Inference on Optimal Policy Values and Other Irregular Functionals via Softmax Smoothing
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| Format: | Preprint |
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2025
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| author | Whitehouse, Justin Chen, Qizhao Austern, Morgane Syrgkanis, Vasilis |
| author_facet | Whitehouse, Justin Chen, Qizhao Austern, Morgane Syrgkanis, Vasilis |
| contents | Constructing confidence intervals for the value of an (unknown) optimal treatment policy is a fundamental 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. Many existing works circumvent non-differentiability by making the unrealistic assumption of zero probability of treatment non-response, i.e. that every unit responds (either positively or negatively) to an assigned treatment. Further, works that don't circumvent this restriction rely on refitting nuisance models a number of times proportional to the sample size. In this paper, we construct and analyze a simple, softmax smoothing-based estimator for the value of an optimal treatment policy. Our estimator applies in both static and dynamic treatment regimes, only requires fitting a constant number of nuisance models, and is statistically efficient when there is zero probability of non-response to treatment. Also, while our estimator does not require making semi-parametric restrictions, it can exploit them when they exist. We further show how our softmax smoothing approach can be used to estimate general parameters that are specified as a maximum of scores involving nuisance components, and look at conditional Balke and Pearl bounds and $L^1$ calibration error as salient examples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_11780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inference on Optimal Policy Values and Other Irregular Functionals via Softmax Smoothing Whitehouse, Justin Chen, Qizhao Austern, Morgane Syrgkanis, Vasilis Econometrics Machine Learning Statistics Theory Methodology Constructing confidence intervals for the value of an (unknown) optimal treatment policy is a fundamental 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. Many existing works circumvent non-differentiability by making the unrealistic assumption of zero probability of treatment non-response, i.e. that every unit responds (either positively or negatively) to an assigned treatment. Further, works that don't circumvent this restriction rely on refitting nuisance models a number of times proportional to the sample size. In this paper, we construct and analyze a simple, softmax smoothing-based estimator for the value of an optimal treatment policy. Our estimator applies in both static and dynamic treatment regimes, only requires fitting a constant number of nuisance models, and is statistically efficient when there is zero probability of non-response to treatment. Also, while our estimator does not require making semi-parametric restrictions, it can exploit them when they exist. We further show how our softmax smoothing approach can be used to estimate general parameters that are specified as a maximum of scores involving nuisance components, and look at conditional Balke and Pearl bounds and $L^1$ calibration error as salient examples. |
| title | Inference on Optimal Policy Values and Other Irregular Functionals via Softmax Smoothing |
| topic | Econometrics Machine Learning Statistics Theory Methodology |
| url | https://arxiv.org/abs/2507.11780 |