Modern semiparametric estimation often relies on flexible black-box machine learning methods to estimate nuisance functions, raising a fundamental question: how do nuisance estimation errors propagate into inference for low-dimensional target parameters? The dominant paradigm, exemplified by double machine learning (DML), yields error bounds in which nuisance estimation errors enter multiplicatively. While widely adopted, it remains unclear whether this multiplicative-rate dependence is optimal for black-box models. In this paper, we start by revisiting the partial linear model $Y = μ_0(X)+T\cdotβ_0+\varepsilon$ under a structure-agnostic setting, where the nuisance function $μ_0$ is estimated using a generic machine learning model, with approximation error $δ^a_μ$ and stochastic error $δ_μ^s$. We show that the standard DML rate is not optimal in the regime where the auxiliary function $\mathbb{E}[T|X=x]$ cannot be consistently estimated. We propose a new estimator for $β_0$ that achieves a sharper rate of $n^{-1/2}+δ^a_μ+(δ_μ^s)^2$ and establish a matching lower bound demonstrating its optimality. Our results reveal a new principle: the first-order stochastic error of nuisance estimation can be eliminated without imposing any additional assumptions. This also leads to a revised tuning strategy favoring under-smoothing, where $δ^a_μ\asymp(δ_μ^s)^2$, rather than the classical bias-variance trade-off $δ^a_μ\asymp δ_μ^s$. Under mild additional conditions, the estimator is asymptotically normal with minimal asymptotic variance. The proposed method extends to a broad class of semi-parametric linear functional estimation problems, including average treatment effect estimation. Our results imply that popular orthogonal score methods in semiparametric estimation with black-box nuisance learners can be substantially improved.

翻译：现代半参数估计常依赖灵活的黑箱机器学习方法估计干扰函数，由此引发一个根本性问题：干扰估计误差如何传导至低维目标参数的推断？以双重机器学习为代表的主流范式给出了误差界，其中干扰估计误差以乘积形式进入。尽管该范式被广泛采用，但对于黑箱模型而言，这种乘法速率依赖是否为最优尚不明确。本文首先在结构无关设定下重新审视部分线性模型$Y = μ_0(X)+T\cdotβ_0+\varepsilon$，其中干扰函数$μ_0$由通用机器学习模型估计，包含逼近误差$δ^a_μ$与随机误差$δ_μ^s$。研究表明，当辅助函数$\mathbb{E}[T|X=x]$无法被一致估计时，标准双重机器学习速率并非最优。我们提出$β_0$的新估计量，实现更优速率$n^{-1/2}+δ^a_μ+(δ_μ^s)^2$，并通过匹配下界证明其最优性。结果揭示新原理：无需附加假设即可消除干扰估计的一阶随机误差。这引导出偏好欠平滑的调参策略，即满足$δ^a_μ\asymp(δ_μ^s)^2$，而非经典偏差-方差权衡$δ^a_μ\asymp δ_μ^s$。在温和附加条件下，该估计量渐进正态且具有最小渐近方差。所提方法可推广至包含平均处理效应估计在内的广义半参数线性泛函估计问题。结果表明，现有采用黑箱干扰学习器的半参数估计中流行的正交得分方法可被显著改进。