It's Hard to Be Normal: The Impact of Noise on Structure-agnostic Estimation

Jikai Jin; Lester Mackey; Vasilis Syrgkanis

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arXiv · 2507.02275

It's Hard to Be Normal: The Impact of Noise on Structure-agnostic Estimation

Jikai Jin·Lester Mackey·Vasilis Syrgkanis

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Abstract

Structure-agnostic causal inference studies how well one can estimate a treatment effect given black-box machine learning estimates of nuisance functions (like the impact of confounders on treatment and outcomes). Here, we find that the answer depends in a surprising way on the distribution of the treatment noise. Focusing on the partially linear model of \citet{robinson1988root}, we first show that the widely adopted double machine learning (DML) estimator is minimax rate-optimal for Gaussian treatment noise, resolving an open problem of \citet{mackey2018orthogonal}. Meanwhile, for independent non-Gaussian treatment noise, we show that DML is always suboptimal by constructing new practical procedures with higher-order robustness to nuisance errors. These \emph{ACE} procedures use structure-agnostic cumulant estimators to achieve $r$-th order insensitivity to nuisance errors whenever the $(r+1)$-st treatment cumulant is non-zero. We complement these core results with novel minimax guarantees for binary treatments in the partially linear model. Finally, using synthetic demand estimation experiments, we demonstrate the practical benefits of our higher-order robust estimators.

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stat.ML cs.LG econ.EM math.ST stat.ME stat.TH

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BibTeX RIS

Jikai Jin, Lester Mackey, Vasilis Syrgkanis. 2025-07-03. It's Hard to Be Normal: The Impact of Noise on Structure-agnostic Estimation. https://arxiv.org/abs/2507.02275

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