Nonsmooth Implicit Differentiation for Machine Learning and Optimization
This work addresses the challenge of implicit differentiation in nonsmooth machine learning and optimization, providing a foundational tool for researchers and practitioners, though it is incremental as it extends classical theory to nonsmooth contexts.
The authors tackled the problem of training complex learning architectures by establishing a nonsmooth implicit function theorem with an operational calculus, enabling formal subdifferentiation for a wide class of nonsmooth problems, such as deep equilibrium networks and neural nets with conic optimization layers, and demonstrated its sharpness through numerical experiments showing pathological gradient dynamics without proper hypotheses.
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.