Estimation.CATE.Orthogonal­Learning.Local­Emp­Process

DR-Learner bounded-loss Rademacher modulus. Consider the doubly robust orthogonal-learning system for conditional treatment effects, cross-fitted via a one-shot sample split, whose truth-identifying candidate belongs to the target set, has a measurable evaluation map, and recovers the population CATE function through that evaluation map. Suppose the candidate evaluation maps are uniformly bounded, the observed outcome is almost-surely bounded, the realised nuisance outcome-regression is uniformly bounded on both treatment arms, and the realised propensity stays away from 0 and 1 by a fixed margin; suppose also that the resulting doubly robust loss is continuous in the candidate over the target set at that realised nuisance, a sequence R n bounds the Rademacher complexity of the centred loss class on the estimation fold, the truth-identifying candidate still minimizes the clamped population loss at the realised nuisance, and the confidence level δ lies strictly above 0 and at most 1. Then there is a nonnegative constant b for which the system obeys the local empirical-process modulus condition at the explicit rate ρ n = √(2 R n + 2b · √(2 log(1/δ) / |B(n)|)) on nonempty estimation folds, with boundary value ρ n = √(2b) on empty folds.

DR-Learner localized modulus, sharp Foster–Syrgkanis form. For the DR-Learner CATE estimation system, suppose the evaluation functional is measurable in its parameter and recovers the true CATE at a parameter θ₀ in the constraint set, and that the evaluation functional, outcome, and fixed nuisance conditional-mean are uniformly bounded while the nuisance's propensity score satisfies ε-overlap. Assume the centred DR-loss is continuous in θ, a clamped version of θ₀ minimizes it, its population Rademacher complexity along a dense index sequence is controlled by a sub-root envelope ψ with respect to a seminorm that is invariant under almost-everywhere modification, and the same Rademacher upper bound extends to loss differences across the whole constraint set, together with Lipschitz and diameter control of the centred loss increments — nonnegative Lipschitz constant L, a diameter bound Rmax dominating every critical radius criticalRadius (ψ m), and the sub-root fixed-point property, plus boundedness and integrability of the empirical star-hull Rademacher process needed by the localization bridge and a confidence level in $(0,1]$ together with the Foster–Syrgkanis critical-radius domination inequality across dyadic shell counts. Then there is a nonnegative envelope b such that the DR-Learner system satisfies the localized empirical-process modulus predicate at rate ρ n = (8L+3)·criticalRadius (ψ |B(n)|) on nonempty validation folds, falling back to √(2b) when the fold is empty.

DR-Learner end-to-end high-probability oracle inequality. Consider the same doubly robust orthogonal-learning system, one-shot sample split, and boundedness / overlap / continuity / Rademacher package as in localEmpProcessModulus_drLearner — truth-identifying admissibility and evaluation correctness, uniform bounds on the candidate evaluations, the outcome, and the realised nuisance regression together with propensity overlap, and loss continuity, a Rademacher bound R n, the clamped-loss minimizer property, and a confidence level δ in (0, 1] . Assume in addition that a plug-in empirical-risk-minimisation estimator sequence τhat, evaluated against the same realised nuisance on every cross-fitting fold, attains the empirical risk up to an optimization slack r_opt n, that the population risk is strongly convex at the realised nuisance with modulus σ > 0, and that the truth-identifying candidate satisfies the first-order optimality inequality for the population risk's directional derivative at the truth. Then there is a nonnegative constant b such that, for every sample size n, with P.μ-probability at least 1 − δ the estimation error obeys ‖τhat n ω − θ₀‖² ≤ (4(1+σ)/σ²) · ρ_{n,δ}² + (4/σ) · Bias_n + (4/σ) · r_opt n, where ρ_{n,δ} is the modulus rate realised by localEmpProcessModulus_drLearner.

