Training a few neural network parts cuts most errors quickly
Phase Transition from Random Features to Fully Trainable Networks
Machine Learning
Summary
Building computer models that learn patterns can be slow or expensive if everything is adjustable. The authors show that making only a small portion of a neural network trainable leads to most of the improvement you get from training it fully. They find a clear jump when increasing trainable parts, with the biggest gains early on. Their math explains why this happens and matches real experiments, even for deep networks.
What this means in practice
- •For machine learning engineers: Design smaller trainable subsets of neural networks to reduce training costs while retaining most performance gains.
- •For scientific computing teams: Use partly trainable networks for efficient function approximation in computational simulations, balancing accuracy and compute effort.
Authors
Xin He, Huan Lao, Xiliang Lu, Shijun Zhang
Abstract
Constructing accurate and computationally tractable approximation spaces is a fundamental task in scientific computing. Partially trainable neural networks bridge fixed random features and fully adaptive representations. This paper quantifies a phase transition pattern in approximation bounds and observes analogous behavior in training errors: a rapid but smooth initial improvement followed by diminishing returns, with a sharper transition under greater initialization mismatch. Under polynomial spectral decay with exponent $α>1$ and leverage compatibility, we establish a squared $L_2$ approximation rate $\mathcal{O}((κ_P\log(m/δ)/m)^α)$ for two-layer networks of width $m$ and trainable proportion $P$, where $κ_P=(P/κ_1^*+(1-P)/κ_0^*)^{-1}$. The bound holds uniformly over the unit ball of a reproducing kernel Hilbert space (RKHS) with probability at least $1-δ$. The constants $κ_0^*$ and $κ_1^*$ quantify the fixed and trainable groups' sampling mismatches relative to the target RKHS. For spherical biased-ReLU features in dimension $d$, we verify these assumptions with $α=(d+2)/(d-1)$ and $κ_0^*=4\sqrt{2}τ(d+1)/d$ for initialization scale $τ\ge1$, and $κ_1^*=1$ under ridge leverage sampling. For $d=5$ and $τ=10$, a trainable proportion of approximately $4.1\%$ captures $90\%$ of the reduction in the approximation bound from $P=0$ to $P=1$. A structured three-layer example illustrates how a few trainable first-layer neurons exploit fixed downstream blocks through nonlinear composition. Numerical experiments across diverse settings, including networks with up to 15 layers, exhibit similar phase transitions in training errors.