An Envelope for Davis-Yin Splitting and Strict Saddle Point Avoidance

TL;DR

This paper introduces a new envelope function for Davis-Yin Splitting that aligns with gradient descent and proves that, starting from random points, these methods almost surely avoid strict saddle points, extending prior results.

Contribution

It generalizes the envelope concept to DYS and proves saddle point avoidance for splitting methods, extending existing gradient descent results.

Findings

New envelope function for DYS matching gradient descent.

DYS and related methods avoid strict saddle points from random initializations.

Extension of saddle point avoidance results from gradient descent to splitting methods.

Abstract

It is known that operator splitting methods based on Forward Backward Splitting (FBS), Douglas-Rachford Splitting (DRS), and Davis-Yin Splitting (DYS) decompose a difficult optimization problems into simpler subproblems under proper convexity and smoothness assumptions. In this paper, we identify an envelope (an objective function) whose gradient descent iteration under a variable metric coincides with DYS iteration. This result generalizes the Moreau envelope for proximal-point iteration and the envelopes for FBS and DRS iterations identified by Patrinos, Stella, and Themelis. Based on the new envelope and the Stable-Center Manifold Theorem, we further show that, when FBS or DRS iterations start from random points, they avoid all strict saddle points with probability one. This results extends the similar results by Lee et al. from gradient descent to splitting methods.