Many real-world problems, such as those with fairness constraints, involve complex expectation constraints and large datasets, necessitating the design of efficient stochastic methods to solve them. Most existing research focuses on cases with no {constraint} or easy-to-project constraints or deterministic constraints. In this paper, we consider nonconvex nonsmooth stochastic optimization problems with expectation constraints, for which we build a novel exact penalty model. We first show the relationship between the penalty model and the original problem. Then on solving the penalty problem, we present a single-loop SPIDER-type stochastic subgradient method, which utilizes the subgradients of both the objective and constraint functions, as well as the constraint function value at each iteration. Under certain regularity conditions (weaker than Slater-type constraint qualification or strong feasibility assumed in existing works), we establish an iteration complexity result of $O(\epsilon^{-4})$ to reach a near-$\epsilon$ stationary point of the penalized problem in expectation, matching the lower bound for such tasks. Building on the exact penalization, an $(\epsilon,\epsilon)$-KKT point of the original problem is obtained. For a few scenarios, our complexity of either the {objective} sample subgradient or the constraint sample function values can be lower than the state-of-the-art results by a factor of $\epsilon^{-2}$. Moreover, on solving two fairness-constrained problems and a multi-class Neyman-Pearson classification problem, our method is significantly (up to 466 times) faster than the state-of-the-art algorithms, including switching subgradient method and inexact proximal point methods.
In this paper, we present a unified algorithm for stochastic optimization that makes use of a "momentum" term; in other words, the stochastic gradient depends not only on the current true gradient of the objective function, but also on the true gradient at the previous iteration. Our formulation includes the Stochastic Heavy Ball (SHB) and the Stochastic Nesterov Accelerated Gradient (SNAG) algorithms as special cases. In addition, in our formulation, the momentum term is allowed to vary as a function of time (i.e., the iteration counter). The assumptions on the stochastic gradient are the most general in the literature, in that it can be biased, and have a conditional variance that grows in an unbounded fashion as a function of time. This last feature is crucial in order to make the theory applicable to "zero-order" methods, where the gradient is estimated using just two function evaluations. We present a set of sufficient conditions for the convergence of the unified algorithm. These conditions are natural generalizations of the familiar Robbins-Monro and Kiefer-Wolfowitz-Blum conditions for standard stochastic gradient descent. We also analyze another method from the literature for the SHB algorithm with a time-varying momentum parameter, and show that it is impracticable.
We study the sample complexity of stochastic convex optimization when problem parameters, e.g., the distance to optimality, are unknown. We pursue two strategies. First, we develop a reliable model selection method that avoids overfitting the validation set. This method allows us to generically tune the learning rate of stochastic optimization methods to match the optimal known-parameter sample complexity up to $\log\log$ factors. Second, we develop a regularization-based method that is specialized to the case that only the distance to optimality is unknown. This method provides perfect adaptability to unknown distance to optimality, demonstrating a separation between the sample and computational complexity of parameter-free stochastic convex optimization. Combining these two methods allows us to simultaneously adapt to multiple problem structures. Experiments performing few-shot learning on CIFAR-10 by fine-tuning CLIP models and prompt engineering Gemini to count shapes indicate that our reliable model selection method can help mitigate overfitting to small validation sets.
In this paper, we study a class of stochastic and finite-sum convex optimization problems with deterministic constraints. Existing methods typically aim to find an $\epsilon$-$expectedly\ feasible\ stochastic\ optimal$ solution, in which the expected constraint violation and expected optimality gap are both within a prescribed tolerance $\epsilon$. However, in many practical applications, constraints must be nearly satisfied with certainty, rendering such solutions potentially unsuitable due to the risk of substantial violations. To address this issue, we propose stochastic first-order methods for finding an $\epsilon$-$surely\ feasible\ stochastic\ optimal$ ($\epsilon$-SFSO) solution, where the constraint violation is deterministically bounded by $\epsilon$ and the expected optimality gap is at most $\epsilon$. Our methods apply an accelerated stochastic gradient (ASG) scheme or a modified variance-reduced ASG scheme $only\ once$ to a sequence of quadratic penalty subproblems with appropriately chosen penalty parameters. We establish first-order oracle complexity bounds for the proposed methods in computing an $\epsilon$-SFSO solution. As a byproduct, we also derive first-order oracle complexity results for sample average approximation method in computing an $\epsilon$-SFSO solution of the stochastic optimization problem using our proposed methods to solve the sample average problem.
