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.
To obtain Clarke hardness, we use the seesaw gadget introduced by [Tian and So, 2025], where a scalar variable t and the branch−|t|/2are added
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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.