arXiv:2411.03006v4 Announce Type: replace-cross Abstract: Neural networks with piecewise linear activation functions, such as rectified linear units (ReLU) or maxout, are among the most fundamental models in modern machine learning. We make a step towards proving lower bounds on the size of such neural networks by linking their representative capabilities to the notion of the extension complexity $\mathrm{xc}(P)$ of a polytope $P$. This is a well-studied quantity in combinatorial optimization and polyhedral geometry describing the number of inequalities needed to model $P$ as a linear program.
This paper represents a focused effort to establish theoretical lower bounds on the complexity of neural networks, a fundamental step in understanding their computational limits.
Understanding the intrinsic complexity of neural networks via concepts like extension complexity is critical for guiding future AI research, optimization, and the design of more efficient architectures.
This research provides a new theoretical lens, linking neural network complexity to established concepts in combinatorial optimization, which could lead to more principled approaches to network design and the development of more efficient AI.
- · AI researchers
- · Machine learning theoreticians
- · Combinatorial optimization researchers
- · Developers relying solely on empirical trial-and-error
- · Purely heuristic AI optimization methods
It provides a foundational mathematical tool to analyze the efficiency of neural networks.
This could lead to breakthroughs in designing more theoretically optimal and resource-efficient AI models.
The insights gained might inform the development of novel AI training algorithms or prompt entirely new network architectures that circumvent current limitations.
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Read at arXiv cs.LG