Generalization and Stability of Interpolating Neural Networks with Minimal Width

We investigate the generalization and optimization of $k$-homogeneous shallow neural-network classifiers in the interpolating regime. The study focuses on analyzing the performance of the model when it is capable of perfectly classifying the input data with a positive margin $\gamma$. When using gradient descent with logistic-loss minimization, we show that the training loss converges to zero at a rate of $\tilde O(1/\gamma^{2/k} T)$ given a polylogarithmic number of neurons. This suggests that gradient descent can find a perfect classifier for $n$ input data within $\tilde{\Omega}(n)$ iterations. Additionally, through a stability analysis we show that with $m=\Omega(\log^{4/k} (n))$ neurons and $T=\Omega(n)$ iterations, the test loss is bounded by $\tilde{O}(1/\gamma^{2/k} n)$. This is in contrast to existing stability results which require polynomial width and yield suboptimal generalization rates. Central to our analysis is the use of a new self-bounded weak convexity property, which leads to a generalized local quasi-convexity property for sufficiently parameterized neural-network classifiers. Eventually, despite the objective's non-convexity, this leads to convergence and generalization-gap bounds that are similar to those in the convex setting of linear logistic regression.