Generalization in Supervised Learning Through Riemannian Contraction

• URL: http://arxiv.org/abs/2201.06656v1
• Date: Mon, 17 Jan 2022 23:08:47 GMT
• Title: Generalization in Supervised Learning Through Riemannian Contraction
• Authors: Leo Kozachkov, Patrick M. Wensing, Jean-Jacques Slotine
• Abstract summary: In a supervised learning setting, we show that if an metric 0 is contracting in someian rate $lambda, it is uniformly uniformly rate with $math. The results hold for gradient and deterministic loss surfaces, in both continuous and stable $-time. They can be shown to be optimal in certain linear settings, such as over Descent$ convex or strongly convex loss surfaces.
• Score: 4.3604518673788135
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove that Riemannian contraction in a supervised learning setting implies generalization. Specifically, we show that if an optimizer is contracting in some Riemannian metric with rate $\lambda > 0$, it is uniformly algorithmically stable with rate $\mathcal{O}(1/\lambda n)$, where $n$ is the number of labelled examples in the training set. The results hold for stochastic and deterministic optimization, in both continuous and discrete-time, for convex and non-convex loss surfaces. The associated generalization bounds reduce to well-known results in the particular case of gradient descent over convex or strongly convex loss surfaces. They can be shown to be optimal in certain linear settings, such as kernel ridge regression under gradient flow.

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