Deep Relaxation of Controlled Stochastic Gradient Descent via Singular Perturbations

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS

SIAM Journal on Control and Optimization Pub Date : 2024-07-24 DOI:10.1137/23m1544878

Martino Bardi, Hicham Kouhkouh

引用次数: 0

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2229-2253, August 2024.
Abstract. We consider a singularly perturbed system of stochastic differential equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate the entropic gradient descent in the optimization of deep neural networks via homogenization. We embed it in a much larger class of two-scale stochastic control problems and rely on convergence results for Hamilton–Jacobi–Bellman equations with unbounded data proved recently by ourselves (ESAIM Control Optim. Calc. Var. 2023). We show that the limit of the value functions is itself the value function of an effective control problem with extended controls and that the trajectories of the perturbed system converge in a suitable sense to the trajectories of the limiting effective control system. These rigorous results improve the understanding of the convergence of the algorithms used by Chaudhari et al., as well as of their possible extensions where some tuning parameters are modeled as dynamic controls.

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通过奇异扰动实现受控随机梯度下降的深度放松

SIAM 控制与优化期刊》第 62 卷第 4 期第 2229-2253 页，2024 年 8 月。摘要。我们考虑了 Chaudhari 等人（Res. Math. Sci. 2018）提出的奇异扰动随机微分方程系统，以通过同质化近似深度神经网络优化中的熵梯度下降。我们将其嵌入到一类更大的二尺度随机控制问题中，并依赖于我们自己最近证明的具有无约束数据的汉密尔顿-雅各比-贝尔曼方程的收敛结果（ESAIM Control Optim. Calc. Var. 2023）。我们证明了值函数的极限本身就是具有扩展控制的有效控制问题的值函数，并且扰动系统的轨迹在适当意义上收敛于极限有效控制系统的轨迹。这些严谨的结果加深了人们对 Chaudhari 等人所使用算法的收敛性的理解，也加深了人们对将某些调整参数建模为动态控制的可能扩展的理解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.