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

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.

We investigate problems where a model is trained twice on the same dataset: the predictions given by the model at the end of the first training stage are used in the second training stage. We conduct a precise theoretical analysis of the performance of the model after the two stages. These results are then applied to the problem of data selection, where a small number of data points need to be selected from a pre-existing pool of data, to be used for training. A classical algorithm for finding the most important data points to select consists in training a model on a first few data points, then leveraging its predictions to select the remaining points. The analysis reveals the existence of an optimal number of samples to use in the first step of this algorithm.