In this work, we provide an extended discussion of a new approach to explainable Reinforcement Learning called Diverse Near-Optimal Alternatives (DNA), first proposed at L4DC 2025. DNA seeks a set of reasonable "options" for trajectory-planning agents, optimizing policies to produce qualitatively diverse trajectories in Euclidean space. In the spirit of explainability, these distinct policies are used to "explain" an agent's options in terms of available trajectory shapes from which a human user may choose. In particular, DNA applies to value function-based policies on Markov decision processes where agents are limited to continuous trajectories. Here, we describe DNA, which uses reward shaping in local, modified Q-learning problems to solve for distinct policies with guaranteed epsilon-optimality. We show that it successfully returns qualitatively different policies that constitute meaningfully different "options" in simulation, including a brief comparison to related approaches in the stochastic optimization field of Quality Diversity. Beyond the explanatory motivation, this work opens new possibilities for exploration and adaptive planning in RL.
Accurate automatic screening of minors in unconstrained images demands models that are robust to distribution shift and resilient to the children under-representation in publicly available data. To overcome these issues, we propose a multi-task architecture with dedicated under/over-age discrimination tasks based on a frozen FaRL vision-language backbone joined with a compact two-layer MLP that shares features across one age-regression head and four binary under-age heads for age thresholds of 12, 15, 18, and 21 years, focusing on the legally critical age range. To address the severe class imbalance, we introduce an $\alpha$-reweighted focal-style loss and age-balanced mini-batch sampling, which equalizes twelve age bins during stochastic optimization. Further improvement is achieved with an age gap that removes edge cases from the loss. Moreover, we set a rigorous evaluation by proposing the Overall Under-Age Benchmark, with 303k cleaned training images and 110k test images, defining both the "ASORES-39k" restricted overall test, which removes the noisiest domains, and the age estimation wild shifts test "ASWIFT-20k" of 20k-images, stressing extreme pose ($>$45{\deg}), expression, and low image quality to emulate real-world shifts. Trained on the cleaned overall set with resampling and age gap, our multiage model "F" lowers the root-mean-square-error on the ASORES-39k restricted test from 5.733 (age-only baseline) to 5.656 years and lifts under-18 detection from F2 score of 0.801 to 0.857 at 1% false-adult rate. Under the domain shift to the wild data of ASWIFT-20k, the same configuration nearly sustains 0.99 recall while boosting F2 from 0.742 to 0.833 with respect to the age-only baseline, demonstrating strong generalization under distribution shift. For the under-12 and under-15 tasks, the respective boosts in F2 are from 0.666 to 0.955 and from 0.689 to 0.916, respectively.
The basic question of delineating those statistical problems that are solvable without making any assumptions on the underlying data distribution has long animated statistics and learning theory. This paper characterizes when a convex M-estimation or stochastic optimization problem is solvable in such an assumption-free setting, providing a precise dividing line between solvable and unsolvable problems. The conditions we identify show, perhaps surprisingly, that Lipschitz continuity of the loss being minimized is not necessary for distribution free minimization, and they are also distinct from classical characterizations of learnability in machine learning.
The proliferation of distributed renewable energy triggers the peer-to-peer (P2P) energy market formations. To make profits, prosumers equipped with photovoltaic (PV) panels and even the energy storage system (ESS) can actively participate in the real-time P2P energy market and trade energy. However, in real situations, system states such as energy demands and renewable energy power generation are highly uncertain, making it difficult for prosumers to make optimal real-time decisions. Moreover, severe problems with the physical network can arise from the real-time P2P energy trading, such as bus voltage violations and line overload. To handle these problems, this work first formulates the real-time P2P energy trading problem as a spatio-temporally constrained stochastic optimization problem by considering ESS and the spatial physical network constraints. To deal with the uncertainties online, a modified Lyapunov optimization method is innovatively proposed to approximately reformulate the stochastic optimization problem into an online one by relaxing the time-coupling constraints. Compared with the state-of-the-art online methods, the proposed one renders more flexibility and better performance for the real-time P2P energy market operation. Additionally, to protect the prosumers' privacy, an online distributed algorithm based on the consensus alternating direction method of multipliers (ADMM) is developed to solve the reformulated online problem by decoupling the spatial constraints. The theoretical near-optimal performance guarantee of the proposed online distributed algorithm is derived, and its performance can be further improved by minimizing the performance gap. Simulation results demonstrate that the proposed online distributed algorithm can guarantee the fast, stable, and safe long-term operation of the real-time P2P energy market.
