凸网络仍难以验证：Lipschitz常数的维度-精度障碍
Convex Networks Remain Hard to Certify: Dimension-Accuracy Barriers for Lipschitz Constants
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中文总结 AI 辅助
该研究证明输入凸神经网络的全局欧氏Lipschitz常数判定问题是NP完全且W[1]-难的，不存在维度分离的多项式精度保证，解决了相关公开问题。
中文摘要 AI 辅助
输入凸神经网络允许在其输入上进行全局可处理的最小化，因此人们可能期望其全局正则性在低输入维度下是可处理的。我们证明了与该期望相悖的精确且精度敏感的障碍。给定一个无偏单隐层ReLU网络 $f(x)=\sum_{r=1}^n \mathrm{ReLU}(a_r^\top x)$，其输出权重为单位正数，判定其全局欧氏Lipschitz常数是否至少为一个有理阈值是NP完全的，且当按输入维度 $d$ 参数化时是W[1]-难的。在单位球上以及第一层权重为最多9个非零整数的情况下，同样成立。更明确地说，除非FPT等于W[1]，否则不存在确定性乘法近似方案能在 $g(d)\mathrm{poly}(\mathcal{B},1/\varepsilon)$ 时间内运行。在指数时间假设下，此类算法无法在 $g(d)(\mathcal{B}+1/\varepsilon)^{o(d/\log d)}$ 时间内运行。因此，精度与维度之间无法存在多项式依赖关系。该精确结果解决了COLT 2025提出的一个公开问题的欧氏情形，以及ICLR 2026针对一般两层网络的参数化困难理论中遗留的问题。该近似障碍是针对生成元呈现的zonotope的，补充了已知的 $(1/\varepsilon)^{O(d)}$ 时间方案和针对半空间呈现的多面体的类似障碍。我们的提升选择归约具有逆多项式径向间隙，这通过有理循环zonogon的定量定理证明。等价地，该结果适用于欧氏zonotope半径和按秩参数化的半正定二元二次最大化。凸性使最小化变得容易，但它并未使全局敏感性成为固定参数可处理的，也不允许维度分离的完全多项式精度保证。
英文摘要
Input-convex neural networks permit globally tractable minimization over their inputs, so one might expect their global regularity to be tractable in low input dimension. We prove exact and accuracy-sensitive barriers to this expectation. Given a bias-free one-hidden-layer ReLU network $f(x)=\sum_{r=1}^n \mathrm{ReLU}(a_r^\top x)$ with unit positive output weights, deciding whether its global Euclidean Lipschitz constant is at least a rational threshold is NP-complete and W[1]-hard when parameterized by the input dimension $d$. The same holds on the unit ball and with integral first-layer weights having at most nine nonzeros. More sharply, no deterministic multiplicative approximation scheme runs in $g(d)\mathrm{poly}(\mathcal B,1/\varepsilon)$ time unless FPT equals W[1]. Under the Exponential Time Hypothesis, no such algorithm runs in $g(d)(\mathcal B+1/\varepsilon)^{o(d/\log d)}$ time. Thus accuracy cannot have a polynomial dependence separated from dimension. The exact result resolves the Euclidean case of an open problem posed at COLT 2025 and left open by the ICLR 2026 parameterized hardness theory for general two-layer networks. The approximation barrier is specific to generator-presented zonotopes, complementing known $(1/\varepsilon)^{O(d)}$-time schemes and an analogous barrier for halfspace-presented polytopes. Our lifted-selector reduction has an inverse-polynomial radial gap, proved through a quantitative theorem for rational cyclic zonogons. Equivalently, the results apply to Euclidean zonotope radius and positive-semidefinite binary quadratic maximization parameterized by rank. Convexity makes minimization easy, but it does not make global sensitivity fixed-parameter tractable or permit a dimension-separated fully polynomial accuracy guarantee.
