# Sharp Structure-Agnostic Minimax Risk for Partial Linear Models

arXiv: https://arxiv.org/abs/2609.07997 · HTML: https://mlsanity.com/paper/2609.07997/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 36 references that are in the map.
- First posted on arXiv 7 Sep 2026.

## Builds on (most cited first, 14 of 14)

- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [Orthogonal Machine Learning: Power and Limitations](https://mlsanity.com/paper/1711.00342/) (2017)
- [The Fundamental Limits of Structure-Agnostic Functional Estimation](https://mlsanity.com/paper/2305.04116/) (2023)
- [A Simple and General Debiased Machine Learning Theorem with Finite Sample Guarantees](https://mlsanity.com/paper/2105.15197/) (2021)
- [On Doubly Robust Inference for Double Machine Learning in Semiparametric Regression](https://mlsanity.com/paper/2107.06124/) (2021)
- [It's Hard to Be Normal: The Impact of Noise on Structure-agnostic Estimation](https://mlsanity.com/paper/2507.02275/) (2025)
- [Wild refitting for black box prediction](https://mlsanity.com/paper/2506.21460/) (2025)
- [Sharp Structure-Agnostic Lower Bounds for General Linear Functional Estimation](https://mlsanity.com/paper/2512.17341/) (2025)
- [Interleaved Resampling and Refitting: Data and Compute-Efficient Evaluation of Black-Box Predictors](https://mlsanity.com/paper/2603.14218/) (2026)
- [Optimally taming biases in black-box models for efficient semiparametric estimation](https://mlsanity.com/paper/2606.06368/) (2026)
- [On the Asymptotic Inadmissibility of Double Machine Learning Estimators Under Structure-Agnostic Models](https://mlsanity.com/paper/2606.22391/) (2026)
- [Perturbing the Derivative: Doubly Wild Refitting for Model-Free Evaluation of Opaque Machine Learning Predictors](https://mlsanity.com/paper/2511.18789/) (2025)
- [Upper Confidence Bounds for the Prediction Error of Kernel Ridge Regression via Gaussian Refitting](https://mlsanity.com/paper/2607.28846/) (2026)
- [Optimal use of a black-box learner in semiparametric estimation](https://mlsanity.com/paper/2607.21541/) (2026)

## Abstract (arXiv)

We characterize the sharp structure-agnostic minimax risk for coefficient estimation in the partial linear model when the outcome and treatment nuisances are learned by two distinct black-box learners, which resolves the open problem in double machine learning posed by Gu (2025). For each nuisance (q∈μ,π), we characterize the available learner by an approximation-error budget (aq) and a stochastic-error budget (sq), with the latter controlled through localized Rademacher complexity. Writing (mathcal En) for the minimax mean-squared error, we show that [mathcal Enasymp1wedgeleftfrac1n+left(aμ aπ+minleftaπ sμ+sπ2,,aμ sπ+sμ2rightright)2right.] The main new ingredient is a novel lower bound for the general two-learner problem. Our proof constructs four finite-mixture testing experiments using orthogonal code functions. Across these experiments, the hidden perturbations are placed outside both learner classes, outside only the treatment learner class, outside only the outcome learner class, or inside both learner classes. These four configurations capture, respectively, the interaction between the two approximation errors, the two asymmetric interactions between one learner's approximation error and the other learner's learning error, and the joint estimation difficulty of learning both nuisances. Combining the four resulting lower bounds yields the displayed rate, which matches the latest upper bound in Gu (2026). Our result shows that standard double machine learning can overstate the intrinsic difficulty of target estimation and provides a target-specific principle for learner selection: approximation error and stochastic complexity must be jointly balanced across the two nuisance learners rather than optimized separately.