Prediction-Powered Causal Inference by Automatic Debiased Machine Learning and Semi-Supervised Riesz Regression

This study investigates semiparametric efficient estimation of causal and structural parameters in a semi-supervised setting. In our setting, unlabeled auxiliary regressors are available in addition to labeled observations consisting of outcomes and regressors. Our goal is to construct estimators of causal and structural parameters whose asymptotic variances are smaller than those of estimators constructed using only labeled data. We refer to this framework as prediction-powered causal inference (PPCI). We first derive the efficient influence function and the efficiency bound, which imply that the use of auxiliary regressors can attain a smaller asymptotic variance than the efficiency bound attainable from labeled observations alone. Then, by combining the efficient influence function with the debiased machine learning (DML) framework, we propose methods that we call DML-PPCI. If we construct an estimating-equation estimator, we refer to the method as EE-DML-PPCI; if we construct a targeted-learning estimator, we refer to the method as TMLE-DML-PPCI. The asymptotic variances of both estimators match our derived efficiency bound. In the construction of the estimators, estimation of the efficient influence function plays an important role. In our study, the efficient influence function is also a Neyman orthogonal score, which depends on the Riesz representer and the regression function. For Riesz representer estimation, we develop semi-supervised generalized Riesz regression with convergence rate guarantees.

Estimating the expected output of wide random MLPs more efficiently than sampling

By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.