Optimization of Inf-Convolution Regularized Nonconvex Composite Problems

TL;DR

This paper investigates nonconvex composite problems involving inf-convolution with Legendre functions, establishing conditions under which stationary points of splitting models correspond to those of the original problem, with applications to distributed deep learning.

Contribution

It analytically links stationary points of splitting formulations to original inf-convolution problems in nonconvex settings, with implications for distributed neural network training.

Findings

Established equivalence between stationary points of splitting and original models.

Characterized stationary points of the penalty objective in distributed training.

Demonstrated practical relevance on deep neural network training tasks.

Abstract

In this work, we consider nonconvex composite problems that involve inf-convolution with a Legendre function, which gives rise to an anisotropic generalization of the proximal mapping and Moreau-envelope. In a convex setting such problems can be solved via alternating minimization of a splitting formulation, where the consensus constraint is penalized with a Legendre function. In contrast, for nonconvex models it is in general unclear that this approach yields stationary points to the infimal convolution problem. To this end we analytically investigate local regularity properties of the Moreau-envelope function under prox-regularity, which allows us to establish the equivalence between stationary points of the splitting model and the original inf-convolution model. We apply our theory to characterize stationary points of the penalty objective, which is minimized by the elastic averaging SGD (EASGD) method for distributed training. Numerically, we demonstrate the practical relevance of the proposed approach on the important task of distributed training of deep neural networks.