Estimating optimal dynamic policies from offline data is a fundamental problem in dynamic decision making. In the context of causal inference, the problem is known as estimating the optimal dynamic treatment regime. Even though there exists a plethora of methods for estimation, constructing confidence intervals for the value of the optimal regime and structural parameters associated with it is inherently harder, as it involves non-linear and non-differentiable functionals of un-known quantities that need to be estimated. Prior work resorted to sub-sample approaches that can deteriorate the quality of the estimate. We show that a simple soft-max approximation to the optimal treatment regime, for an appropriately fast growing temperature parameter, can achieve valid inference on the truly optimal regime. We illustrate our result for a two-period optimal dynamic regime, though our approach should directly extend to the finite horizon case. Our work combines techniques from semi-parametric inference and $g$-estimation, together with an appropriate triangular array central limit theorem, as well as a novel analysis of the asymptotic influence and asymptotic bias of softmax approximations.

翻译：从离线数据估计最优动态策略是动态决策中的基本问题。在因果推断背景下，该问题被称为最优动态治疗方案估计。尽管已有大量估计方法，但为最优策略的价值及其相关结构参数构建置信区间本质上更为困难，因为这涉及需要估计的未知量的非线性、不可微泛函。先前研究采用子样本方法，但可能降低估计质量。我们证明，对于适当快速增长的温度参数，最优治疗策略的简单softmax近似能够实现真正最优策略的有效推断。我们以两期最优动态策略为例说明结果，但该方法应可直接推广至有限时域情形。本工作结合半参数推断与g估计技术，配合适当的三角阵列中心极限定理，以及对softmax近似渐进影响与渐进偏差的新颖分析。