Parameterized Complexity of Stationarity Testing for Piecewise-Affine Functions and Shallow CNN Losses
arXiv:2605.10219v2 Announce Type: replace-cross Abstract: We study the parameterized complexity of testing approximate first-order stationarity at a prescribed point for continuous piecewise-affine (PA) functions, a basic task in nonsmooth optimization. PA functions form a canonical model for nonsmooth stationarity testing and capture the local polyhedral geometry that appears in ReLU-type training losses. Recent work by Tian and So (SODA 2025) shows that testing approximate stationarity notions for PA functions is computationally intractable in the worst case, and identifies fixed-dimensional
This is a theoretical computer science paper, which by its nature is foundational and not directly tied to immediate events.
For a sophisticated reader, this type of research indicates the fundamental limits and complexities of optimization in specific AI models.
This paper refines understanding of the computational difficulty in ensuring stationarity for certain AI functions, which doesn't immediately change practices.
Researchers gain a deeper understanding of the computational challenges in optimizing piecewise-affine functions and shallow CNNs.
Future algorithm development for non-smooth optimization in AI might incorporate these complexity insights to design more efficient or specialized methods.
Over a very long period, this foundational work could indirectly contribute to the design of more robust or theoretically grounded AI models.
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