Before revision:  Specifically, in settings in which the data is collected via a (contextual) bandit algorithm, we prove that certain adaptively weighted M-estimators are uniformly asymptotically normal and demonstrate empirically that we can use their asymptotic distribution to construct reliable confidence regions for a variety of inferential targets.
After revision:  Specifically, in settings in which the data is collected via a (contextual) bandit algorithm, we prove that certain adaptively weighted M-estimators , modified with particular adaptive weights, can be used to construct asymptotically valid confidence regions for a variety of inferential targets.