No-Regret Algorithms for Heavy-Tailed Linear Bandits

We analyze the problem of linear bandits under heavy tailed noise. Most of the work on linear bandits has been based on the assumption of bounded or sub-Gaussian noise. This assumption however is often violated in common scenarios such as financial markets. We present two algorithms to tackle this problem: one based on dynamic truncation and one based on a median of means estimator. We show that, when the noise admits only a 1 + epsilon moment, these algorithms are still able to achieve regret in (O) over tilde (T2+epsilon 2(1+epsilon)) and (O) over tilde (T1+2 epsilon/1+3 epsilon) respectively. In particular, they guarantee sublinear regret as long as the noise has finite variance. We also present empirical results showing that our algorithms achieve a better performance than the current state of the art for bounded noise when the L-infinity bound on the noise is large yet the 1+epsilon moment of the noise is small.