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Exact Worst-case Performance of First-order Methods for Composite Convex Optimization

Adrien B. Taylor (adrien.taylor***at***uclouvain.be)
Julien M. Hendrickx (julien.hendrickx***at***uclouvain.be)
François Glineur (francois.glineur***at***uclouvain.be)

Abstract: We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this worst-case. We achieve this by reducing the computation of the worst-case to solving a convex semidefinite program, generalizing previous works on performance estimation by Drori and Teboulle [13] and the authors [43]. We use these developments to obtain a tighter analysis of the proximal point algorithm and of several variants of fast proximal gradient, conditional gradient, subgradient and alternating projection methods. In particular, we present a new analytical worst-case guarantee for the proximal point algorithm that is twice better than previously known, and improve the standard worst-case guarantee for the conditional gradient method by more than a factor of two. We also show how the optimized gradient method proposed by Kim and Fessler in [22] can be extended by incorporating a projection or a proximal operator, which leads to an algorithm that converges in the worst-case twice as fast as the standard accelerated proximal gradient method [2].

Keywords: composite convex optimization, first-order methods, performance estimation, ex- act worst-case analysis, convergence rate, proximal point algorithm, conditional gradient method, optimized gradient method, semidefinite programming, convex interpolation

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: Accepted in SIOPT

Download: [PDF]

Entry Submitted: 06/23/2016
Entry Accepted: 06/23/2016
Entry Last Modified: 02/06/2017

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