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

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 $H^1(\gamma)$, 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.