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- W3036288194 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 $epsilonsim p^{-beta}$ where $p$ is the size of the training set. We find that $betasimfrac{1}{d}$ 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 {it 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 $xiin (0,2)$ 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 $betarightarrowfrac{1}{3}$ as $drightarrowinfty$. (iii) We confirm these findings for the {it 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 $lambda^{-frac{2(d-1)}{3d-3+xi}}$." @default.
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- W3036288194 date "2020-06-17" @default.
- W3036288194 modified "2023-09-22" @default.
- W3036288194 title "How isotropic kernels learn simple invariants." @default.
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