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Estimating optimal dynamic policies from offline data is a fundamentalproblem in dynamic decision making. In the context of causal inference, theproblem is known as estimating the optimal dynamic treatment regime. Eventhough there exists a plethora of methods for estimation, constructingconfidence intervals for the value of the optimal regime and structuralparameters associated with it is inherently harder, as it involves non-linearand non-differentiable functionals of unknown quantities that need to beestimated. Prior work resorted to sub-sample approaches that can deterioratethe quality of the estimate. We show that a simple soft-max approximation tothe optimal treatment regime, for an appropriately fast growing temperatureparameter, can achieve valid inference on the truly optimal regime. Weillustrate our result for a two-period optimal dynamic regime, though ourapproach should directly extend to the finite horizon case. Our work combinestechniques from semi-parametric inference and $g$-estimation, together with anappropriate triangular array central limit theorem, as well as a novel analysisof the asymptotic influence and asymptotic bias of softmax approximations.

Inference on Optimal Dynamic Policies via Softmax Approximation