Testing Stationarity Concepts for ReLU Networks: Hardness, Regularity, and Robust Algorithms

We study the computational problem of the stationarity test for the empirical loss of neural networks with ReLU activation functions. Our contributions are: Hardness: We show that checking a certain first-order approximate stationarity concept for a piecewise linear function is co-NP-hard. This implies that testing a certain stationarity concept for a modern nonsmooth neural network is in general computationally intractable. As a corollary, we prove that testing so-called first-order minimality for functions in abs-normal form is co-NP-complete, which was conjectured by Griewank and Walther (2019, SIAM J. Optim., vol. 29, p284). Regularity: We establish a necessary and sufficient condition for the validity of an equality-type subdifferential chain rule in terms of Clarke, Fr\'echet, and limiting subdifferentials of the empirical loss of two-layer ReLU networks. This new condition is simple and efficiently checkable. Robust algorithms: We introduce an algorithmic scheme to test near-approximate stationarity in terms of both Clarke and Fr\'echet subdifferentials. Our scheme makes no false positive or false negative error when the tested point is sufficiently close to a stationary one and a certain qualification is satisfied. This is the first practical and robust stationarity test approach for two-layer ReLU networks.