On conditional versus marginal bias in multi-armed bandits
This work addresses bias assessment in adaptive data analysis for researchers in statistics and machine learning, offering incremental insights into conditional inference settings.
The paper tackles the problem of bias in multi-armed bandits by characterizing the sign of conditional bias for monotone functions like sample means, showing that conditional and marginal bias can differ depending on the conditioning event.
The bias of the sample means of the arms in multi-armed bandits is an important issue in adaptive data analysis that has recently received considerable attention in the literature. Existing results relate in precise ways the sign and magnitude of the bias to various sources of data adaptivity, but do not apply to the conditional inference setting in which the sample means are computed only if some specific conditions are satisfied. In this paper, we characterize the sign of the conditional bias of monotone functions of the rewards, including the sample mean. Our results hold for arbitrary conditioning events and leverage natural monotonicity properties of the data collection policy. We further demonstrate, through several examples from sequential testing and best arm identification, that the sign of the conditional and marginal bias of the sample mean of an arm can be different, depending on the conditioning event. Our analysis offers new and interesting perspectives on the subtleties of assessing the bias in data adaptive settings.