Online Convex Optimization Against Adversaries with Memory and Application to Statistical Arbitrage

TL;DR

This paper extends online learning with memory to the broader online convex optimization framework, proposing algorithms with optimal regret bounds and applying them to financial statistical arbitrage.

Contribution

It introduces two algorithms for online convex optimization with memory, achieving optimal regret bounds for convex and strongly convex losses, including non-Lipschitz cases.

Findings

Algorithms attain optimal regret bounds for convex and strongly convex losses.

Application to statistical arbitrage demonstrates practical effectiveness.

Extended framework captures temporal constraints in online learning.

Abstract

The framework of online learning with memory naturally captures learning problems with temporal constraints, and was previously studied for the experts setting. In this work we extend the notion of learning with memory to the general Online Convex Optimization (OCO) framework, and present two algorithms that attain low regret. The first algorithm applies to Lipschitz continuous loss functions, obtaining optimal regret bounds for both convex and strongly convex losses. The second algorithm attains the optimal regret bounds and applies more broadly to convex losses without requiring Lipschitz continuity, yet is more complicated to implement. We complement our theoretic results with an application to statistical arbitrage in finance: we devise algorithms for constructing mean-reverting portfolios.