Asymptotic Theory of Iterated Empirical Risk Minimization, with Applications to Active Learning

TL;DR

This paper develops an asymptotic theory for iterated empirical risk minimization, revealing fundamental tradeoffs and behaviors in active learning scenarios with data reuse and prediction-dependent losses.

Contribution

It provides the first sharp asymptotic analysis of iterated ERM with data reuse, especially in active learning, and uncovers key tradeoffs and phenomena like double descent.

Findings

Explicit asymptotic predictions for second-stage test error.

Removal of oracle and sample-splitting assumptions in active learning.

Identification of a fundamental tradeoff in labeling budget allocation.

Abstract

We study a class of iterated empirical risk minimization (ERM) procedures in which two successive ERMs are performed on the same dataset, and the predictions of the first estimator enter as an argument in the loss function of the second. This setting, which arises naturally in active learning and reweighting schemes, introduces intricate statistical dependencies across samples and fundamentally distinguishes the problem from classical single-stage ERM analyses. For linear models trained with a broad class of convex losses on Gaussian mixture data, we derive a sharp asymptotic characterization of the test error in the high-dimensional regime where the sample size and ambient dimension scale proportionally. Our results provide explicit, fully asymptotic predictions for the performance of the second-stage estimator despite the reuse of data and the presence of prediction-dependent losses. We apply this theory to revisit a well-studied pool-based active learning problem, removing oracle and sample-splitting assumptions made in prior work. We uncover a fundamental tradeoff in how the labeling budget should be allocated across stages, and demonstrate a double-descent behavior of the test error driven purely by data selection, rather than model size or sample count.