# Training Neural Networks as Learning Data-adaptive Kernels: Provable Representation and Approximation Benefits

## TL;DR

This paper demonstrates that neural networks trained with gradient flow adaptively learn a kernel representation, enabling better approximation and generalization compared to fixed basis methods, with formal proofs and convergence results.

## Contribution

It introduces a dynamic RKHS framework showing neural networks learn an adaptive kernel and perform optimal projections, formalizing their representation and approximation advantages.

## Key findings

- Neural networks learn an adaptive RKHS during training.
- Gradient flow performs the global least-squares projection onto the adaptive RKHS.
- Neural network functions converge to kernel ridgeless regression with an adaptive kernel.

## Abstract

Consider the problem: given the data pair $(\mathbf{x}, \mathbf{y})$ drawn from a population with $f_*(x) = \mathbf{E}[\mathbf{y} | \mathbf{x} = x]$, specify a neural network model 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. Firstly, 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. Secondly, we prove that as the RKHS is data-adaptive and task-specific, the residual for $f_*$ lies in a subspace that is potentially much smaller than the orthogonal complement of the RKHS. The result formalizes the representation and approximation benefits of neural networks. Lastly, we show that the neural network function computed by gradient flow converges to the kernel ridgeless regression with an adaptive kernel, in the limit of vanishing regularization. The adaptive kernel viewpoint provides new angles of studying the approximation, representation, generalization, and optimization advantages of neural networks.