Generalization and Stability of Interpolating Neural Networks with Minimal Width

• URL: http://arxiv.org/abs/2302.09235v2
• Date: Mon, 27 Mar 2023 21:00:54 GMT
• Title: Generalization and Stability of Interpolating Neural Networks with Minimal Width
• Authors: Hossein Taheri, Christos Thrampoulidis
• Abstract summary: We investigate the generalization and optimization of shallow neural-networks trained by gradient in the interpolating regime. We prove the training loss number minimizations $m=Omega(log4 (n))$ neurons and neurons $Tapprox n$. With $m=Omega(log4 (n))$ neurons and $Tapprox n$, we bound the test loss training by $tildeO (1/)$.
• Score: 37.908159361149835
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate the generalization and optimization properties of shallow neural-network classifiers trained by gradient descent in the interpolating regime. Specifically, in a realizable scenario where model weights can achieve arbitrarily small training error $\epsilon$ and their distance from initialization is $g(\epsilon)$, we demonstrate that gradient descent with $n$ training data achieves training error $O(g(1/T)^2 /T)$ and generalization error $O(g(1/T)^2 /n)$ at iteration $T$, provided there are at least $m=\Omega(g(1/T)^4)$ hidden neurons. We then show that our realizable setting encompasses a special case where data are separable by the model's neural tangent kernel. For this and logistic-loss minimization, we prove the training loss decays at a rate of $\tilde O(1/ T)$ given polylogarithmic number of neurons $m=\Omega(\log^4 (T))$. Moreover, with $m=\Omega(\log^{4} (n))$ neurons and $T\approx n$ iterations, we bound the test loss by $\tilde{O}(1/n)$. Our results differ from existing generalization outcomes using the algorithmic-stability framework, which necessitate 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 resemble those found in the convex setting of linear logistic regression.

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