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

In many online learning scenarios the loss functions are not memoryless, but rather depend on history. Our first contribution is a complete characterization of sufficient and necessary conditions for learning with memory, accompanied with a novel algorithm for this framework that attains the optimal O( √ T )-regret. This improves previous online learning algorithms that guaranteed O(T ) regret and required more stringent conditions. As an application of the new technique, we address the classical problem in finance of constructing mean reverting portfolios. We design an efficient online learning algorithm for this problem, and provide guarantees for its performance. We complement our theoretical findings with an empirical study that verifies our theoretical results on financial data.