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

arXiv:2610.21201v1 Announce Type: new 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.

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.

Machine Learning (cs.LG)

arXiv:2610.21201 [cs.LG]

(or arXiv:2610.21201v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.21201 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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

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