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.

通过对常用凸松弛方法进行深入研究，我们发现：（i）更高级的松弛方法允许更多单变量函数被精确分析的ReLU网络表达，（ii）更精确的松弛方法能够允许指数级规模的解空间编码相同函数的ReLU网络，以及（iii）即使使用最精确的单神经元松弛方法，也无法构建能够精确分析多变量凸、单调的分段线性函数的ReLU网络。

ReLU网络在凸松弛下的表达能力