Online Decision Making with High-Dimensional Covariates

Big data have enabled decision makers to tailor decisions at the individual level in a variety of domains, such as personalized medicine and online advertising. Doing so 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 polylogarithmically 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 and physicians in correctly dosing a majority of patients.