When is a Convolutional Filter Easy To Learn?
This provides theoretical guarantees for gradient-based learning in convolutional networks beyond Gaussian assumptions, which is incremental but addresses a key bottleneck in understanding non-Gaussian data.
The paper analyzes the convergence of gradient descent for learning a convolutional filter with ReLU activation, showing it can learn the filter in polynomial time under non-Gaussian input distributions, with convergence rates depending on input smoothness and patch closeness.
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.