Risk-Aware Linear Bandits: Theory and Applications in Smart Order Routing

Motivated by practical considerations in machine learning for financial decision-making, such as risk aversion and large action space, we consider risk-aware bandits optimization with applications in smart order routing (SOR). Specifically, based on preliminary observations of linear price impacts made from the NASDAQ ITCH dataset, we initiate the study of risk-aware linear bandits. In this setting, we aim at minimizing regret, which measures our performance deficit compared to the optimum's, under the mean-variance metric when facing a set of actions whose rewards are linear functions of (initially) unknown parameters. Driven by the variance-minimizing globally-optimal (G-optimal) design, we propose the novel instance-independent Risk-Aware Explore-then-Commit (RISE) algorithm and the instance-dependent Risk-Aware Successive Elimination (RISE++) algorithm. Then, we rigorously analyze their near-optimal regret upper bounds to show that, by leveraging the linear structure, our algorithms can dramatically reduce the regret when compared to existing methods. Finally, we demonstrate the performance of the algorithms by conducting extensive numerical experiments in the SOR setup using both synthetic datasets and the NASDAQ ITCH dataset. Our results reveal that 1) The linear structure assumption can indeed be well supported by the Nasdaq dataset; and more importantly 2) Both RISE and RISE++ can significantly outperform the competing methods, in terms of regret, especially in complex decision-making scenarios.