Tight Sample Complexity of Learning One-hidden-layer Convolutional Neural Networks
This work provides a tight sample complexity analysis for training CNNs, which is significant for machine learning practitioners seeking efficient learning guarantees, though it is incremental in refining existing theoretical bounds.
The paper tackles the problem of learning one-hidden-layer convolutional neural networks with non-overlapping filters, proposing an approximate gradient descent algorithm that achieves linear convergence to ground-truth parameters with tight sample complexity matching information-theoretic lower bounds for linear activations.
We study the sample complexity of learning one-hidden-layer convolutional neural networks (CNNs) with non-overlapping filters. We propose a novel algorithm called approximate gradient descent for training CNNs, and show that, with high probability, the proposed algorithm with random initialization grants a linear convergence to the ground-truth parameters up to statistical precision. Compared with existing work, our result applies to general non-trivial, monotonic and Lipschitz continuous activation functions including ReLU, Leaky ReLU, Sigmod and Softplus etc. Moreover, our sample complexity beats existing results in the dependency of the number of hidden nodes and filter size. In fact, our result matches the information-theoretic lower bound for learning one-hidden-layer CNNs with linear activation functions, suggesting that our sample complexity is tight. Our theoretical analysis is backed up by numerical experiments.