Deep Online Convex Optimization by Putting Forecaster to Sleep

Methods from convex optimization such as accelerated gradient descent are widely used as building blocks for deep learning algorithms. However, the reasons for their empirical success are unclear, since neural networks are not convex and standard guarantees do not apply. This paper develops the first rigorous link between online convex optimization and error backpropagation on convolutional networks. The first step is to introduce circadian games, a mild generalization of convex games with similar convergence properties. The main result is that error backpropagation on a convolutional network is equivalent to playing out a circadian game. It follows immediately that the waking-regret of players in the game (the units in the neural network) controls the overall rate of convergence of the network. Finally, we explore some implications of the results: (i) we describe the representations learned by a neural network game-theoretically, (ii) propose a learning setting at the level of individual units that can be plugged into deep architectures, and (iii) propose a new approach to adaptive model selection by applying bandit algorithms to choose which players to wake on each round.