标题： 通过平滑方法对最优策略值和其他不规则泛函进行推断 显示英文标题

标题： Inference on Optimal Policy Values and Other Irregular Functionals via Smoothing

摘要： 构建最优治疗策略值的置信区间是因果推理中的一个重要问题。 深入了解最优策略值可以指导制定奖励最大化的个性化治疗方案。 然而，由于定义最优值的函数不可微，因此用于执行推理的标准半参数方法无法直接应用。 现有的处理这种不可微性的方法大致分为两大阵营。 一类是基于构建最优值的平滑近似值的估计器。 这些方法计算量轻，但通常对结果回归做出不切实际的参数假设。 另一类是直接消除非平滑目标偏差的方法。这些方法并未对干扰函数做出参数假设，但它们要么需要计算大量难以处理的干扰估计，要么假设不切实际的 $L^\infty$ 个干扰收敛率，要么做出强边际假设以阻止对某种治疗无反应。 在本文中，我们重新探讨了构建不可微函数的平滑近似的问题。 通过仔细控制一阶偏差和二阶余数，我们证明了基于 softmax 平滑的估计器可用于估计指定为涉及干扰成分的得分最大值的参数。 具体而言，这包括作为特例的最优治疗策略的值。 我们的估计器获得了 $\sqrt{n}$ 的收敛率，避免了参数限制/不切实际的边际假设，并且通常具有统计有效性。 显示英文摘要

摘要： 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.