Averaging techniques such as Ruppert--Polyak averaging and exponential movering averaging (EMA) are powerful approaches to accelerate optimization procedures of stochastic gradient descent (SGD) optimization methods such as the popular ADAM optimizer. However, depending on the specific optimization problem under consideration, the type and the parameters for the averaging need to be adjusted to achieve the smallest optimization error. In this work we propose an averaging approach, which we refer to as parallel averaged ADAM (PADAM), in which we compute parallely different averaged variants of ADAM and during the training process dynamically select the variant with the smallest optimization error. A central feature of this approach is that this procedure requires no more gradient evaluations than the usual ADAM optimizer as each of the averaged trajectories relies on the same underlying ADAM trajectory and thus on the same underlying gradients. We test the proposed PADAM optimizer in 13 stochastic optimization and deep neural network (DNN) learning problems and compare its performance with known optimizers from the literature such as standard SGD, momentum SGD, Adam with and without EMA, and ADAMW. In particular, we apply the compared optimizers to physics-informed neural network, deep Galerkin, deep backward stochastic differential equation and deep Kolmogorov approximations for boundary value partial differential equation problems from scientific machine learning, as well as to DNN approximations for optimal control and optimal stopping problems. In nearly all of the considered examples PADAM achieves, sometimes among others and sometimes exclusively, essentially the smallest optimization error. This work thus strongly suggest to consider PADAM for scientific machine learning problems and also motivates further research for adaptive averaging procedures within the training of DNNs.
Open-weight LLM zoos provide access to numerous high-quality models, but selecting the appropriate model for specific tasks remains challenging and requires technical expertise. Most users simply want factually correct, safe, and satisfying responses without concerning themselves with model technicalities, while inference service providers prioritize minimizing operating costs. These competing interests are typically mediated through service level agreements (SLAs) that guarantee minimum service quality. We introduce MESS+, a stochastic optimization algorithm for cost-optimal LLM request routing while providing rigorous SLA compliance guarantees. MESS+ learns request satisfaction probabilities of LLMs in real-time as users interact with the system, based on which model selection decisions are made by solving a per-request optimization problem. Our algorithm includes a novel combination of virtual queues and request satisfaction prediction, along with a theoretical analysis of cost optimality and constraint satisfaction. Across a wide range of state-of-the-art LLM benchmarks, MESS+ achieves an average of 2x cost savings compared to existing LLM routing techniques.
Constrained stochastic nonlinear optimization problems have attracted significant attention for their ability to model complex real-world scenarios in physics, economics, and biology. As datasets continue to grow, online inference methods have become crucial for enabling real-time decision-making without the need to store historical data. In this work, we develop an online inference procedure for constrained stochastic optimization by leveraging a method called Sketched Stochastic Sequential Quadratic Programming (SSQP). As a direct generalization of sketched Newton methods, SSQP approximates the objective with a quadratic model and the constraints with a linear model at each step, then applies a sketching solver to inexactly solve the resulting subproblem. Building on this design, we propose a new online inference procedure called random scaling. In particular, we construct a test statistic based on SSQP iterates whose limiting distribution is free of any unknown parameters. Compared to existing online inference procedures, our approach offers two key advantages: (i) it enables the construction of asymptotically valid confidence intervals; and (ii) it is matrix-free, i.e. the computation involves only primal-dual SSQP iterates $(\boldsymbol{x}_t, \boldsymbol{\lambda}_t)$ without requiring any matrix inversions. We validate our theory through numerical experiments on nonlinearly constrained regression problems and demonstrate the superior performance of our random scaling method over existing inference procedures.
