Awakening of the Buddha: Subspace Learning During Population-Loss Plateaus

2026-09-30

AI Summary

This study investigates whether neural networks continue to learn effectively during loss plateaus. Focusing on two-layer ReLU and SwiGLU networks, it establishes a framework that decouples loss stagnation from representation learning by integrating population gradient descent, the Average Gradient Outer Product (AGOP), spectral analysis, and Hermite polynomial theory. Theoretically, the work demonstrates that networks optimize predictive representations through subspace alignment during plateaus, providing high-probability alignment guarantees and bounds on mean squared error reduction. Empirical results confirm significant subspace alignment during plateau phases and subsequent loss breakthroughs. By revealing the mechanism of “apparent stagnation yet actual learning,” this research offers new perspectives for understanding the implicit representational dynamics of deep networks.

Abstract

Population loss can remain nearly constant while a neural network learns a substantially more predictive representation. We establish this separation for two-layer ReLU and leaky-ReLU networks trained on Gaussian inputs by simultaneous fixed-step population gradient descent on all parameters. For structured additive teachers whose links are positive mixtures of Gaussian-damped cubics in $H^1(γ)$, we give explicit conditions under which small IID Gaussian initialization yields a high-probability guarantee: at a checkpoint during a high-loss plateau, minimum alignment between the rank-$r$ teacher subspace and the leading $r$-dimensional eigenspace of the predictor's average gradient outer product (AGOP) increases by at least $1/2$, and the minimum refit MSE under unchanged coefficient budgets decreases by more than $0.399$, both relative to initialization. The same trajectory subsequently attains a trained loss below every value in the plateau window. A complementary result treats unequal-weight cubic teachers and small additive Sobolev perturbations using projected-feature refits. For SwiGLU networks with an exactly fitted intercept, we prove leading-AGOP alignment during a loss plateau at fixed width and dimension as Gaussian initialization vanishes, for square-integrable teachers with nonzero Hermite content of degree one, two, or three. A rank-one cubic specialization also gives simultaneous unrestricted-refit gains at a prescribed width. An approximation lower bound further shows that certain interaction targets retain nonzero error when ridge neurons are restricted to shared orthogonal axes within the teacher subspace. Population-moment experiments with ReLU students across 21 teachers and 50 initializations per teacher complement the analysis.

Innovation

• subspace learning
• loss plateau
• AGOP alignment
• representation learning
• gradient descent