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Deep learning is formulated as a discrete-time optimal control problem. Thisallows one to characterize necessary conditions for optimality and developtraining algorithms that do not rely on gradients with respect to the trainableparameters. In particular, we introduce the discrete-time method of successiveapproximations (MSA), which is based on the Pontryagin's maximum principle, fortraining neural networks. A rigorous error estimate for the discrete MSA isobtained, which sheds light on its dynamics and the means to stabilize thealgorithm. The developed methods are applied to train, in a rather principledway, neural networks with weights that are constrained to take values in adiscrete set. We obtain competitive performance and interestingly, very sparseweights in the case of ternary networks, which may be useful in modeldeployment in low-memory devices.

An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networks