algorithm via experiment on the Neyman-Pearson (NP) classification, fair classification, and federated safe reinforcement learning tasks.

We study \emph{online episodic Constrained Markov Decision Processes} (CMDPs) under both stochastic and adversarial constraints. We provide a novel algorithm whose guarantees greatly improve those of the state-of-the-art best-of-both-worlds algorithm introduced by Stradi et al. (2025). In the stochastic regime, \emph{i.e.}, when the constraints are sampled from fixed but unknown distributions, our method achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation without relying on Slater's condition, thereby handling settings where no strictly feasible solution exists. Moreover, we provide guarantees on the stronger notion of \emph{positive} constraint violation, which does not allow to recover from large violation in the early episodes by playing strictly safe policies. In the adversarial regime, \emph{i.e.}, when the constraints may change arbitrarily between episodes, our algorithm ensures sublinear constraint violation without Slater's condition, and achieves sublinear $α$-regret with respect to the \emph{unconstrained} optimum, where $α$ is a suitably defined multiplicative approximation factor. We further validate our results through synthetic experiments, showing the practical effectiveness of our algorithm.

This paper introduces a novel approach to contextual stochastic optimization, integrating operations research and machine learning to address decision-making under uncertainty. Traditional methods often fail to leverage contextual information, which underscores the necessity for new algorithms. In this study, we utilize neural networks with combinatorial optimization layers to encode policies. Our goal is to minimize the empirical cost, which is estimated from past data on uncertain parameters and contexts. To that end, we present a surrogate learning problem and a generic primal-dual algorithm that is applicable to various combinatorial settings in stochastic optimization. Our approach extends classic Fenchel--Young loss results and introduces a new regularization method using sparse perturbations on the distribution simplex. This allows for tractable updates in the original space and can accommodate diverse objective functions. We establish sublinear convergence for the exact linear-parametric version and provide a bound on the non-optimality of the resulting policy in terms of the empirical cost. Experiments on three contextual stochastic optimization problems show that our algorithm is efficient and scalable, achieving performance comparable to state-of-the-art baselines with significantly reduced computational requirements.

Bilevel optimization has recently attracted significant attention in machine learning due to its wide range of applications and advanced hierarchical optimization capabilities. In this paper, we propose a plug-and-play framework, named PnPBO, for developing and analyzing stochastic bilevel optimization methods. This framework integrates both modern unbiased and biased stochastic estimators into the single-loop bilevel optimization framework introduced in [9], with several improvements. In the implementation of PnPBO, all stochastic estimators for different variables can be independently incorporated, and an additional moving average technique is applied when using an unbiased estimator for the upper-level variable. In the theoretical analysis, we provide a unified convergence and complexity analysis for PnPBO, demonstrating that the adaptation of various stochastic estimators (including PAGE, ZeroSARAH, and mixed strategies) within the PnPBO framework achieves optimal sample complexity, comparable to that of single-level optimization. This resolves the open question of whether the optimal complexity bounds for solving bilevel optimization are identical to those for single-level optimization. Finally, we empirically validate our framework, demonstrating its effectiveness on several benchmark problems and confirming our theoretical findings.

We revisit a classical assumption for analyzing stochastic gradient algorithms where the squared norm of the stochastic subgradient (or the variance for smooth problems) is allowed to grow as fast as the squared norm of the optimization variable. We contextualize this assumption in view of its inception in the 1960s, its seemingly independent appearance in the recent literature, its relationship to weakest-known variance assumptions for analyzing stochastic gradient algorithms, and its relevance in deterministic problems for non-Lipschitz nonsmooth convex optimization. We build on and extend a connection recently made between this assumption and the Halpern iteration. For convex nonsmooth, and potentially stochastic, optimization, we analyze horizon-free, anytime algorithms with last-iterate rates. For problems beyond simple constrained optimization, such as convex problems with functional constraints or regularized convex-concave min-max problems, we obtain rates for optimality measures that do not require boundedness of the feasible set.

Reinforcement learning (RL) has successfully solved various deterministic and stochastic planning problems. However, conventional RL struggles with complex real-world constraints, particularly when feasibility is explicit and depends on the current state or trajectory. In this work, we address stochastic sequential decision-making with state-dependent constraints through a real-world case study of the master stowage planning problem in container shipping, which aims to optimize revenue and costs under demand uncertainty and operational constraints. We propose a deep RL framework with an encoder-decoder model that integrates problem instance, solution, and uncertainty information to guide planning. We introduce differentiable projection layers that enforce convex polyhedral constraints, while Jacobian corrections offset the projections to yield unbiased policy gradient estimates. Experiments show that our model efficiently finds adaptive, feasible solutions that generalize across distribution shifts and scale to longer planning horizons, outperforming state-of-the-art baselines in constrained RL and stochastic programming. As such, our policies enable adaptive, uncertainty-aware planning that can support resilient and sustainable supply chains.

We derive first-order Pontryagin optimality conditions for stochastic optimal control with deterministic controls for systems modeled by rough differential equations (RDE) driven by Gaussian rough paths. This Pontryagin Maximum Principle (PMP) applies to systems following stochastic differential equations (SDE) driven by Brownian motion, yet it does not rely on forward-backward SDEs and involves the same Hamiltonian as the deterministic PMP. The proof consists of first deriving various integrable error bounds for solutions to nonlinear and linear RDEs by leveraging recent results on Gaussian rough paths. The PMP then follows using standard techniques based on needle-like variations. As an application, we propose the first indirect shooting method for nonlinear stochastic optimal control and show that it converges 10x faster than a direct method on a stabilization task.

Stochastic optimization is a widely used approach for optimization under uncertainty, where uncertain input parameters are modeled by random variables. Exact or approximation algorithms have been obtained for several fundamental problems in this area. However, a significant limitation of this approach is that it requires full knowledge of the underlying probability distributions. Can we still get good (approximation) algorithms if these distributions are unknown, and the algorithm needs to learn them through repeated interactions? In this paper, we resolve this question for a large class of ''monotone'' stochastic problems, by providing a generic online learning algorithm with $\sqrt{T\log(T)}$ regret relative to the best approximation algorithm (under known distributions). Importantly, our online algorithm works in a semi-bandit setting, where in each period, the algorithm only observes samples from the random variables that were actually probed. Moreover, our result extends to settings with censored and binary feedback, where the policy only observes truncated or thresholded versions of the probed variables. Our framework applies to several fundamental problems such as prophet inequality, Pandora's box, stochastic knapsack, single-resource revenue management and sequential posted pricing.