Testing approximate first-order stationarity for continuous PA functions is XP-tractable but W[1]-hard (with ETH lower bounds) when parameterized by dimension d, and the same parameterized complexity classification holds for stationarity testing in shallow ReLU CNN weight space.
The general definitions follow the standard nonsmooth-analysis viewpoint; see, for example, Clarke [1990], Rockafellar and Wets
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Parameterized Complexity of Stationarity Testing for Piecewise-Affine Functions and Shallow CNN Losses
Testing approximate first-order stationarity for continuous PA functions is XP-tractable but W[1]-hard (with ETH lower bounds) when parameterized by dimension d, and the same parameterized complexity classification holds for stationarity testing in shallow ReLU CNN weight space.