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 tractability as an open direction. We address this direction from the viewpoint of parameterized complexity, with the ambient dimension $d$ as the parameter. In this paper, we give XP algorithms in fixed dimension for the tractable sides, and prove W[1]-hardness for the complementary sides. Moreover, lower bounds under the Exponential Time Hypothesis rule out algorithms running in time $ρ(d)\size^{o(d)}$ for any computable function $ρ$, where $\size$ denotes the total binary encoding length of the stationarity-testing instance. As a further consequence, our results yield the corresponding parameterized complexity picture for testing local minimality of continuous PA functions. We further extend our hardness results to a family of shallow ReLU CNN training losses, with stationarity tested in the trainable weight space. Thus, the same parameterized-complexity picture also appears for simple CNN training losses.

翻译：我们研究了连续分段仿射(PA)函数在指定点处检验近似一阶驻点性质的参数化复杂性，这是非光滑优化中的一项基础任务。PA函数构成了非光滑驻点检验的典范模型，并刻画了ReLU型训练损失中出现的局部多面体几何结构。Tian与So（SODA 2025）的最新研究表明，在一般情况下检验PA函数的近似驻点性质是计算不可处理的，并指出"固定维数可处理性"是一个待研究方向。本文从参数化复杂性角度探讨该方向，以环境维数$d$为参数。我们针对可处理情形给出了固定维数下的XP算法，并对互补情形证明了W[1]-困难性。此外，基于指数时间假说的下界排除了任何可计算函数$ρ(d)$下运行时间为$ρ(d)\size^{o(d)}$的算法，其中$\size$表示驻点检验实例的总二进制编码长度。进一步，我们的结果还可推出连续PA函数局部极小性检验的对应参数化复杂性图景。我们将困难性结果推广至一类浅层ReLU CNN训练损失，并在可训练权值空间中检验驻点性质。因此，简单的CNN训练损失同样呈现出相同的参数化复杂性图景。