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We provide quantitative bounds measuring the $L^2$ difference in functionspace between the trajectory of a finite-width network trained on finitely manysamples from the idealized kernel dynamics of infinite width and infinite data.An implication of the bounds is that the network is biased to learn the topeigenfunctions of the Neural Tangent Kernel not just on the training set butover the entire input space. This bias depends on the model architecture andinput distribution alone and thus does not depend on the target function whichdoes not need to be in the RKHS of the kernel. The result is valid for deeparchitectures with fully connected, convolutional, and residual layers.Furthermore the width does not need to grow polynomially with the number ofsamples in order to obtain high probability bounds up to a stopping time. Theproof exploits the low-effective-rank property of the Fisher Information Matrixat initialization, which implies a low effective dimension of the model (farsmaller than the number of parameters). We conclude that local capacity controlfrom the low effective rank of the Fisher Information Matrix is stillunderexplored theoretically.

Spectral Bias Outside the Training Set for Deep Networks in the Kernel Regime