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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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A unified and constructive framework for the universality of neural networks
Abstract One of the reasons why many neural networks are capable of replicating complicated tasks or functions is their universal approximation property. Though the past few decades have seen tremendous advances in theories of neural networks, a single constructive and elementary framework for neural network universality remains unavailable. This paper is an effort to provide a unified and constructive framework for the universality of a large class of activation functions including most of the existing ones. At the heart of the framework is the concept of neural network approximate identity (nAI). The main result is as follows: any nAI activation function is universal in the space of continuous functions on compacta. It turns out that most of the existing activation functions are nAI, and thus universal. The framework induces several advantages over the contemporary counterparts. First, it is constructive with elementary means from functional analysis, probability theory, and numerical analysis. Second, it is one of the first unified and constructive attempts that is valid for most of the existing activation functions. Third, it provides new proofs for most activation functions. Fourth, for a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with a predetermined number of neurons and the values of weights/biases. Fifth, the framework allows us to abstractly present the first universal approximation with a favorable non-asymptotic rate. Sixth, our framework also provides insights into the developments, and hence providing constructive derivations, of some of the existing approaches.
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- PAR ID:
- 10517351
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- IMA Journal of Applied Mathematics
- Volume:
- 89
- Issue:
- 1
- ISSN:
- 0272-4960
- Format(s):
- Medium: X Size: p. 197-230
- Size(s):
- p. 197-230
- Sponsoring Org:
- National Science Foundation
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