Worst-Case Convergence Analysis of Inexact Gradient and Newton Methods Through Semidefinite Programming Performance Estimation

Etienne de Klerk (1) , François Glineur (2) , Adrien Taylor (3)

1 Tilburg University - Department of Econometrics and Operations Research
2 UCL - Université Catholique de Louvain = Catholic University of Louvain
3 SIERRA - Statistical Machine Learning and Parsimony

We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton's method for self-concordant functions, including the case of inexact search directions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori and Teboulle [it Math. Program., 145 (2014), pp. 451--482], and extends recent performance estimation results for the method of Cauchy by the authors [it Optim. Lett., 11 (2017), pp. 1185--1199]. To illustrate the applicability of the tools, we demonstrate a novel complexity analysis of short step interior point methods using inexact search directions. As an example in this framework, we sketch how to give a rigorous worst-case complexity analysis of a recent interior point method by Abernethy and Hazan [it PMLR, 48 (2016), pp. 2520--2528].

Mots clés: performance estimation problems; gradient method; inexact search direction; semidefinite programming; interior point methods

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]

Etienne de Klerk, François Glineur, Adrien Taylor. Worst-Case Convergence Analysis of Inexact Gradient and Newton Methods Through Semidefinite Programming Performance Estimation. SIAM Journal on Optimization, 2020, 30 (3), pp.2053-2082. ⟨10.1137/19M1281368⟩. ⟨hal-02956367⟩