This paper presents an adaptive framework for edge inference based on a dynamically configurable transformer-powered deep joint source channel coding (DJSCC) architecture. Motivated by a practical scenario where a resource constrained edge device engages in goal oriented semantic communication, such as selectively transmitting essential features for object detection to an edge server, our approach enables efficient task aware data transmission under varying bandwidth and channel conditions. To achieve this, input data is tokenized into compact high level semantic representations, refined by a transformer, and transmitted over noisy wireless channels. As part of the DJSCC pipeline, we employ a semantic token selection mechanism that adaptively compresses informative features into a user specified number of tokens per sample. These tokens are then further compressed through the JSCC module, enabling a flexible token communication strategy that adjusts both the number of transmitted tokens and their embedding dimensions. We incorporate a resource allocation algorithm based on Lyapunov stochastic optimization to enhance robustness under dynamic network conditions, effectively balancing compression efficiency and task performance. Experimental results demonstrate that our system consistently outperforms existing baselines, highlighting its potential as a strong foundation for AI native semantic communication in edge intelligence applications.
Motivated by stochastic optimization, we introduce the problem of learning from samples of contextual value distributions. A contextual value distribution can be understood as a family of real-valued distributions, where each sample consists of a context $x$ and a random variable drawn from the corresponding real-valued distribution $D_x$. By minimizing a convex surrogate loss, we learn an empirical distribution $D'_x$ for each context, ensuring a small L\'evy distance to $D_x$. We apply this result to obtain the sample complexity bounds for the learning of an $\epsilon$-optimal policy for stochastic optimization problems defined on an unknown contextual value distribution. The sample complexity is shown to be polynomial for the general case of strongly monotone and stable optimization problems, including Single-item Revenue Maximization, Pandora's Box and Optimal Stopping.
In this paper, we consider discrete-time non-linear stochastic dynamical systems with additive process noise in which both the initial state and noise distributions are uncertain. Our goal is to quantify how the uncertainty in these distributions is propagated by the system dynamics for possibly infinite time steps. In particular, we model the uncertainty over input and noise as ambiguity sets of probability distributions close in the $\rho$-Wasserstein distance and aim to quantify how these sets evolve over time. Our approach relies on results from quantization theory, optimal transport, and stochastic optimization to construct ambiguity sets of distributions centered at mixture of Gaussian distributions that are guaranteed to contain the true sets for both finite and infinite prediction time horizons. We empirically evaluate the effectiveness of our framework in various benchmarks from the control and machine learning literature, showing how our approach can efficiently and formally quantify the uncertainty in linear and non-linear stochastic dynamical systems.
Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian manifold structure and performing optimization intrinsically on the manifold. This approach typically involves computing a search direction in the tangent space and updating variables via a retraction operation. However, as the size of the variables increases, the computational cost of the retraction can become prohibitively high, limiting the applicability of Riemannian optimization to large-scale problems. To address this challenge and enhance scalability, we propose a novel approach that restricts each update on a random submanifold, thereby significantly reducing the per-iteration complexity. We introduce two sampling strategies for selecting the random submanifolds and theoretically analyze the convergence of the proposed methods. We provide convergence results for general nonconvex functions and functions that satisfy Riemannian Polyak-Lojasiewicz condition as well as for stochastic optimization settings. Additionally, we demonstrate how our approach can be generalized to quotient manifolds derived from the orthogonal manifold. Extensive experiments verify the benefits of the proposed method, across a wide variety of problems.
