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.

本文研究了使用修正线性单元（ReLU）训练简单神经网络的计算复杂度，并分析了训练数据维数对计算复杂度的影响。我们提供了参数化复杂度的结果，并且针对各种损失函数分析了两层ReLU网络的训练问题。

由数据维度参数化的ReLU网络训练的计算复杂度