Multi-Task Best Arm Identification with Risk Constraints

Abstract

Best Arm Identification (BAI) is a challenging problem in sequential decision-making with numerous realworld applications. Existing approaches typically assume all arms are feasible or address expectation-based constraints, often relying on strong assumptions, loose sample complexity bounds, and non-optimal solutions. This paper introduces a multi-task BAI problem with risk constraints in the fixed-confidence setting, where each arm is evaluated across multiple performance metrics. The agent seeks to optimize one metric while ensuring that the quantiles of other metrics remain below specified thresholds for each task. We first derive a tight, instance-dependent lower bound on sample complexity. Using this bound, we define optimality conditions for the static optimal sampling ratio, demonstrating how it balances trade-offs between tasks, constraints, and the interplay of optimality and feasibility. We then propose a Track-and-Stop strategy with asymptotically optimal sample complexity, complemented by a computationally efficient strategy that iteratively refines the optimality conditions. Numerical experiments confirm that our algorithm performs competitively.

Keywords: Best arm identification, risk constraints, fixed-confidence, sample complexity, multi-task