Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principle

We study adaptive estimation and inference in linear inverse problems defined by conditional moment restrictions. These problems are typically ill-posed, so estimators require regularization, yet the optimal regularization strength depends on a source parameter that is almost never known. We develop a discrepancy principle that chooses this parameter from empirical losses and prescribed bounds on statistical error, without knowing the source parameter. The framework covers Regularized DeepIV (RDIV) and the Tikhonov Regularized Adversarial Estimator (TRAE). Under the stated realizability conditions and comparable fitting and calibration error bounds, both one-step procedures attain strong- and weak-error bounds that match known-source rates for source parameters up to one. We also extend the framework to iterative procedures under realizability of the population iterates, widening the range of source parameters covered by the rate comparison. Finally, we use these adaptive estimators to construct an asymptotically normal doubly robust estimator that adapts to the well-posedness of the primal and dual problems.