Adaptive Online Learning and Portfolio Optimization: Regret Minimization, Robustness, and Efficient Algorithms

Financial markets are dynamic, volatile, and often adversarial, making the design of portfolio strategies that are both adaptive and theoretically sound a central challenge in quantitative finance. This dissertation develops new algorithms for online portfolio selection on both active and passive sides, within the framework of online convex optimization (OCO), with a focus on balancing adaptivity and robustness through regret minimization. On the active side, we propose the Strongly Adaptive Optimistic Follow-the-Regularized-Leader (SAOFTRL) framework. This framework combines optimistic gradient prediction with a strongly adaptive meta-algorithm (SA-meta) to enhance stability and robustness. We show that SAOFTRL achieves a novel regret bound that combines the advantages of predictive signals and SA-meta, ensuring robustness in worst-case scenarios. We further reinterpret SAOFTRL as a mean-variance portfolio framework with a shrinkage estimator and adaptive lookback windows, thereby ensuring reliable performance in the average case. For practical deployment, we design an efficient algorithm based on the Successive Convex Approximation (SCA) method, reducing computational complexity without compromising the regret bound. Empirical studies show that compared to existing benchmarks, SAOFTRL achieves faster convergence speed and better returns. On the passive side, we introduce the Adaptive Passive-Aggressive Strategy (APAS), a general-purpose online regression framework developed for enhanced index tracking. APAS adaptively adjusts portfolio weights to optimize cumulative returns while minimizing low tracking error. We establish that APAS converges to the optimal solution with a regret bound of O( T ), even under non-convex loss functions. To ensure scalability, we design an efficient SCA-based implementation capable of handling large-scale problems. Experiments on both synthetic and real market datasets demonstrate that APAS consistently achieves lower tracking error and higher returns compared to traditional methods. These results shed light on the trade-off between adaptivity and robustness in online portfolio optimization. The proposed methods advance both the theoretical understanding and the practical application of online learning in financial decision-making.