An Efficient Interior-Point Method for Online Convex Optimization

TL;DR

This paper introduces an efficient interior-point method for online convex optimization that achieves near-optimal regret bounds, is adaptive over sub-intervals, and has computational complexity comparable to solving linear systems.

Contribution

The paper presents a new interior-point algorithm for online convex optimization with optimal regret bounds and adaptive properties, improving computational efficiency.

Findings

Achieves regret of O(√T log T), near the theoretical minimum.

Algorithm is adaptive to sub-intervals, not just the entire time horizon.

Runs efficiently by solving linear systems of size n per iteration.

Abstract

A new algorithm for regret minimization in online convex optimization is described. The regret of the algorithm after $T$ time periods is $O(\sqrt{T \log T})$ - which is the minimum possible up to a logarithmic term. In addition, the new algorithm is adaptive, in the sense that the regret bounds hold not only for the time periods $1,\ldots,T$ but also for every sub-interval $s,s+1,\ldots,t$. The running time of the algorithm matches that of newly introduced interior point algorithms for regret minimization: in $n$-dimensional space, during each iteration the new algorithm essentially solves a system of linear equations of order $n$, rather than solving some constrained convex optimization problem in $n$ dimensions and possibly many constraints.