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The classical universal approximation (UA) theorem for neural networks establishes mild conditions under which a feedforward neural network can approximate a continuous functionfwith arbitrary accuracy. A recent result shows that neural networks also enjoy a more generalintervaluniversal approximation (IUA) theorem, in the sense that the abstract interpretation semantics of the network using the interval domain can approximate the direct image map off(i.e., the result of applyingfto a set of inputs) with arbitrary accuracy. These theorems, however, rest on the unrealistic assumption that the neural network computes over infinitely precise real numbers, whereas their software implementations in practice compute over finite-precision floating-point numbers. An open question is whether the IUA theorem still holds in the floating-point setting. This paper introduces the first IUA theorem forfloating-pointneural networks that proves their remarkable ability toperfectly capturethe direct image map of any rounded target functionf, showing no limits exist on their expressiveness. Our IUA theorem in the floating-point setting exhibits material differences from the real-valued setting, which reflects the fundamental distinctions between these two computational models. This theorem also implies surprising corollaries, which include (i) the existence ofprovably robustfloating-point neural networks; and (ii) thecomputational completenessof the class of straight-line programs that use only floating-point additions and multiplications for the class of all floating-point programs that halt.
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Interval universal approximation for neural networks
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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- Award ID(s):
- 1652140
- PAR ID:
- 10602520
- Publisher / Repository:
- Association for Computing Machinery (ACM)
- Date Published:
- Journal Name:
- Proceedings of the ACM on Programming Languages
- Volume:
- 6
- Issue:
- POPL
- ISSN:
- 2475-1421
- Format(s):
- Medium: X Size: p. 1-29
- Size(s):
- p. 1-29
- Sponsoring Org:
- National Science Foundation
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