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Inference on Optimal Dynamic Policies via Softmax Approximation

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  • Qizhao Chen
  • Morgane Austern
  • Vasilis Syrgkanis

Abstract

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 unknown 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.

Suggested Citation

  • Qizhao Chen & Morgane Austern & Vasilis Syrgkanis, 2023. "Inference on Optimal Dynamic Policies via Softmax Approximation," Papers 2303.04416, arXiv.org, revised Dec 2023.
  • Handle: RePEc:arx:papers:2303.04416
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    References listed on IDEAS

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    1. Erica E. M. Moodie & Thomas S. Richardson & David A. Stephens, 2007. "Demystifying Optimal Dynamic Treatment Regimes," Biometrics, The International Biometric Society, vol. 63(2), pages 447-455, June.
    2. S. A. Murphy, 2003. "Optimal dynamic treatment regimes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 331-355, May.
    3. Emmanuel Rio, 2009. "Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions," Journal of Theoretical Probability, Springer, vol. 22(1), pages 146-163, March.
    4. Victor Chernozhukov & Whitney Newey & Rahul Singh & Vasilis Syrgkanis, 2020. "Adversarial Estimation of Riesz Representers," Papers 2101.00009, arXiv.org, revised Apr 2024.
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    Cited by:

    1. Gyungbae Park, 2024. "Debiased Machine Learning when Nuisance Parameters Appear in Indicator Functions," Papers 2403.15934, arXiv.org.

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