发表机构
FAMU–FSU College of Engineering; Florida State University（佛罗里达农工大学-佛罗里达州立大学工程学院; 佛罗里达州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明两层隐藏层ReLU网络中最小化隐藏神经元数量的问题是NP-难解的，为ReLU网络设计采用启发式逼近方法提供了理论依据。
AI 中文摘要
神经网络架构优化中的一个基本问题是，能否高效计算出在规定容差内逼近目标函数所需的最小隐藏神经元数量。本文针对两层隐藏层ReLU网络，在$L^p(\boldsymbol{R}^d,\boldsymbol{R}^m)$逼近约束下解决了该问题。对于每个固定的$d\boldsymbol{\text{≥}}1$、$m\boldsymbol{\text{≥}}1$且$1\boldsymbol{\text{≤}}p\boldsymbol{<}\boldsymbol{\text{∞}}$，我们证明精确计算最优值是NP-难解的。即使目标由有理ReLU网络表示（其实现非零、逐分量非负、紧支撑、全局Lipschitz且分段仿射连续），该结论仍然成立。从3-SAT问题出发的多项式时间归约会产生架构间隙：不可满足公式对应的最优值为0，而可满足公式对应的最优值至少为$d+2$。证明过程构造了由两层隐藏层ReLU网络实现的紧支撑多面体平截头体函数，并建立了盒-平截头体函数有限线性组合的$L^p$密度。这些结果为在ReLU神经网络设计中采用启发式逼近方法提供了理论依据，表明在多项式时间内达到最小配置是计算上无法实现的。
英文摘要
A fundamental question in neural network architecture optimization is whether the minimum hidden-neuron count required to approximate a target function within a prescribed tolerance can be computed efficiently. This paper resolves this question for two-hidden-layer ReLU networks under an $L^p(\mathbb{R}^d,\mathbb{R}^m)$ approximation constraint. For every fixed $d \ge 1$, $m \ge 1$, and $1 \le p < \infty$, we prove that computing the optimum exactly is NP-hard. The result holds even when the target is represented by a rational ReLU network whose realization is nonzero, componentwise nonnegative, compactly supported, globally Lipschitz, and continuous piecewise affine. The polynomial-time reduction from 3-SAT produces an architecture gap in which unsatisfiable formulas yield an optimum of zero, whereas satisfiable formulas yield an optimum of at least $d+2$. The proof constructs compactly supported polyhedral frustum functions realized by two-hidden-layer ReLU networks and establishes the $L^p$-density of finite linear combinations of box-frustum functions. The results offer theoretical justification for employing heuristic approximation methods in the design of ReLU neural networks, illustrating that attaining a minimal configuration within polynomial time is computationally unachievable.
Journal refLee, S., NP-Hardness of Minimizing Neurons in Two-Hidden-Layer ReLU Neural Networks, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 48, no. 11, 2026
DOI：10.1109/TPAMI.2026.3711513
