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.

Neural Slack Variables for Shape Constraints

Enforcing functional inequality constraints such as monotonicity and convexity in neural networks is a fundamental challenge in many industrial and scientific applications. Classical one-sided penalty methods, along with primal-dual methods gated by complementary slackness, provide constraint gradients only at violated locations, resulting in fragile satisfaction. Architectures that guarantee feasibility by construction, on the other hand, remain largely limited to elementary cases and impose additional inductive biases. We introduce neural slack variables, a deep learning native primal-side approach that converts constraint enforcement into a regression problem by coupling the primary network with a jointly learned auxiliary network. The auxiliary network serves as a valid target for the primary network's constraint quantities, inducing feasibility and regularity. Neural slack variables achieve zero measured violations on dense-grid monotonicity and convexity test cases, where penalty and primal-dual baselines leave residual violations, and enable arbitrage-free learning of volatility surfaces, an open industrial challenge in quantitative finance.