Hard Way, The Only Way: Provable Merits of Training on Hardest Examples

Abstract

Despite a surge of empirical interest in the ``unreasonable effectiveness'' of drastic data pruning for AI systems, a unified theoretical understanding of this phenomenon remains elusive. In this work, we provide a precise high-dimensional theory of learning under data pruning via weighted empirical risk minimization (ERM). As in previous work, we analyze a setting where pruning is performed along a one-dimensional subspace of the input space. Our analysis precisely characterizes the limiting test error curves in the regime where the input dimension N and sample size P tend to infinity with a fixed ratio N/P. The resulting theory recovers several recent results in the literature as special cases and provides a general lens to evaluate how pruning strategies interact with the loss function and label noise. Crucially, we characterize the optimal pruning strategy in the high-sample regime where P greatly exceeds N. We demonstrate that strategic pruning can accelerate the decay of classification test error from the classical sqrt(N/P) rate, typical of vanilla ERM with loss functions without exponential tail (e.g., quadratic loss), to the much faster Bayes-optimal scaling of order N/P. In this setup, we prove that, surprisingly, this optimal performance is achieved by a drastic pruning strategy: discarding almost all training data except for a small proportion, consisting strictly of the 2N hardest training examples.