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Deep learning as optimal control problems: models and numerical methods

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arxiv 1904.05657 v3 pith:CET7SUJN submitted 2019-04-11 math.OC cs.LGcs.NAmath.NA

classification math.OCcs.LGcs.NAmath.NA
keywords learningconditionscontroldeepoptimaloptimalityalgorithmsdifferential
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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.

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  1. An optimal control approach for neural network architecture adaptation with a posteriori error estimation

    cs.LG 2026-07 conditional novelty 7.0 of 10

    The paper derives a posteriori error estimates for neural network depth adaptation by formulating training as an optimal control problem and using dual weighted residuals to insert layers where error is highest.

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