Inference on Optimal Policy Values and Other Irregular Functionals via Smoothing

Constructing confidence intervals for the value of an optimal treatment policy is an important problem in causal inference. Insight into the optimal policy value can guide the development of reward-maximizing, individualized treatment regimes. However, because the functional that defines the optimal value is non-differentiable, standard semi-parametric approaches for performing inference fail to be directly applicable. Existing approaches for handling this non-differentiability fall roughly into two camps. In one camp are estimators based on constructing smooth approximations of the optimal value. These approaches are computationally lightweight, but typically place unrealistic parametric assumptions on outcome regressions. In another camp are approaches that directly de-bias the non-smooth objective. These approaches don't place parametric assumptions on nuisance functions, but they either require the computation of intractably-many nuisance estimates, assume unrealistic $L^\infty$ nuisance convergence rates, or make strong margin assumptions that prohibit non-response to a treatment. In this paper, we revisit the problem of constructing smooth approximations of non-differentiable functionals. By carefully controlling first-order bias and second-order remainders, we show that a softmax smoothing-based estimator can be used to estimate parameters that are specified as a maximum of scores involving nuisance components. In particular, this includes the value of the optimal treatment policy as a special case. Our estimator obtains $\sqrt{n}$ convergence rates, avoids parametric restrictions/unrealistic margin assumptions, and is often statistically efficient.