Statistical Inference with M-Estimators on Adaptively Collected Data
This addresses the need for reliable scientific analysis in sequential decision-making applications such as advertising and mobile health, offering a general solution beyond existing incremental methods for simple models.
The paper tackles the problem of conducting valid statistical inference on data collected by adaptive bandit algorithms, which classical methods fail to handle, and develops theory showing that modified M-estimators can provide asymptotically valid confidence regions for complex models like logistic regression.
Bandit algorithms are increasingly used in real-world sequential decision-making problems. Associated with this is an increased desire to be able to use the resulting datasets to answer scientific questions like: Did one type of ad lead to more purchases? In which contexts is a mobile health intervention effective? However, classical statistical approaches fail to provide valid confidence intervals when used with data collected with bandit algorithms. Alternative methods have recently been developed for simple models (e.g., comparison of means). Yet there is a lack of general methods for conducting statistical inference using more complex models on data collected with (contextual) bandit algorithms; for example, current methods cannot be used for valid inference on parameters in a logistic regression model for a binary reward. In this work, we develop theory justifying the use of M-estimators -- which includes estimators based on empirical risk minimization as well as maximum likelihood -- on data collected with adaptive algorithms, including (contextual) bandit algorithms. Specifically, we show that M-estimators, modified with particular adaptive weights, can be used to construct asymptotically valid confidence regions for a variety of inferential targets.