Optimized Piecewise Affine Abstractions of Neural Networks with Learnable Activation Functions

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arXiv:2602.06737v2 Announce Type: replace Abstract: We present a generalized framework for the range verification of neural networks featuring non-linear activation functions. Our approach first constructs an ``optimized piecewise affine abstraction" of the network that replaces each non-linear activation function by a piecewise affine (PWA) function plus a bounded error. Such PWA functions are readily amenable to existing neural network verification techniques using specializations of linear arithmetic SMT solvers and mixed-integer optimization approaches. However, there are infinitely many ways to abstract each node, with a natural tradeoff between the number of pieces used, the global error bound, and the complexity of the resulting verification problem. We propose a dynamic programming (DP) algorithm to systematically compute the optimized PWA abstraction for general activation functions, guaranteeing tighter output bounds. The algorithm combines a local DP approximation at each node with a global error bound, yielding a variant of the knapsack problem for deciding how to allocate a fixed budget on the total number of pieces across units so as to minimize the worst-case error bound between the network and its approximation. Although the knapsack problem is itself NP-hard, we can use pseudo-polynomial DP algorithms as well as approximation schemes to solve it efficiently. Crucially, our approach is broadly applicable to diverse networks consisting of non-linear activations, including standard Multi-Layer Perceptrons (MLPs) and recently proposed architectures such as …

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