Minimax off-policy evaluation and learning with subgaussian and differentiable importance weighting

In this work, we study the statistical properties of the off-policy estimation problem, i.e., estimating expectations under a target policy using samples collected from a different policy. We begin by presenting a novel minimax concentration lower bound that highlights the fundamental limits of off-policy estimation. We then analyze two well-known importance weighting (IW) techniques: vanilla IW and self-normalized importance weighting (SN). For both methods, we derive concentration and anti-concentration results, showing that their concentration rates are provably suboptimal compared to our lower bound. Observing that this undesired behavior arises from the heavy-tailed nature of the IW and SN estimators, we propose a new class of parametric estimators based on a transformation using the power mean (PM), which is no longer heavy-tailed. We study the theoretical properties of the PM estimator in terms of bias and variance. We show that, with suitable (possibly data-driven) tuning of its parameters, the PM estimator satisfies two key properties under certain conditions: ( i ) it achieves a subgaussian concentration rate that matches our lower bound and ( ii ) it maintains differentiability with respect to the target policy. Finally, we validate our approach through numerical simulations on both synthetic datasets and contextual bandits, comparing it against standard off-policy evaluation and learning baselines. 1