On Some Versions of Subspace Optimization Methods with Inexact Gradient Information

TL;DR

This paper proposes modified subspace optimization methods for convex and quasar-convex problems that effectively handle inexact gradient information, analyzing their convergence under various inexactness conditions.

Contribution

It introduces new modifications of subspace optimization methods that accommodate inexact gradients and extends convergence analysis to quasar-convex functions.

Findings

Methods converge under different inexactness conditions.

Extension of convergence results to quasar-convex functions.

Analysis of inexactness impact on convergence rates.

Abstract

It is well-known that accelerated gradient first order methods possess optimal complexity estimates for the class of convex smooth minimization problems. In many practical situations, it makes sense to work with inexact gradients. However, this can lead to the accumulation of corresponding inexactness in the theoretical estimates of the rate of convergence. We propose some modification of the methods for convex optimization with inexact gradient based on the subspace optimization such as Nemirovski's Conjugate Gradients and Sequential Subspace Optimization. We research the method convergence for different condition of inexactness both in gradient value and accuracy of subspace optimization problems. Besides this, we investigate generalization of this result to the class of quasar-convex (weakly-quasi-convex) functions.