Product-bias DR-Learner oracle inequality. Consider a CATE estimation system satisfying the underlying backdoor identification assumptions, with an evaluation functional that is measurable in its parameter and recovers the true CATE at a parameter θ₀ belonging to the constraint set, and suppose the evaluation functional, the outcome, and a fixed plug-in nuisance's conditional-mean function are uniformly bounded while that nuisance's propensity score satisfies ε-overlap. Assume further the DR-loss is continuous in the parameter, its localized empirical process admits a Rademacher complexity bound, and a clamped version of θ₀ still minimizes the loss, the confidence level lies in $(0,1]$, the overlap parameter is positive, and both the true and the plug-in nuisance lie in the ε-overlap slice, and that the estimator sequence stays within the constraint set as an approximate sample-split plug-in empirical-risk minimizer (slack r_opt) whose excess population risk satisfies a strong-convexity-type lower bound with constant σ>0 and a first-order orthogonality inequality at the true nuisance's directional derivative. Finally, a battery of boundedness and integrability conditions bounds the directional-derivative envelope and controls the two nuisance-error terms entering the second-order bias decomposition. Then for every sample fold n, with probability at least 1-δ the squared estimation error of the DR-Learner is bounded by an oracle/Rademacher term plus an explicit second-order product-bias term (2B/ε)·Σ_a ‖Δμ_a‖₂·‖Δe‖₂ in the two nuisance L²-errors, plus an optimization slack.

DR-Learner oracle inequality at a random cross-fitted nuisance. Consider a CATE estimation system satisfying the backdoor identification assumptions, with an evaluation functional measurable in its parameter and recovering the true CATE at a parameter θ₀ in the constraint set, and suppose the overlap parameter is positive while the true nuisance and every realization of the random, fold-A-estimated nuisance ĥ n ω lie in the ε-overlap slice. Assume the estimator sequence stays in the constraint set as an approximate sample-split plug-in empirical-risk minimizer with respect to the random nuisance (slack r_opt), whose excess population risk at each realized ĥ n ω obeys a strong-convexity-type lower bound with constant σ>0, together with a first-order orthogonality inequality at the true nuisance's directional derivative, and that on a high-probability event the centred excess empirical risk at the realized estimator and nuisance is controlled by a rate ρ n. Finally, a battery of boundedness and integrability conditions governs the directional-derivative envelope and bounds the two random nuisance-error terms. Then for every n, with probability at least 1-δ the squared estimation error is bounded by the oracle/Rademacher term (4(1+σ)/σ²)·(ρ n)² plus a random second-order product-bias term (4/σ)·(2B/ε)·Σ_a ‖(ĥ n ω).μ_fn a − μ_val a‖₂·‖(ĥ n ω).e_fn − e_val‖₂ evaluated at the realized nuisance, plus an optimization slack (4/σ)·r_opt n.

DR-Learner localized high-probability oracle inequality. For the DR-Learner CATE estimation system, suppose the evaluation functional is measurable in its parameter and recovers the true CATE at a parameter θ₀ in the constraint set, and that the evaluation functional, outcome, and fixed nuisance conditional-mean are uniformly bounded while the nuisance's propensity score satisfies ε-overlap. Assume the centred DR-loss is continuous in θ, a clamped version of θ₀ minimizes it, its population Rademacher complexity along a dense index sequence is controlled by a sub-root envelope ψ with respect to a seminorm invariant under almost-everywhere modification, and the same Rademacher upper bound extends to loss differences across the whole constraint set, together with Lipschitz and diameter control of the centred loss increments — nonnegative Lipschitz constant L, a diameter bound Rmax dominating every critical radius, and the sub-root fixed-point property, plus boundedness and integrability of the empirical star-hull Rademacher process needed by the localization bridge and a confidence level in $(0,1]$ together with the Foster–Syrgkanis critical-radius domination inequality across dyadic shell counts. Finally, suppose the estimator sequence is an approximate sample-split plug-in empirical-risk minimizer at the fixed nuisance h (slack r_opt), whose excess population risk obeys a strong-convexity-type lower bound with constant σ>0, together with a first-order orthogonality inequality at the truth's directional derivative. Then there is a nonnegative envelope b such that, for every n, with probability at least 1-δ the squared estimation error is bounded by the oracle term (4(1+σ)/σ²)·(ρ n)² at the sharp localized Foster–Syrgkanis rate ρ n = (8L+3)·criticalRadius (ψ |B(n)|) (falling back to √(2b) on an empty validation fold), plus the nuisance bias term (4/σ)·Bias_n and an optimization slack (4/σ)·r_opt n.