发表机构
Ashoka University; Technische Universität Berlin; Max Planck Institute for Mathematics in the Sciences; University of Technology Nuremberg; University College London; Freie Universität Berlin（阿肖克大学; 柏林工业大学; 马克斯·普朗克数理科学研究所; 纽伦堡工业大学; 伦敦大学学院; 柏林自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明了zonotope上$L_p$-范数最大化相对于维度是W[1]-困难的，对应两层ReLU输入凸神经网络的$L_p$-Lipschitz常数计算也具相同困难性，解决了COLT'25的未解决问题。
AI 中文摘要
Lipschitz常数是量化神经网络对微小输入扰动敏感性的标准方法，但即使对于浅层ReLU网络，计算它们也很困难。我们针对两层输入凸神经网络（ICNNs）研究该问题，这是一种受限架构，其中非负输出权重确保凸性。计算这些网络的$L_p$-Lipschitz常数等价于在zonotope上最大化对偶范数。虽然zonotope上的$L_1$-和$L_\fty$-范数最大化分别存在固定参数和多项式时间算法，但其余$L_p$-范数的参数化复杂性是未解决的问题。我们证明，对于每个固定的$p\in (1,\infty)\cap \mathbb{Q}$，在$\mathbb{R}^d$中的zonotope上最大化$L_p$-范数相对于维度$d$是W[1]-困难的。此外，我们的困难性结果意味着，在指数时间假设（ETH）下，蛮力枚举算法对于该问题本质上是最优的。通过对偶性，相同的困难性结果适用于计算两层ReLU ICNNs的$L_p$-Lipschitz常数。我们的证明首先针对$L_2$-范数建立结果，然后使用合适的泰勒近似将构造转移到任意固定的$p\in (1,\infty)\cap \mathbb{Q}$。这些结果解决了关于zonotope范数最大化和两层ICNN Lipschitz常数的参数化复杂性状态的相应问题。我们的论文解决了COLT'25上提出的一个未解决问题，同时有几篇独立的同期论文解决了相同问题。我们的论文优先清晰阐述证明背后的基础数学和概念直觉，此外还明确描述了我们的研究过程，包括大型语言模型（LLMs）的使用。
英文摘要
Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult even for shallow ReLU networks. We study this problem for two-layer input-convex neural networks (ICNNs), a restricted architecture where nonnegative output weights enforce convexity. Computing the $L_p$-Lipschitz constant for these networks is equivalent to maximizing the dual norm over a zonotope. While $L_1$- and $L_\infty$-norm maximization on zonotopes admit fixed-parameter and polynomial-time algorithms, respectively, the parameterized complexity of the remaining $L_p$-norms was open. We prove that, for every fixed $p\in (1,\infty)\cap \mathbb{Q}$, maximizing the $L_p$-norm over a zonotope in $\mathbb{R}^d$ is W[1]-hard with respect to the dimension $d$. Moreover, our hardness results imply that brute-force enumeration algorithms are essentially optimal for this problem under the Exponential Time Hypothesis. By duality, the same hardness results hold for computing the $L_p$-Lipschitz constant of two-layer ReLU ICNNs. Our proof first establishes the result for the $L_2$-norm and then transfers the construction to arbitrary fixed $p\in (1,\infty)\cap\mathbb{Q}$ using a suitable Taylor approximation. These results resolve the corresponding questions regarding the parameterized complexity status for zonotope norm maximization and two-layer ICNN Lipschitz constants. Our paper resolves an open problem posted at COLT'25. There are several independent concurrent papers resolving the same problem. Our paper prioritizes a clear exposition of the underlying mathematics and conceptual intuitions behind the proof. Additionally, we explicitly describe our research process including the use of LLMs.
