On the Bias, Risk, and Consistency of Sample Means in Multi-armed Bandits

The sample mean is among the most well-studied estimators in statistics, having many desirable properties such as unbiasedness and consistency. However, when analyzing data collected using a multi-armed bandit (MAB) experiment, the sample mean is biased and much remains to be understood about its properties. For example, when is it consistent, how large is its bias, and can we bound its mean squared error? This paper delivers a thorough and systematic treatment of the bias, risk, and consistency of MAB sample means. Specifically, we identify four distinct sources of selection bias (sampling, stopping, choosing, and rewinding) and analyze them both separately and together. We further demonstrate that a new notion of effective sample size can be used to bound the risk of the sample mean under suitable loss functions. We present several carefully designed examples to provide intuition on the different sources of selection bias we study. Our treatment is nonparametric and algorithm-agnostic, meaning that it is not tied to a specific algorithm or goal. In a nutshell, our proofs combine variational representations of information-theoretic divergences with new martingale concentration inequalities.

Automated summary

This paper extensively discusses the bias, risk, and consistency of sample means in multi-armed bandit experiments, identifying four distinct sources of selection bias. A new notion of effective sample size is introduced to bound the risk of the sample mean, with carefully designed examples provided for better understanding of the various sources of selection bias studied. The proofs in the paper combine variational representations of information-theoretic divergences with new martingale concentration inequalities.