Understanding the computational complexity of training simple neural networks with rectified linear units (ReLUs) has recently been a subject of intensive research. Closing gaps and complementing results from the literature, we present several results on the parameterized complexity of training two-layer ReLU networks with respect to various loss functions. After a brief discussion of other parameters, we focus on analyzing the influence of the dimension $d$ of the training data on the computational complexity. We provide running time lower bounds in terms of W[1]-hardness for parameter $d$ and prove that known brute-force strategies are essentially optimal (assuming the Exponential Time Hypothesis). In comparison with previous work, our results hold for a broad(er) range of loss functions, including $\ell^p$-loss for all $p\in[0,\infty]$. In particular, we extend a known polynomial-time algorithm for constant $d$ and convex loss functions to a more general class of loss functions, matching our running time lower bounds also in these cases.

中文翻译：

---

数据维度参数化的ReLU网络训练的计算复杂性

近年来，了解使用线性化精算单元（ReLU）训练简单神经网络的计算复杂性已成为深入研究的主题。缩小差距并从文献中补充结果，我们提出了关于训练两层ReLU网络针对各种损失函数的参数化复杂性的一些结果。在简要讨论了其他参数之后，我们着重分析训练数据的维数d $对计算复杂度的影响。我们为参数$ d $提供了基于W [1]硬度的运行时间下限，并证明了已知的蛮力策略本质上是最优的（假设指数时间假设）。与先前的工作相比，我们的结果适用于更广泛的损失函数，包括所有$ p \ in [0，\ infty] $的$ \ ell ^ p $-损失。