title: Implicit Bias of MSE Gradient Optimization in Underparameterized Neural Networks
abstract: We study the dynamics of a neural network in function space when optimizing the mean squared error via gradient flow.  We show that in the underparameterized regime the network learns eigenfunctions of an integral operator $T_K$ determined by the Neural Tangent Kernel at rates corresponding to their eigenvalues.  For example, for uniformly distributed data on the sphere $S^{d - 1}$ and rotation invariant weight distributions, the eigenfunctions of $T_K$ are the spherical harmonics.  Our results can be understood as describing a spectral bias in the underparameterized regime. The proofs use the concept of ``Damped Deviations'' where deviations of the NTK matter less for eigendirections with large eigenvalues.  Aside from the underparameterized regime, the damped deviations point-of-view allows us to extend certain results in the literature in the overparameterized setting. 
authors: Benjamin Bowman Guido Montufar
hIndex: Guido Montufar 20