Finite-Sample Failures and Condition-Number Diagnostics in Double Machine Learning

Standard Double Machine Learning (DML; Chernozhukov et al., 2018) confidence intervals can exhibit substantial finite-sample coverage distortions when the underlying score equations are ill-conditioned, even if nuisance functions are estimated with state-of-the-art methods. Focusing on the partially linear regression (PLR) model, we show that a simple, easily computed condition number for the orthogonal score, denoted kappa_DML := 1 / |J_theta|, largely determines when DML inference is reliable. Our first result derives a nonasymptotic, Berry-Esseen-type bound showing that the coverage error of the usual DML t-statistic is of order n^{-1/2} + sqrt(n) * r_n, where r_n is the standard DML remainder term summarizing nuisance estimation error. Our second result provides a refined linearization in which both estimation error and confidence interval length scale as kappa_DML / sqrt(n) + kappa_DML * r_n, so that ill-conditioning directly inflates both variance and bias. These expansions yield three conditioning regimes - well-conditioned, moderately ill-conditioned, and severely ill-conditioned - and imply that informative, shrinking confidence sets require kappa_DML = o_p(sqrt(n)) and kappa_DML * r_n -> 0. We conduct Monte Carlo experiments across overlap levels, nuisance learners (OLS, Lasso, random forests), and both low- and high-dimensional (p > n) designs. Across these designs, kappa_DML is highly predictive of finite-sample performance: well-conditioned designs with kappa_DML < 1 deliver near-nominal coverage with short intervals, whereas severely ill-conditioned designs can exhibit large bias and coverage around 40% for nominal 95% intervals, despite flexible nuisance fitting. We propose reporting kappa_DML alongside DML estimates as a routine diagnostic of score conditioning, in direct analogy to condition-number checks and weak-instrument diagnostics in IV settings.