Imitation-regularized offline learning with small or missing action probabilities
We study the problem of offline learning in automated decision systems under the contextual bandits model. We are given logged historical data consisting of contexts, (randomized) actions, and (nonnegative) rewards. A common goal is to evaluate what would happen if different actions were taken in the same contexts, so as to optimize the action policies accordingly. The typical approach to this problem, inverse probability weighted estimation (IPWE) , requires logged action probabilities, which may be missing in practice due to engineering complications. Even when available, small action probabilities cause large uncertainty in IPWE, rendering the corresponding results insignificant. To solve both problems, we show how one can use policy improvement (PIL) objectives, regularized by policy imitation (IML). We motivate and analyze PIL as an extension to Clipped-IPWE, by showing that both are lower-bound surrogates to the vanilla IPWE. We also formally connect IML to IPWE variance estimation  and natural policy gradients. Without probability logging, our PIL-IML interpretations justify and improve, by reward-weighting, the state-of- art cross-entropy (CE) loss that predicts the action items among all action candidates available in the same contexts. With probability logging, our main theoretical contribution connects IML-underfitting to the existence of either confounding variables or model mis-specification. We show the value and accuracy of our insights by simulations based on Simpson’s paradox, standard UCI multiclassto- bandit conversions and on the Criteo counterfactual analysis challenge dataset.