Black-Box Assisted Regression: Phase Transitions and Minimax Optimality

Pre-trained black-box predictive functions encode knowledge distilled from massive datasets and extensive computation. However, when the available input features differ from those the black box expects, direct use is infeasible. We introduce a method for transferring predictive knowledge from the black box to a new, heterogeneous input space. Our approach decomposes the target regression function into a transferable component, which the black box can inform, and a non-transferable component, which captures information unique to the new space. We propose a two-step neural network procedure, estimating the transferable component from abundant unlabeled feature pairs that bridge the two input spaces and the non-transferable component from limited labels. We derive prediction risk bounds that improve on those of a non-transfer alternative when the non-transferable component is small or smooth, and the procedure adapts to either case. Under additional conditions, the worst-case risk of our estimator is of strictly smaller polynomial order than the minimax risk of estimation from the labeled data alone. We extend the framework to multiple black boxes, each on its own input space, and show that aggregation can reduce prediction error relative to the best single black box. Simulated and real data demonstrate the practical value of the method.

We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate $\tilde O(n^{-1/3})$, and establishing a matching risk bound of order $\tilde O(n^{-1/3})$. We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.

Distribution shift between training and deployment is a pervasive challenge for modern AI systems. In many cases, the target marginals of covariates and response are known or specified through population-level observations, boundary conditions, properties of simulator configurations, or alignment-time distributional constraints. Such knowledge may provide valuable side information for regression estimation. We study this problem in the multivariate linear regression setting with a stable conditional mean $E[Y\mid X]$ across source and target, and identify the hybrid-loss estimator, which jointly incorporates both target marginals, as a benchmark target-aware estimator. Its direct computation, however, requires solving a coupled nonlinear optimization that is expensive at scale. Our main contribution is to develop and evaluate two computationally tractable alternatives: a constrained moment-matching estimator and a two-stage estimator that augments ordinary least squares with a calibration step. For all three estimators, we derive and compare closed-form asymptotic mean squared errors, yielding conditions under which the tractable alternatives match or closely approximate the hybrid benchmark, and regimes in which they do not. Monte Carlo experiments across three controlled shift regimes validate the theoretical results, investigate the accuracy-runtime tradeoffs among the three estimators, and translate into guidance on estimator choice. In particular, the two-stage estimator nearly matches the hybrid benchmark in the high signal-to-noise regime at essentially no additional cost, providing theoretical grounding for empirical observations in nonlinear settings.