Online Decision-Making with High-Dimensional Covariates

Abstract

Big data has enabled decision-makers to tailor decisions at the individual-level in a variety of domains such as personalized medicine and online advertising. This involves learning a model of decision rewards conditional on individual-specific covariates. In many practical settings, these covariates are high-dimensional; however, typically only a small subset of the observed features are predictive of a decision’s success. We formulate this problem as a K-armed contextual bandit with high-dimensional covariates, and present a new efficient bandit algorithm based on the LASSO estimator. We prove that our algorithm’s cumulative expected regret scales at most poly-logarithmically in the covariate dimension d; to the best of our knowledge, this is the first such bound for a contextual bandit. The key step in our analysis is proving a new tail inequality that guarantees the convergence of the LASSO estimator despite the non-i.i.d. data induced by the bandit policy. Furthermore, we illustrate the practical relevance of our algorithm by evaluating it on a simplified version of a medication dosing problem. A patient’s optimal medication dosage depends on the patient’s genetic profile and medical records; incorrect initial dosage may result in adverse consequences such as stroke or bleeding. We show that our algorithm outperforms existing bandit methods as well as physicians to correctly dose a majority of patients.

Keywords: contextual bandits, adaptive treatment allocation, online learning, high-dimensional statistics, LASSO, personalized decision-making