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Expressiveness of Multi-Neuron Convex Relaxations in Neural Network Certification

Yuhao Mao, Yani Zhang, Martin Vechev

TL;DR

This paper investigates the expressive power of multi-neuron convex relaxations used for neural network certification. It shows a universal convex barrier: even multi-neuron relaxations with finite resources cannot provably complete certification for general ReLU networks, extending the single-neuron barrier to all such relaxations. On the positive side, it proves completeness is achievable via equivalence-preserving network transformations or convex polytope partitioning, and it highlights that multi-neuron relaxations can reduce partition complexity relative to single-neuron methods. The results provide a theoretical foundation for future certified robustness work, including training and verification strategies that employ multi-neuron relaxations as core subroutines. Practically, these findings guide the design of complete verifiers and suggest new directions for robust training leveraging multi-neuron relaxations.

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.

Expressiveness of Multi-Neuron Convex Relaxations in Neural Network Certification

TL;DR

This paper investigates the expressive power of multi-neuron convex relaxations used for neural network certification. It shows a universal convex barrier: even multi-neuron relaxations with finite resources cannot provably complete certification for general ReLU networks, extending the single-neuron barrier to all such relaxations. On the positive side, it proves completeness is achievable via equivalence-preserving network transformations or convex polytope partitioning, and it highlights that multi-neuron relaxations can reduce partition complexity relative to single-neuron methods. The results provide a theoretical foundation for future certified robustness work, including training and verification strategies that employ multi-neuron relaxations as core subroutines. Practically, these findings guide the design of complete verifiers and suggest new directions for robust training leveraging multi-neuron relaxations.

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.

Paper Structure

This paper contains 39 sections, 25 theorems, 47 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Single-Neuron Convex Barrier
  3. This Work: Quantifying the Expressiveness and Completeness of Multi-Neuron Relaxations
  4. Key Contributions
  5. Background
  6. Convex Relaxations for Certification
  7. ReLU Network Analysis with Layerwise and Cross-Layer Convex Relaxations
  8. Single-Neuron and Multi-Neuron Relaxations
  9. Layerwise Multi-Neuron Incompleteness
  10. Cross-Layer Multi-Neuron Incompleteness
  11. Making Multi-Neuron Verifiers Complete
  12. Completeness via Network Transformations
  13. Completeness via Convex Polytope Partitioning
  14. Discussion
  15. Result Contexts
  16. ...and 24 more sections

Key Result

Lemma 3.0

Let $L\in {\mathbb{N}}$ and let $X$ be a convex polytope. Consider a ReLU network $f =f_L\circ \cdots \circ f_1$. Denote the variable of the $j$-th hidden layer of $f$ by ${\bm{v}}^{(j)}$, for $j\in [L-1]$, and the variable of the output layer by ${\bm{v}}^{(L)}$. For $1\leq i<L$, let ${\mathcal{C}}

Figures (6)

  • Figure 1: Triangle relaxation of a ReLU with input $x \in [-1, 1]$.
  • Figure 2: Blue area shows how the input box transforms under $W_1$ and ReLU; shaded area is the feasible set computed by ${\mathcal{P}}_1$.
  • Figure 3: A network encoding $f=\max(x_1, x_2)$.
  • Figure 4: A partition of the input set where every part remains as a convex polytope through the layers.
  • Figure 5: Visualization of the single-neuron and multi-neuron relaxations for a network encoding $f(x)=0$.
  • ...and 1 more figures

Theorems & Definitions (44)

  • Lemma 3.0
  • Lemma 3.0
  • Theorem 3.1
  • Lemma 4.0
  • Theorem 4.1
  • Theorem 5.1
  • Corollary 5.1
  • Proposition 5.1
  • Definition 5.2
  • Definition 5.3
  • ...and 34 more