Best of both worlds: Stochastic & adversarial best-arm identification
This work addresses the fundamental challenge of designing robust bandit algorithms for best-arm identification, providing impossibility results and near-optimal solutions for practitioners who face unknown reward types.
The paper studies best-arm identification in bandits with both stochastic and adversarial rewards, showing that a learner cannot be optimal in both settings simultaneously. It provides a lower bound characterizing the optimal rate for stochastic problems under adversarial robustness and proposes a parameter-free algorithm that matches this bound up to log factors while remaining robust to adversarial rewards.
We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.