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

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 subproblem 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 result extends the similar results by Lee et al. from gradient descent to splitting methods. 4When h = 0, (5) simplifies to Douglas-Rachford Splitting iteration,When g = 0, P γg reduces to Id and thus (5) simplifies towhich is Forward-Backward Splitting iteration slightly generalized by including the linear operator L.When f = 0, P γf reduces to Id and (5) simplifies to Backward-Forward Splitting,When f = g = 0, (5) simplifies to gradient descent iteration Derivation of envelopeNow we show that, (5) can be written as gradient descent iteration of an envelope function under the following assumption.Assumption 1.

A Bregman Forward-Backward Linesearch Algorithm for Nonconvex Composite Optimization: Superlinear Convergence to Nonisolated Local Minima

“…In future work we plan to address the following issues: (i) reducing the working assumptions by also accounting for boundary points, (ii) extending existing superlinear direction schemes such as those proposed in [48,85,3,81] for either convex or smooth problems to the more general setting of this paper, (iii) assessing the performance of such schemes in the Bella framework with numerical simulations on nonconvex nonsmooth problems such as low-rank matrix completion, sparse nonnegative matrix factorization, phase retrieval, and deep learning, and (iv) guaranteeing saddle point avoidance, in the spirit of [67,46,52].…”

Three Operator Splitting with a Nonconvex Loss Function

“…Existing work on TOS applied to nonconvex problems limits itself to the setting where at least two terms in (1) have Lipschitz continuous gradients. Under this assumption, Liu & Yin (2019) identify an envelope function for TOS, which permits one to interpret TOS as gradient descent for this envelope under a variable metric. Their envelope generalizes the well-known Moreau envelope as well as the envelopes for Douglas-Rachford and Forward-Backward splitting introduced in (Patrinos et al, 2014) and (Themelis et al, 2018).…”

Perturbed Proximal Descent to Escape Saddle Points for Non-convex and Non-smooth Objective Functions

“…A recent preprint [15] approaches the problem of finding local minima using the forward-backward envelope technique developed in [19], where the assumption about the smoothness of objective function is weakened to local smoothness instead of global smoothness.…”