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

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.

Learning Decision-Sufficient Representations for Linear Optimization

We study how to construct compressed datasets that suffice to recover optimal decisions in linear programs with an unknown cost vector $c$ lying in a prior set $\mathcal{C}$. Recent work by Bennouna et al. provides an exact geometric characterization of sufficient decision datasets (SDDs) via an intrinsic decision-relevant dimension $d^\star$. However, their algorithm for constructing minimum-size SDDs requires solving mixed-integer programs. In this paper, we establish hardness results showing that computing $d^\star$ is NP-hard and deciding whether a dataset is globally sufficient is coNP-hard, thereby resolving a recent open problem posed by Bennouna et al. To address this worst-case intractability, we introduce pointwise sufficiency, a relaxation that requires sufficiency for an individual cost vector. Under nondegeneracy, we provide a polynomial-time cutting-plane algorithm for constructing pointwise-sufficient decision datasets. In a data-driven regime with i.i.d.\ costs, we further propose a cumulative algorithm that aggregates decision-relevant directions across samples, yielding a stable compression scheme of size at most $d^\star$. This leads to a distribution-free PAC guarantee: with high probability over the training sample, the pointwise sufficiency failure probability on a fresh draw is at most $\tilde{O}(d^\star/n)$, and this rate is tight up to logarithmic factors. Finally, we apply decision-sufficient representations to contextual linear optimization, obtaining compressed predictors with generalization bounds scaling as $\tilde{O}(\sqrt{d^\star/n})$ rather than $\tilde{O}(\sqrt{d/n})$, where $d$ is the ambient cost dimension.