Stochastic Differential Equations for Modeling First Order Optimization Methods

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-04-11 DOI:10.1137/21m1435665

M. Dambrine, Ch. Dossal, B. Puig, A. Rondepierre

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引用次数: 0

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1402-1426, June 2024.
Abstract. In this article, a family of SDEs are derived as a tool to understand the behavior of numerical optimization methods under random evaluations of the gradient. Our objective is to transpose the introduction of continuous versions through ODEs to understand the asymptotic behavior of a discrete optimization scheme to the stochastic setting. We consider a continuous version of the stochastic gradient scheme and of a stochastic inertial system. This article first studies the quality of the approximation of the discrete scheme by an SDE when the step size tends to 0. Then, it presents new asymptotic bounds on the values [math], where [math] is a solution of the SDE and [math], when [math] is convex and under integrability conditions on the noise. Results are provided under two sets of hypotheses: first considering [math] and convex functions and then adding some geometrical properties of [math]. All of these results provide insight on the behavior of these inertial and perturbed algorithms in the setting of stochastic algorithms.

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一阶优化方法建模的随机微分方程

SIAM 优化期刊》，第 34 卷第 2 期，第 1402-1426 页，2024 年 6 月。摘要本文导出了一系列 SDEs，作为理解梯度随机评估下数值优化方法行为的工具。我们的目的是通过 ODEs 将连续版本的引入转置到随机环境中，以理解离散优化方案的渐近行为。我们考虑了随机梯度方案的连续版本和随机惯性系统。本文首先研究了当步长趋近于 0 时，离散方案与 SDE 的近似质量。然后，本文提出了[math]（其中[math]是 SDE 的一个解）和[math]（其中[math]是凸的，并且在噪声的可整性条件下）值的新渐近约束。我们提供了两组假设下的结果：首先考虑 [math] 和凸函数，然后添加 [math] 的一些几何特性。所有这些结果都有助于深入了解这些惯性算法和扰动算法在随机算法环境下的行为。

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 Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.