Debiased Machine Learning: Identification, Estimation, and Shape Constraints

We develop a general framework of identification and estimation for automatic debiased machine learning (DML) where the parameter of interest $θ_0$ is identified by a moment condition involving a nuisance $γ_0$ that may be high dimensional. DML leverages machine learning to estimate $γ_0$ while correcting for regularization and overfitting biases that may otherwise transmit to biased estimation of $θ_0$. We establish conditions under which the Riesz representer $α_0$, which is at the core of DML, is identified, and show that the identification occurs precisely when $α_0$ uniquely optimizes a quadratic functional. This characterization enables us to develop a general estimation procedure for $α_0$ that allows for generic $γ_0$ including those defined by models with endogeneity and encompasses both classical sieves and modern architectures such as deep neural networks. To improve estimation precision and mitigate the curse of dimensionality, we incorporate shape constraints on $γ_0$ by embedding them into a possibly nonlinear parameter space. We illustrate our estimation procedure through simulations and empirical applications.

Optimal use of a black-box learner in semiparametric estimation

Consider the partial linear model $Y = μ_0(X) + β_0 \cdot T + \varepsilon$ and $T = π_0(X) + u$ in the structure-agnostic setting, where we are blind to the structure $μ_0$ and $π_0$ and estimate the nuisances by a black-box hypothesis class. The learnability of the class is characterized by the estimation error $δ_s$ in the absence of model misspecification and its $L_2$ mis-specification error $δ_{a, μ}$ and $δ_{a, π}$ for $μ_0$ and $π_0$, respectively. We propose a novel estimator of the target linear coefficient $θ_0 = β_0$ with error rate \[ \frac{1}{\sqrt{n}} + δ_{a, μ} \cdot δ_{a, π} + [δ_s]^2. \] A matching lower bound is also established, implying that this rate is unimprovable. Compared with the product rate yielded by double machine learning (DML), our estimator removes the suboptimal term $\max(δ_{a, μ}, δ_{a, π})\cdot δ_s$ at no extra cost or assumption. Building on the underlying insights, which are neither tailored to the one-learner setting nor the partial linear model, we propose Transductive Adversarial Moment-calibrated Editing (TAME), which locally edits debiasing weights induced by black-box regression estimates on the inference sample through adversarial conditional moment calibration. TAME can be combined with any initial black-box estimates and can strictly improve on DML guarantees when the nuisance difficulties are imbalanced. We discuss how to fully exploit the advantages introduced by TAME, including the gains from using two learners, the resulting under-smoothing principle for model selection, and extensions to other linear functional estimation problems.

Optimally taming biases in black-box models for efficient semiparametric estimation

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.