A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz Functions

TL;DR

This paper provides a unified analysis of the proximal subgradient method for minimizing composite functions that are weakly convex, nonsmooth, and non-Lipschitz, establishing new complexity bounds and extending to stochastic algorithms.

Contribution

It introduces a unified framework leveraging the Moreau envelope to relate error bounds, growth conditions, and subgradient behavior for broad classes of weakly convex functions.

Findings

Derived novel iteration complexity results.

Established connections between error bounds and subgradient growth.

Extended analysis to stochastic proximal subgradient algorithms.

Abstract

This paper presents a unified analysis for the proximal subgradient method (Prox-SubGrad) type approach to minimize an overall objective of $f(x)+r(x)$, subject to convex constraints, where both $f$ and $r$ are weakly convex, nonsmooth, and non-Lipschitz. Leveraging on the properties of the Moreau envelope of weakly convex functions, we are able to relate error-bound conditions, the growth conditions of the subgradients of the objective, and the behavior of the proximal subgradient iterates on some remarkably broad classes of objective functions. Various existing as well as new bounding conditions are studied, leading to novel iteration complexity results. The terrain of our exploration expands to stochastic proximal subgradient algorithms.