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

arXiv: https://arxiv.org/abs/1804.08739 · HTML: https://mlsanity.com/paper/1804.08739/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Apr 2018.

## Built on by (most cited first, 1 of 1)

- [Stochastic Gradient Langevin Dynamics with Variance Reduction](https://mlsanity.com/paper/2102.06759/) (2021)

## Abstract (arXiv)

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.