In plain words: Using physics-style math, this theory explains why the shapes of a trained network's weight matrices reveal how good each layer is, without any training or test data. Its new quality score matches the older score closely on a small toy network and large modern models.
Abstract · SETOL: A Semi-Empirical Theory of (Deep) Learning
We present a SemiEmpirical Theory of Learning (SETOL) that explains the remarkable performance of State-Of-The-Art (SOTA) Neural Networks (NNs). We provide a formal explanation of the origin of the fundamental quantities in the phenomenological theory of Heavy-Tailed Self-Regularization (HTSR): the heavy-tailed power-law layer quality metrics, alpha and alpha-hat. In prior work, these metrics have been shown to predict trends in the test accuracies of pretrained SOTA NN models, importantly, without needing access to either testing or training data. Our SETOL uses techniques from statistical mechanics as well as advanced methods from random matrix theory and quantum chemistry. The derivation suggests new mathematical preconditions for ideal learning, including a new metric, ERG, which is equivalent to applying a single step of the Wilson Exact Renormalization Group. We test the assumptions and predictions of SETOL on a simple 3-layer multilayer perceptron (MLP), demonstrating excellent agreement with the key theoretical assumptions. For SOTA NN models, we show how to estimate the individual layer qualities of a trained NN by simply computing the empirical spectral density (ESD) of the layer weight matrices and plugging this ESD into our SETOL formulas. Notably, we examine the performance of the HTSR alpha and the SETOL ERG layer quality metrics, and find that they align remarkably well, both on our MLP and on SOTA NNs.
Charles H Martin, Christopher Hinrichs
arXiv:2507.17912 · cs.LG, cond-mat.stat-mech · submitted Jul 23, 2025 · updated Jul 27, 2025
abstract · pdf · html · 139 pages, 28 figures. Code for experiments available at https://github.com/charlesmartin14/SETOL_experiments