最小化拟凸函数的随机模型方法
Stochastic Model-Based Methods for Minimizing Paraconvex Functions
- Inria（法国国家数字科学研究所）
- INSA Lyon（里昂国立应用科学学院）
- Department of Mathematics and Statistics, University of Quy Nhon（归仁大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文研究最小化拟凸函数的随机算法，建立了随机次梯度与近端次梯度方法的收敛性，并扩展到基于随机模型的一类算法，其核心原理是将算法视为广义Moreau包络上的扰动下降方法。
中文摘要 AI 辅助
本文研究了最小化拟凸函数的随机算法，拟凸函数是一类推广弱凸函数的函数类，例如包含Hölder光滑函数以及凸函数与Hölder光滑映射的复合函数。我们首先分别建立了无约束和复合拟凸优化中随机次梯度方法和随机近端次梯度方法的收敛性。随后，分析被扩展到一大类通过满足特定近似质量和连续性假设的随机模型访问目标函数的算法。收敛分析背后的主要原理是，所考虑的算法可以被解释为原始拟凸问题的广义Moreau包络上的扰动下降方法。
英文摘要
This paper studies stochastic algorithms for minimizing paraconvex functions, a function class that generalizes weakly convex functions and includes, for instance, Hölder smooth functions and compositions of convex functions with Hölder smooth maps. We first establish the convergence of the stochastic subgradient method and the stochastic proximal subgradient method for unconstrained and composite paraconvex optimization, respectively. The analysis is then extended to a broad family of algorithms that access the objective through stochastic models satisfying certain approximation-quality and continuity assumptions. The main principle that underlies the convergence analysis is that the algorithms under consideration can be interpreted as perturbed descent methods on the generalized Moreau envelope of the original paraconvex problem.
