Nonsmooth Implicit Differentiation for Machine-Learning and Optimization

Jerome Bolte, Tam Le, Edouard Pauwels, Antonio Silveti-Falls

Keywords: Implicit differentiation, algorithmic differentiation, nonsmooth optimization, deep learning, conservative fields.

TL;DR: We present a nonsmooth implicit function theorem with an operational calculus which is compatible with algorithmic differentiation and optimization.

Abstract: In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a nonsmooth form of the classical invertibility condition is fulfilled. This approach allows for formal subdifferentiation: for instance, replacing derivatives by Clarke Jacobians in the usual differentiation formulas is fully justified for a wide class of nonsmooth problems. Moreover this calculus is entirely compatible with algorithmic differentiation (e.g., backpropagation). We provide several applications such as training deep equilibrium networks, training neural nets with conic optimization layers, or hyperparameter-tuning for nonsmooth Lasso-type models. To show the sharpness of our assumptions, we present numerical experiments showcasing the extremely pathological gradient dynamics one can encounter when applying implicit algorithmic differentiation without any hypothesis.

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