The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality

• URL: http://arxiv.org/abs/2105.08675v1
• Date: Tue, 18 May 2021 17:05:26 GMT
• Title: The Computational Complexity of ReLU Network Training Parameterized by Data Dimensionality
• Authors: Vincent Froese, Christoph Hertrich, Rolf Niedermeier
• Abstract summary: We analyze the influence of the dimension $d$ of the training data on the computational complexity. We prove that known brute-force strategies are essentially optimal. In particular, we extend a known-time algorithm for constant $d$ and convex loss functions to a more general class of loss functions.
• Score: 8.940054309023525
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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