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arXiv · 2410.06816

Expressiveness of Multi-Neuron Convex Relaxations in Neural Network Certification

Yuhao Mao·Yani Zhang·Martin Vechev
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Abstract

Neural network certification methods heavily rely on convex relaxations to provide robustness guarantees. However, these relaxations are often imprecise: even the most accurate single-neuron relaxation is incomplete for general ReLU networks, a limitation known as the *single-neuron convex barrier*. While multi-neuron relaxations have been heuristically applied to address this issue, two central questions arise: (i) whether they overcome the convex barrier, and if not, (ii) whether they offer theoretical capabilities beyond those of single-neuron relaxations. In this work, we present the first rigorous analysis of the expressiveness of multi-neuron relaxations. Perhaps surprisingly, we show that they are inherently incomplete, even when allocated sufficient resources to capture finitely many neurons and layers optimally. This result extends the single-neuron barrier to a *universal convex barrier* for neural network certification. On the positive side, we show that completeness can be achieved by either (i) augmenting the network with a polynomial number of carefully designed ReLU neurons or (ii) partitioning the input domain into convex sub-polytopes, thereby distinguishing multi-neuron relaxations from single-neuron ones which are unable to realize the former and have worse partition complexity for the latter. Our findings establish a foundation for multi-neuron relaxations and point to new directions for certified robustness, including training methods tailored to multi-neuron relaxations and verification methods with multi-neuron relaxations as the main subroutine.

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Yuhao Mao, Yani Zhang, Martin Vechev. 2024-10-09. Expressiveness of Multi-Neuron Convex Relaxations in Neural Network Certification. https://arxiv.org/abs/2410.06816

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