Online Convex Optimization with Memory for Mean-Reverting Portfolios

Summary

This work extends online learning with memory from the experts setting to general online convex optimization. The setting models decisions whose losses depend on a sequence of past actions, capturing temporal constraints that ordinary per-period optimization may miss. The paper presents two algorithms designed to achieve low regret against an adversary whose losses incorporate memory.

One method applies to Lipschitz continuous losses and is described as attaining optimal regret bounds for both convex and strongly convex cases. The other handles a broader class of convex losses without a Lipschitz requirement and also has optimal regret bounds, at the cost of greater implementation complexity. The financial application uses these methods to construct mean-reverting portfolios, connecting the learning framework to statistical arbitrage. The excerpt reports theoretical guarantees and an application, but provides no portfolio data, trading costs, implementation details, or empirical performance figures; practical effectiveness therefore cannot be judged from this description alone.

Key ideas

• Online convex optimization with memory accounts for temporal constraints in sequential decisions.
• The paper proposes two low-regret algorithms for adversarial losses that depend on memory.
• The first algorithm covers Lipschitz losses, including convex and strongly convex cases.
• The second algorithm covers convex losses without requiring Lipschitz continuity, but is more complex to implement.
• The framework is applied to the construction of mean-reverting portfolios for statistical arbitrage.

Tags

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


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









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.

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