Principled Analyses and Design of First-Order Methods with Inexact Proximal Operators

Proximal operations are among the most common primitives appearing in both practical and theoretical (or high-level) optimization methods. This basic operation typically consists in solving an intermediary (hopefully simpler) optimization problem. In this work, we survey notions of inaccuracies that can be used when solving those intermediary optimization problems. Then, we show that worst-case guarantees for algorithms relying on such inexact proximal operations can be systematically obtained through a generic procedure based on semidefinite programming. This methodology is primarily based on the approach introduced by Drori and Teboulle (2014) and on convex interpolation results, and allows producing non-improvable worst-case analyzes. In other words, for a given algorithm, the methodology generates both worst-case certificates (i.e., proofs) and problem instances on which those bounds are achieved. Relying on this methodology, we study numerical worst-case performances of a few basic methods relying on inexact proximal operations including accelerated variants, and design a variant with optimized worst-case behaviour. We further illustrate how to extend the approach to support strongly convex objectives by studying a simple relatively inexact proximal minimization method.