How isotropic kernels perform on simple invariants

• URL: http://arxiv.org/abs/2006.09754v5
• Date: Mon, 14 Dec 2020 20:20:51 GMT
• Title: How isotropic kernels perform on simple invariants
• Authors: Jonas Paccolat, Stefano Spigler and Matthieu Wyart
• Abstract summary: We investigate how the training curve of isotropic kernel methods depends on the symmetry of the task to be learned. We show that for large bandwidth, $beta = fracd-1+xi3d-3+xi$, where $xiin (0,2)$ is the exponent characterizing the stripe of the kernel at the origin.
• Score: 0.5729426778193397
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate how the training curve of isotropic kernel methods depends on the symmetry of the task to be learned, in several settings. (i) We consider a regression task, where the target function is a Gaussian random field that depends only on $d_\parallel$ variables, fewer than the input dimension $d$. We compute the expected test error $\epsilon$ that follows $\epsilon\sim p^{-\beta}$ where $p$ is the size of the training set. We find that $\beta\sim 1/d$ independently of $d_\parallel$, supporting previous findings that the presence of invariants does not resolve the curse of dimensionality for kernel regression. (ii) Next we consider support-vector binary classification and introduce the stripe model where the data label depends on a single coordinate $y(\underline{x}) = y(x_1)$, corresponding to parallel decision boundaries separating labels of different signs, and consider that there is no margin at these interfaces. We argue and confirm numerically that for large bandwidth, $\beta = \frac{d-1+\xi}{3d-3+\xi}$, where $\xi\in (0,2)$ is the exponent characterizing the singularity of the kernel at the origin. This estimation improves classical bounds obtainable from Rademacher complexity. In this setting there is no curse of dimensionality since $\beta\rightarrow 1 / 3$ as $d\rightarrow\infty$. (iii) We confirm these findings for the spherical model for which $y(\underline{x}) = y(|\underline{x}|)$. (iv) In the stripe model, we show that if the data are compressed along their invariants by some factor $\lambda$ (an operation believed to take place in deep networks), the test error is reduced by a factor $\lambda^{-\frac{2(d-1)}{3d-3+\xi}}$.

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