When is a Convolutional Filter Easy To Learn?

TL;DR

This paper provides a theoretical analysis of the convergence of gradient descent for learning convolutional filters with ReLU activation, applicable to non-Gaussian inputs, and supports the theory with experiments.

Contribution

It offers the first recovery guarantee for gradient-based learning of convolutional filters on non-Gaussian input distributions, without relying on specific input forms.

Findings

Gradient descent can learn convolutional filters in polynomial time.

Convergence depends on input distribution smoothness and patch similarity.

Experiments support the theoretical convergence guarantees.

Abstract

We analyze the convergence of (stochastic) gradient descent algorithm for learning a convolutional filter with Rectified Linear Unit (ReLU) activation function. Our analysis does not rely on any specific form of the input distribution and our proofs only use the definition of ReLU, in contrast with previous works that are restricted to standard Gaussian input. We show that (stochastic) gradient descent with random initialization can learn the convolutional filter in polynomial time and the convergence rate depends on the smoothness of the input distribution and the closeness of patches. To the best of our knowledge, this is the first recovery guarantee of gradient-based algorithms for convolutional filter on non-Gaussian input distributions. Our theory also justifies the two-stage learning rate strategy in deep neural networks. While our focus is theoretical, we also present experiments that illustrate our theoretical findings.