End-to-end learning has become a widely applicable and studied problem in training predictive ML models to be aware of their impact on downstream decision-making tasks. These end-to-end models often outperform traditional methods that separate training from the optimization and only myopically focus on prediction error. However, the computational complexity of end-to-end frameworks poses a significant challenge, particularly for large-scale problems. While training an ML model using gradient descent, each time we need to compute a gradient we must solve an expensive optimization problem. We present a meta-optimization method that learns efficient algorithms to approximate optimization problems, dramatically reducing computational overhead of solving the decision problem in general, an aspect we leverage in the training within the end-to-end framework. Our approach introduces a neural network architecture that near-optimally solves optimization problems while ensuring feasibility constraints through alternate projections. We prove exponential convergence, approximation guarantees, and generalization bounds for our learning method. This method offers superior computational efficiency, producing high-quality approximations faster and scaling better with problem size compared to existing techniques. Our approach applies to a wide range of optimization problems including deterministic, single-stage as well as two-stage stochastic optimization problems. We illustrate how our proposed method applies to (1) an electricity generation problem using real data from an electricity routing company coordinating the movement of electricity throughout 13 states, (2) a shortest path problem with a computer vision task of predicting edge costs from terrain maps, (3) a two-stage multi-warehouse cross-fulfillment newsvendor problem, as well as a variety of other newsvendor-like problems.
We present JaxSGMC, an application-agnostic library for stochastic gradient Markov chain Monte Carlo (SG-MCMC) in JAX. SG-MCMC schemes are uncertainty quantification (UQ) methods that scale to large datasets and high-dimensional models, enabling trustworthy neural network predictions via Bayesian deep learning. JaxSGMC implements several state-of-the-art SG-MCMC samplers to promote UQ in deep learning by reducing the barriers of entry for switching from stochastic optimization to SG-MCMC sampling. Additionally, JaxSGMC allows users to build custom samplers from standard SG-MCMC building blocks. Due to this modular structure, we anticipate that JaxSGMC will accelerate research into novel SG-MCMC schemes and facilitate their application across a broad range of domains.
Explainable artificial intelligence (XAI) is essential for trustworthy machine learning (ML), particularly in high-stakes domains such as healthcare and finance. Shapley value (SV) methods provide a principled framework for feature attribution in complex models but incur high computational costs, limiting their scalability in high-dimensional settings. We propose Stochastic Iterative Momentum for Shapley Value Approximation (SIM-Shapley), a stable and efficient SV approximation method inspired by stochastic optimization. We analyze variance theoretically, prove linear $Q$-convergence, and demonstrate improved empirical stability and low bias in practice on real-world datasets. In our numerical experiments, SIM-Shapley reduces computation time by up to 85% relative to state-of-the-art baselines while maintaining comparable feature attribution quality. Beyond feature attribution, our stochastic mini-batch iterative framework extends naturally to a broader class of sample average approximation problems, offering a new avenue for improving computational efficiency with stability guarantees. Code is publicly available at https://github.com/nliulab/SIM-Shapley.
We present SKA-SGD (Streaming Krylov-Accelerated Stochastic Gradient Descent), a novel optimization approach that accelerates convergence for ill-conditioned problems by projecting stochastic gradients onto a low-dimensional Krylov subspace. Directly inspired by recent advances in s-step Conjugate Gradient methods with streaming Gauss-Seidel Gram solvers \cite{dambra2025sstep}, our method extends these techniques to the stochastic optimization domain. Our approach combines three key innovations: (1) projection coefficients computed via a single streaming Gauss-Seidel iteration, which is mathematically equivalent to Modified Gram-Schmidt orthogonalization; (2) a Chebyshev polynomial basis for constructing the Krylov subspace, providing superior numerical stability; and (3) efficient implementation for AMD GPUs using HIP. We prove that our streaming approach achieves a backward error near machine precision with $O(s^2)$ complexity rather than $O(s^3)$, where $s$ is the Krylov subspace dimension. Experimental results demonstrate that SKA-SGD significantly outperforms standard SGD and Adam in convergence rate and final error, particularly for problems with condition numbers exceeding $10^3$. GPU performance analysis reveals a crossover point where communication-avoiding benefits outweigh computational overhead, typically occurring at moderate scale ($p \approx 64$ processors) for problem sizes $n \geq 10^6$.
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 risk, 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 demonstrate the linear convergence of our algorithm under certain conditions and provide a bound on the non-optimality of the resulting policy in terms of the empirical risk. Experiments on a contextual stochastic minimum weight spanning tree problem show that our algorithm is efficient and scalable, achieving performance comparable to imitation learning of solutions computed using an expensive Lagrangian-based heuristic.