Learning One Convolutional Layer with Overlapping Patches

TL;DR

This paper introduces Convotron, a provably efficient algorithm for learning a single convolutional layer with overlapping patches, applicable to common computer vision schemes, and robust to noise without special initialization.

Contribution

The paper presents the first provably efficient algorithm for learning one hidden convolutional layer with overlapping patches, extending to general distributions and common vision schemes.

Findings

Convotron converges without special initialization or tuning.

It efficiently learns convolutional layers with overlapping patches.

Learning with a single disjoint patch is also efficient.

Abstract

We give the first provably efficient algorithm for learning a one hidden layer convolutional network with respect to a general class of (potentially overlapping) patches. Additionally, our algorithm requires only mild conditions on the underlying distribution. We prove that our framework captures commonly used schemes from computer vision, including one-dimensional and two-dimensional "patch and stride" convolutions. Our algorithm-- $Convotron$ -- is inspired by recent work applying isotonic regression to learning neural networks. Convotron uses a simple, iterative update rule that is stochastic in nature and tolerant to noise (requires only that the conditional mean function is a one layer convolutional network, as opposed to the realizable setting). In contrast to gradient descent, Convotron requires no special initialization or learning-rate tuning to converge to the global optimum. We also point out that learning one hidden convolutional layer with respect to a Gaussian distribution and just $one$ disjoint patch $P$ (the other patches may be arbitrary) is $easy$ in the following sense: Convotron can efficiently recover the hidden weight vector by updating $only$ in the direction of $P$.