# New Insights into Bootstrapping for Bandits

We investigate the use of bootstrapping in the bandit setting. We first show that the commonly used non-parametric bootstrapping (NPB) procedure can be provably inefficient and establish a near-linear lower bound on the regret incurred by it under the bandit model with Bernoulli rewards. We show that NPB with an appropriate amount of forced exploration can result in sub-linear albeit sub-optimal regret. As an alternative to NPB, we propose a weighted bootstrapping (WB) procedure. For Bernoulli rewards, WB with multiplicative exponential weights is mathematically equivalent to Thompson sampling (TS) and results in near-optimal regret bounds. Similarly, in the bandit setting with Gaussian rewards, we show that WB with additive Gaussian weights achieves near-optimal regret. Beyond these special cases, we show that WB leads to better empirical performance than TS for several reward distributions bounded on $[0,1]$. For the contextual bandit setting, we give practical guidelines that make bootstrapping simple and efficient to implement and result in good empirical performance on real-world datasets.

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