Deep learning as optimal control problems: models and numerical methods
This work provides a theoretical framework for deep learning as optimal control, which is incremental in connecting existing models to numerical methods.
The paper tackles the interpretation of deep learning neural networks as discretizations of optimal control problems with ODE constraints, reviewing optimality conditions and developing algorithms that ensure discrete necessary conditions are met, with numerical comparisons of flows and generalization ability.
We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation. This leads to a class of algorithms for solving the discrete optimal control problem which guarantee that the corresponding discrete necessary conditions for optimality are fulfilled. The differential equation setting lends itself to learning additional parameters such as the time discretisation. We explore this extension alongside natural constraints (e.g. time steps lie in a simplex). We compare these deep learning algorithms numerically in terms of induced flow and generalisation ability.