Regret Minimization and Statistical Inference in Online Decision Making with High-dimensional Covariates
This addresses the challenge of balancing exploration and exploitation in high-dimensional bandit problems, with applications in areas like personalized medicine, though it is incremental in combining existing techniques.
The paper tackles the trade-off between regret minimization and statistical inference in high-dimensional online decision-making, showing that under certain conditions, a pure-greedy algorithm can achieve optimal O(log T) regret and O(T^{1/2})-consistent inference simultaneously.
This paper investigates regret minimization, statistical inference, and their interplay in high-dimensional online decision-making based on the sparse linear context bandit model. We integrate the $\varepsilon$-greedy bandit algorithm for decision-making with a hard thresholding algorithm for estimating sparse bandit parameters and introduce an inference framework based on a debiasing method using inverse propensity weighting. Under a margin condition, our method achieves either $O(T^{1/2})$ regret or classical $O(T^{1/2})$-consistent inference, indicating an unavoidable trade-off between exploration and exploitation. If a diverse covariate condition holds, we demonstrate that a pure-greedy bandit algorithm, i.e., exploration-free, combined with a debiased estimator based on average weighting can simultaneously achieve optimal $O(\log T)$ regret and $O(T^{1/2})$-consistent inference. We also show that a simple sample mean estimator can provide valid inference for the optimal policy's value. Numerical simulations and experiments on Warfarin dosing data validate the effectiveness of our methods.