Deep regularization and direct training of the inner layers of Neural Networks with Kernel Flows

Abstract

We introduce a new regularization method for Artificial Neural Networks (ANNs) based on Kernel Flows (KFs). KFs were introduced as a method for kernel selection in regression/kriging based on the minimization of the loss of accuracy incurred by halving the number of interpolation points in random batches of the dataset. Writing $f_\theta(x) = \big(f^{(n)}_{\theta_n}\circ f^{(n-1)}_{\theta_{n-1}} \circ \dots \circ f^{(1)}_{\theta_1}\big)(x)$ for the functional representation of compositional structure of the ANN, the inner layers outputs $h^{(i)}(x) = \big(f^{(i)}_{\theta_i}\circ f^{(i-1)}_{\theta_{i-1}} \circ \dots \circ f^{(1)}_{\theta_1}\big)(x)$ define a hierarchy of feature maps and kernels $k^{(i)}(x,x')=\exp(- \gamma_i \|h^{(i)}(x)-h^{(i)}(x')\|_2^2)$. When combined with a batch of the dataset these kernels produce KF losses $e_2^{(i)}$ (the $L^2$ regression error incurred by using a random half of the batch to predict the other half) depending on parameters of inner layers $\theta_1,\ldots,\theta_i$ (and $\gamma_i$). The proposed method simply consists in aggregating a subset of these KF losses with a classical output loss. We test the proposed method on CNNs and WRNs without alteration of structure nor output classifier and report reduced test errors, decreased generalization gaps, and increased robustness to distribution shift without significant increase in computational complexity. We suspect that these results might be explained by the fact that while conventional training only employs a linear functional (a generalized moment) of the empirical distribution defined by the dataset and can be prone to trapping in the Neural Tangent Kernel regime (under over-parameterizations), the proposed loss function (defined as a nonlinear functional of the empirical distribution) effectively trains the underlying kernel defined by the CNN beyond regressing the data with that kernel.