Sharp Structure-Agnostic Minimax Risk for Partial Linear Models
Abstract
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 ℰn for the minimax mean-squared error, we show that ℰnasymp1∧{1/n+(aμaπ+min{aπsμ+sπ2, aμsπ+sμ2})2}. 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.
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BibTeX
@misc{hu2026sharp,
title = {{Sharp Structure-Agnostic Minimax Risk for Partial Linear Models}},
author = {Haichen Hu and David Simchi-Levi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07997},
eprint = {2609.07997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07997},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Structure-Agnostic Minimax Risk for Partial Linear Models
AU - Haichen Hu
AU - David Simchi-Levi
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07997
UR - https://arxiv.org/abs/2609.07997
AB - 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 a_q and a stochastic-error budget s_q, with the latter controlled through localized Rademacher complexity. Writing ℰ_n for the minimax mean-squared error, we show that ℰ_nasymp1∧{1/n+(a_μa_π+min{a_πs_μ+s_π^2, a_μs_π+s_μ^2})^2}. 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.
ER -