Stability & Generalisation of Gradient Descent for Shallow Neural Networks without the Neural Tangent Kernel
This work addresses the theoretical understanding of GD stability and generalization for neural networks, providing tighter analyses that avoid common assumptions, which is significant for researchers in machine learning theory, though it is incremental in refining existing bounds.
The paper tackles the problem of understanding generalization and excess risk bounds for gradient descent (GD) in training overparameterized shallow neural networks without relying on the Neural Tangent Kernel (NTK) or Polyak-Lojasiewicz (PL) assumptions. It proves oracle-type bounds showing that GD's performance is controlled by an interpolating network with the shortest path from initialization, and demonstrates consistency with early stopping in regression with label noise.
We revisit on-average algorithmic stability of GD for training overparameterised shallow neural networks and prove new generalisation and excess risk bounds without the NTK or PL assumptions. In particular, we show oracle type bounds which reveal that the generalisation and excess risk of GD is controlled by an interpolating network with the shortest GD path from initialisation (in a sense, an interpolating network with the smallest relative norm). While this was known for kernelised interpolants, our proof applies directly to networks trained by GD without intermediate kernelisation. At the same time, by relaxing oracle inequalities developed here we recover existing NTK-based risk bounds in a straightforward way, which demonstrates that our analysis is tighter. Finally, unlike most of the NTK-based analyses we focus on regression with label noise and show that GD with early stopping is consistent.