Three Operator Splitting with a Nonconvex Loss Function

TL;DR

This paper extends the theoretical analysis of the Three Operator Splitting method to nonconvex problems with a differentiable nonconvex function and two convex functions, providing convergence guarantees without extra smoothness assumptions.

Contribution

It offers new convergence guarantees for TOS in nonconvex settings, including cases with nondifferentiable terms, and extends the analysis to stochastic scenarios.

Findings

Proves convergence of TOS with nonasymptotic bounds.

Handles nondifferentiable convex terms, including indicator functions.

Demonstrates effectiveness on quadratic assignment problems.

Abstract

We consider the problem of minimizing the sum of three functions, one of which is nonconvex but differentiable, and the other two are convex but possibly nondifferentiable. We investigate the Three Operator Splitting method (TOS) of Davis & Yin (2017) with an aim to extend its theoretical guarantees for this nonconvex problem template. In particular, we prove convergence of TOS with nonasymptotic bounds on its nonstationarity and infeasibility errors. In contrast with the existing work on nonconvex TOS, our guarantees do not require additional smoothness assumptions on the terms comprising the objective; hence they cover instances of particular interest where the nondifferentiable terms are indicator functions. We also extend our results to a stochastic setting where we have access only to an unbiased estimator of the gradient. Finally, we illustrate the effectiveness of the proposed method through numerical experiments on quadratic assignment problems.