Minimax Estimation of Conditional Moment Models
This work addresses a foundational challenge in econometrics and machine learning for researchers and practitioners dealing with ill-posed inverse problems, offering a general framework with computationally efficient methods, though it is incremental in building on existing minimax and regularization techniques.
The paper tackles the problem of estimating models with conditional moment restrictions, such as non-parametric instrumental variable regression, by introducing a min-max criterion function that frames estimation as a zero-sum game, and shows that the estimator achieves rates scaling with the critical radius of hypothesis and test function spaces, leading to tight fast rates for various applications like neural networks and random forests.
We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the estimation problem can be thought of as solving a zero-sum game between a modeler who is optimizing over the hypothesis space of the target model and an adversary who identifies violating moments over a test function space. We analyze the statistical estimation rate of the resulting estimator for arbitrary hypothesis spaces, with respect to an appropriate analogue of the mean squared error metric, for ill-posed inverse problems. We show that when the minimax criterion is regularized with a second moment penalty on the test function and the test function space is sufficiently rich, then the estimation rate scales with the critical radius of the hypothesis and test function spaces, a quantity which typically gives tight fast rates. Our main result follows from a novel localized Rademacher analysis of statistical learning problems defined via minimax objectives. We provide applications of our main results for several hypothesis spaces used in practice such as: reproducing kernel Hilbert spaces, high dimensional sparse linear functions, spaces defined via shape constraints, ensemble estimators such as random forests, and neural networks. For each of these applications we provide computationally efficient optimization methods for solving the corresponding minimax problem (e.g. stochastic first-order heuristics for neural networks). In several applications, we show how our modified mean squared error rate, combined with conditions that bound the ill-posedness of the inverse problem, lead to mean squared error rates. We conclude with an extensive experimental analysis of the proposed methods.