Adaptive experimental designs can dramatically improve efficiency in randomized trials. But with adaptively collected data, common estimators based on sample means and inverse propensity-weighted means can be biased or heavy-tailed. This poses statistical challenges, in particular when the experimenter would like to test hypotheses about parameters that were not targeted by the data-collection mechanism. In this paper, we present a class of test statistics that can handle these challenges. Our approach is to adaptively reweight the terms of an augmented inverse propensity-weighting estimator to control the contribution of each term to the estimator’s variance. This scheme reduces overall variance and yields an asymptotically normal test statistic. We validate the accuracy of the resulting estimates and their CIs in numerical experiments and show that our methods compare favorably to existing alternatives in terms of mean squared error, coverage, and CI size.

Adaptive experiments can result in considerable cost savings in multi-armed trials by enabling analysts to quickly focus on the most promising alternatives. Most existing work on adaptive experiments (which include multi-armed bandits) has focused maximizing the speed at which the analyst can identify the optimal arm and/or minimizing the number of draws from sub-optimal arms. In many scientific settings, however, it is not only of interest to identify the optimal arm, but also to perform a statistical analysis of the data collected from the experiment. Naive approaches to statistical inference with adaptive inference fail because many commonly used statistics (such as sample means or inverse propensity weighting) do not have an asymptotically Gaussian limiting distribution centered on the estimate, and so confidence intervals constructed from these statistics do not have correct coverage. But, as shown in this paper, carefully designed data-adaptive weighting schemes can be used to overcome this issue and restore a relevant central limit theorem, enabling hypothesis testing. We validate the accuracy of the resulting confidence intervals in numerical experiments.

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It has become increasingly common for data to be collected adaptively, for example using contextual bandits. Historical data of this type can be used to evaluate other treatment assignment policies to guide future innovation or experiments. However, policy evaluation is challenging if the target policy differs from the one used to collect data, and popular estimators, including doubly robust (DR) estimators, can be plagued by bias, excessive variance, or both. In particular, when the pattern of treatment assignment in the collected data looks little like the pattern generated by the policy to be evaluated, the importance weights used in DR estimators explode, leading to excessive variance.In this paper, we improve the DR estimator by adaptively weighting observations to control its variance. We show that a t-statistic based on our improved estimator is asymptotically normal under certain conditions, allowing us to form confidence intervals and test hypotheses. Using synthetic data and public benchmarks, we provide empirical evidence for our estimator's improved accuracy and inferential properties relative to existing alternatives.