Deep learning with Gaussian continuation

In this paper, we develop a Gaussian continuation framework for deep learning, which is an optimization strategy that involves smoothing the loss function by convolving it with a Gaussian kernel. The width of the kernel is decreased over the optimization steps leading to the degree of smoothing being gradually relaxed during training. This enables gradient-based optimization to more easily traverse suboptimal local minima in non-convex loss landscapes. We carefully study the unique theoretical difficulties for continuation posed by deep learning applications, and how classical assumptions made in theoretical analysis of continuation methods must be revised or qualified. Our analysis shows that if the width of the Gaussian kernel is treated as an optimization variable, it naturally tends to zero in virtually all minimization problems. As a consequence, Gaussian continuation will converge to the minima of the original loss function as long as the base optimizer is capable of escaping saddle points. We demonstrate that Gaussian continuation outperforms baseline methods in training a regression network and a convolutional generative adversarial network (GAN).