A Simple, Optimal and Efficient Algorithm for Online Exp-Concave Optimization

TL;DR

This paper introduces LightONS, a simplified and computationally efficient variant of the Online Newton Step algorithm for exp-concave optimization, maintaining optimal regret while significantly reducing runtime.

Contribution

The paper presents LightONS, a novel algorithm that reduces computational complexity of online exp-concave optimization without sacrificing regret optimality, addressing a key bottleneck in existing methods.

Findings

LightONS achieves $O(d^2 T + d^ ext{omega} \, \sqrt{T \log T})$ runtime.

LightONS maintains the optimal $O(d \log T)$ regret bound.

Application to stochastic optimization yields $ ilde{O}(d^3/\epsilon)$ runtime.

Abstract

Online eXp-concave Optimization (OXO) is a fundamental problem in online learning, where the goal is to minimize regret when loss functions are exponentially concave. The standard algorithm, Online Newton Step (ONS), guarantees an optimal $O(d \log T)$ regret, where $d$ is the dimension and $T$ is the time horizon. Despite its simplicity, ONS may face a computational bottleneck due to the Mahalanobis projection at each round. This step costs $\Omega(d^\omega)$ arithmetic operations for bounded domains, even for simple domains such as the unit ball, where $\omega \in (2,3]$ is the matrix-multiplication exponent. As a result, the total runtime can reach $\tilde{O}(d^\omega T)$, particularly when iterates frequently oscillate near the domain boundary. This paper proposes a simple variant of ONS, called LightONS, which reduces the total runtime to $O(d^2 T + d^\omega \sqrt{T \log T})$ while preserving the optimal regret. Deploying LightONS with the online-to-batch conversion implies a method for stochastic exp-concave optimization with runtime $\tilde{O}(d^3/\epsilon)$, thereby answering an open problem posed by Koren [2013]. The design leverages domain-conversion techniques from parameter-free online learning and defers expensive Mahalanobis projections until necessary, thereby preserving the elegant structure of ONS and enabling LightONS to act as an efficient plug-in replacement in broader scenarios, including gradient-norm adaptivity, parametric stochastic bandits, and memory-efficient OXO.