The Rate of Convergence of Variation-Constrained Deep Neural Networks
This provides insight into why deep neural networks resist overfitting despite large parameter growth, addressing a fundamental issue in machine learning theory.
The paper tackles the problem of understanding the learnability of neural networks through statistical risk, showing that variation-constrained neural networks can achieve a near-parametric convergence rate of n^{-1/2+δ} for sample size n, which is faster than the previously known bound of n^{-1/4}.
Multi-layer feedforward networks have been used to approximate a wide range of nonlinear functions. An important and fundamental problem is to understand the learnability of a network model through its statistical risk, or the expected prediction error on future data. To the best of our knowledge, the rate of convergence of neural networks shown by existing works is bounded by at most the order of $n^{-1/4}$ for a sample size of $n$. In this paper, we show that a class of variation-constrained neural networks, with arbitrary width, can achieve near-parametric rate $n^{-1/2+δ}$ for an arbitrarily small positive constant $δ$. It is equivalent to $n^{-1 +2δ}$ under the mean squared error. This rate is also observed by numerical experiments. The result indicates that the neural function space needed for approximating smooth functions may not be as large as what is often perceived. Our result also provides insight to the phenomena that deep neural networks do not easily suffer from overfitting when the number of neurons and learning parameters rapidly grow with $n$ or even surpass $n$. We also discuss the rate of convergence regarding other network parameters, including the input dimension, network layer, and coefficient norm.