Sharp risk bounds found for estimating effects in partial linear models
Sharp Structure-Agnostic Minimax Risk for Partial Linear Models
Machine Learning
Summary
Estimating relationships in complex data often requires accounting for hidden factors with separate prediction tools. This paper finds the exact limits on how well one can estimate key effects when using two different prediction methods whose errors interact. The authors show that balancing approximation accuracy and randomness in both tools together is crucial, challenging previous beliefs that treated these errors separately. Their results help guide better choices of methods when combining predictions for more accurate effect estimation.
What this means in practice
- •For data science teams: Choose better combined prediction methods for estimating treatment or causal effects in complex datasets by jointly balancing approximation and randomness errors.
- •For financial modelers: Improve risk estimation models by applying joint error balancing of different submodels as guided by the new minimax risk bounds.
A theory result. No direct application yet.
Authors
Haichen Hu, David Simchi-Levi
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 \(q\in\{μ,π\}\), 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 \(\mathcal E_n\) for the minimax mean-squared error, we show that \[\mathcal E_n\asymp1\wedge\left\{\frac1n+\left(a_μa_π+\min\left\{a_πs_μ+s_π^2,\,a_μs_π+s_μ^2\right\}\right)^2\right\}.\] 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.