Neural Taylor Approximations: Convergence and Exploration in Rectifier Networks
This addresses the theoretical gap in optimization for deep learning, offering insights for researchers and practitioners working with non-smooth neural networks.
The paper tackles the lack of convergence guarantees for modern convolutional networks with rectifiers and max-pooling, providing the first such guarantee that matches a lower bound for convex nonsmooth functions. It also applies neural Taylor approximations to analyze why adaptive optimizers like Adam converge better, attributing it to more thorough exploration of activation configurations.
Modern convolutional networks, incorporating rectifiers and max-pooling, are neither smooth nor convex; standard guarantees therefore do not apply. Nevertheless, methods from convex optimization such as gradient descent and Adam are widely used as building blocks for deep learning algorithms. This paper provides the first convergence guarantee applicable to modern convnets, which furthermore matches a lower bound for convex nonsmooth functions. The key technical tool is the neural Taylor approximation -- a straightforward application of Taylor expansions to neural networks -- and the associated Taylor loss. Experiments on a range of optimizers, layers, and tasks provide evidence that the analysis accurately captures the dynamics of neural optimization. The second half of the paper applies the Taylor approximation to isolate the main difficulty in training rectifier nets -- that gradients are shattered -- and investigates the hypothesis that, by exploring the space of activation configurations more thoroughly, adaptive optimizers such as RMSProp and Adam are able to converge to better solutions.