Title: Training Neural Networks as Learning Data-adaptive Kernels: Provable Representation and Approximation Benefits,
Abstract: Consider the problem: given data pair $(\mathbf{x}, \mathbf{y})$ drawn from a
population with $f_*(x) = \mathbf{E}[\mathbf{y} | \mathbf{x} = x]$, specify a
neural network and run gradient flow on the weights over time until reaching
any stationarity. How does $f_t$, the function computed by the neural network
at time $t$, relate to $f_*$, in terms of approximation and representation?
What are the provable benefits of the adaptive representation by neural
networks compared to the pre-specified fixed basis representation in the
classical nonparametric literature? We answer the above questions via a dynamic
reproducing kernel Hilbert space (RKHS) approach indexed by the training
process of neural networks. We show that when reaching any local stationarity,
gradient flow learns an adaptive RKHS representation, and performs the global
least squares projection onto the adaptive RKHS, simultaneously. In addition,
we prove that as the RKHS is data-adaptive and task-specific, the residual for
$f_*$ lies in a subspace that is smaller than the orthogonal complement of the
RKHS, formalizing the representation and approximation benefits of neural
networks.