Convex relaxations are a key component of training and certifying provably safe neural networks. However, despite substantial progress, a wide and poorly understood accuracy gap to standard networks remains, raising the question of whether this is due to fundamental limitations of convex relaxations. Initial work investigating this question focused on the simple and widely used IBP relaxation. It revealed that some univariate, convex, continuous piecewise linear (CPWL) functions cannot be encoded by any ReLU network such that its IBP-analysis is precise. To explore whether this limitation is shared by more advanced convex relaxations, we conduct the first in-depth study on the expressive power of ReLU networks across all commonly used convex relaxations. We show that: (i) more advanced relaxations allow a larger class of univariate functions to be expressed as precisely analyzable ReLU networks, (ii) more precise relaxations can allow exponentially larger solution spaces of ReLU networks encoding the same functions, and (iii) even using the most precise single-neuron relaxations, it is impossible to construct precisely analyzable ReLU networks that express multivariate, convex, monotone CPWL functions.

翻译：凸松弛是训练和认证可证明安全的神经网络的关键组成部分。然而，尽管取得了显著进展，其与标准网络之间的准确率差距仍广泛存在且理解不足，这引发了凸松弛是否具有根本局限性的疑问。针对这一问题的早期研究聚焦于简单且广泛使用的IBP松弛，发现某些单变量凸连续分段线性函数无法被任何ReLU网络精确编码以实现精确的IBP分析。为探究这一局限性是否同样存在于更先进的凸松弛中，我们首次系统研究了ReLU网络在所有常用凸松弛下的表达能力。研究表明：（i）更先进的松弛允许更大类别的单变量函数被表达为可精确分析的ReLU网络；（ii）更精确的松弛可使编码相同函数的ReLU网络的解空间呈指数级扩大；（iii）即使采用最精确的单神经元松弛，也无法构造出表达多元凸单调连续分段线性函数的可精确分析的ReLU网络。