Parameterized Complexity of Stationarity Testing for Piecewise-Affine Functions and Shallow CNN Losses

The paper doesn't just prove hardness; it maps which problem instances remain tractable in fixed dimension, offering a roadmap for when solvers can guarantee efficient stationarity verification versus when they hit computational walls. This distinction between intractable worst-case and solvable restricted regimes is what practitioners actually need.

This work sits in the theoretical foundations layer beneath the recent wave of optimization and learning theory papers. Unlike the bilevel optimization advances (BROS, May 2026) that focus on memory-efficient convergence, or the random feature regression bounds that characterize generalization, this paper addresses a layer below: can we even verify when an optimizer has found a stationary point in nonsmooth settings? The ReLU-centric framing connects to the broader push to understand neural network loss landscapes, though most recent coverage here has focused on generalization and data structure (random walks enabling sparse learning) rather than the verification problem itself.

If follow-up work applies these parameterized hardness boundaries to actual solver implementations (e.g., proximal methods, subgradient algorithms) and shows that the tractable regimes correspond to networks practitioners already use, that confirms the theory has teeth. If the hardness results remain purely asymptotic without practical solver guidance, the impact stays confined to complexity theory.