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Variance-Reduced Proximal Gradient Algorithm for Stochastic Optimization with Constraints: A Non-Asymptotic Instance-Dependent Analysis


Core Concepts
The core message of this paper is to provide a non-asymptotic, instance-dependent analysis of a variance-reduced proximal gradient (VRPG) algorithm for stochastic convex optimization under convex constraints. The algorithm's performance is shown to be governed by the scaled distance between the solutions of the given problem and a certain small perturbation of the given problem, both solved under the given convex constraints.
Abstract
The paper considers the problem of stochastic convex optimization under convex constraints, where the goal is to minimize the expected value of a strongly convex and smooth function f(x, z) over a convex set X, given access to N i.i.d. samples from the underlying distribution P0. The key insights and highlights are: The authors motivate a non-asymptotic, instance-dependent benchmark for the problem, which is based on the scaled distance between the solutions of the original problem and a perturbed version of the problem, both solved under the given constraints. The authors propose a variance-reduced proximal gradient (VRPG) algorithm to solve the constrained stochastic optimization problem. The algorithm proceeds in epochs, where in each epoch, it constructs an approximate problem by recentering the gradient, and then runs a proximal stochastic gradient descent on this approximate problem. The authors provide a non-asymptotic, instance-dependent upper bound on the performance of the VRPG algorithm. Specifically, they show that the algorithm's error is bounded by the square root of the instance-dependent benchmark, up to logarithmic factors in the sample size. The authors further show that as the number of samples N goes to infinity, the VRPG algorithm achieves the renowned local minimax lower bound of Hajek and Le Cam, up to universal constants and a logarithmic factor in the sample size. Overall, the paper provides a refined, instance-dependent analysis of a variance-reduced algorithm for constrained stochastic optimization, which improves upon the existing asymptotic results.
Stats
The paper does not contain any explicit numerical data or statistics. The analysis is theoretical in nature.
Quotes
None.

Key Insights Distilled From

by Koulik Khama... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.00042.pdf
Stochastic Optimization with Constraints

Deeper Inquiries

Can the additional logarithmic factors in the non-asymptotic upper bound be removed, perhaps by using a different variance reduction scheme

The additional logarithmic factors in the non-asymptotic upper bound can potentially be removed by exploring different variance reduction schemes. While the variance reduced proximal gradient algorithm used in the study provides strong guarantees, alternative algorithms or modifications to the existing algorithm could potentially lead to tighter bounds without the additional logarithmic factors. By investigating different approaches to variance reduction or optimization techniques, it may be possible to refine the non-asymptotic upper bound and eliminate the logarithmic factors.

Can the authors' techniques be extended to derive non-asymptotic local minimax lower bounds for the constrained stochastic optimization problem, similar to the work of Cai and Low in the unconstrained setting

Extending the authors' techniques to derive non-asymptotic local minimax lower bounds for constrained stochastic optimization problems, similar to the work of Cai and Low in the unconstrained setting, is a promising direction for future research. By building upon the insights and methodologies presented in this study, researchers could explore the application of perturbed problems, instance-dependent benchmarks, and variance reduction strategies to constrained optimization scenarios. This extension would involve adapting the analysis to account for the specific constraints present in the optimization problem and characterizing the local minimax lower bounds in the constrained setting.

How do the insights from this work on constrained stochastic optimization connect to the broader literature on optimization under uncertainty, such as robust optimization and distributionally robust optimization

The insights from this work on constrained stochastic optimization offer valuable contributions to the broader literature on optimization under uncertainty, including robust optimization and distributionally robust optimization. By considering the variability of noise, the complexity of loss functions, and the geometry of constraint sets, the study provides a nuanced understanding of optimization problems in uncertain environments. These insights can inform the development of robust optimization strategies that account for uncertainty and variability in data, as well as distributionally robust optimization approaches that address ambiguity in probability distributions. The connection between local minimax lower bounds and solutions to perturbed problems can enhance the robustness and reliability of optimization algorithms in uncertain conditions.
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