Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier

TL;DR

This paper establishes a surprising equivalence between simulated annealing and interior point methods using the entropic universal barrier, leading to improved convex optimization algorithms with better time complexity.

Contribution

It reveals a novel connection between simulated annealing and interior point methods, enabling faster algorithms and universal barriers for convex sets via a unified framework.

Findings

Improved time complexity for convex optimization under membership oracle model.

Tightened temperature schedule reduces running time by square root of dimension.

Developed an efficient randomized interior point method with a universal barrier.

Abstract

This paper explores a surprising equivalence between two seemingly-distinct convex optimization methods. We show that simulated annealing, a well-studied random walk algorithms, is directly equivalent, in a certain sense, to the central path interior point algorithm for the the entropic universal barrier function. This connection exhibits several benefits. First, we are able improve the state of the art time complexity for convex optimization under the membership oracle model. We improve the analysis of the randomized algorithm of Kalai and Vempala by utilizing tools developed by Nesterov and Nemirovskii that underly the central path following interior point algorithm. We are able to tighten the temperature schedule for simulated annealing which gives an improved running time, reducing by square root of the dimension in certain instances. Second, we get an efficient randomized interior point method with an efficiently computable universal barrier for any convex set described by a membership oracle. Previously, efficiently computable barriers were known only for particular convex sets.