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To verify safety and robustness of neural networks, researchers have successfully appliedabstract interpretation, primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce theinterval universal approximation(IUA) theorem. IUA shows that neural networks not only can approximate any continuous functionf(universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics off(the result of applyingfto a set of inputs). We call this notion of approximationinterval approximation. Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we callsquashable functions. Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ∞-norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ2-intermediate problem, which is strictly harder thanNP-complete problems, assumingcoNP⊄NP. As a result,IUA is an inherently hard problem: No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.
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This content will become publicly available on January 22, 2026
Expressivity of Neural Networks with Random Weights and Learned Biases
Landmark universal function approximation results for neural networks with trained weights and biases provided the impetus for the ubiquitous use of neural networks as learning models in neuroscience and Artificial Intelligence (AI). Recent work has extended these results to networks in which a smaller subset of weights (e.g., output weights) are tuned, leaving other parameters random. However, it remains an open question whether universal approximation holds when only biases are learned, despite evidence from neuroscience and AI that biases significantly shape neural responses. The current paper answers this question. We provide theoretical and numerical evidence demonstrating that feedforward neural networks with fixed random weights can approximate any continuous function on compact sets. We further show an analogous result for the approximation of dynamical systems with recurrent neural networks. Our findings are relevant to neuroscience, where they demonstrate the potential for behaviourally relevant changes in dynamics without modifying synaptic weights, as well as for AI, where they shed light on recent fine-tuning methods for large language models, like bias and prefix-based approaches.
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- Award ID(s):
- 2238247
- PAR ID:
- 10631315
- Publisher / Repository:
- International Conference on Learning Representations
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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