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

Minor modifications including acknowledgments and references. Code available at https://github.com/mathbarre/InexactProximalOperatorsProximal 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 (Mathematical Programming, 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 provide three new methods with conceptually simple proofs: (i) an optimized relatively inexact proximal point method, (ii) an extension of the hybrid proximal extragradient method of Monteiro and Svaiter (SIAM Journal on Optimization, 2013), and (iii) an inexact accelerated forward-backward splitting supporting backtracking line-search, and both (ii) and (iii) supporting possibly strongly convex objectives. Finally, we use the methodology for studying a recent inexact variant of the Douglas-Rachford splitting due to Eckstein and Yao (Mathematical Programming, 2018). We showcase and compare the different variants of the accelerated inexact forward-backward method on a factorization and a total variation problem