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Title: Interval universal approximation for neural networks

To verify safety and robustness of neural networks, researchers have successfully applied abstract 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 the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (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 of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable 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 Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ 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. more »« less

Ding, Shuo; Zhang, Qirun(
, Proceedings of the ACM on Programming Languages)

Many problems in programming language theory and formal methods are undecidable, so they cannot be solved precisely. Practical techniques for dealing with undecidable problems are often based on decidable approximations. Undecidability implies that those approximations are always imprecise. Typically, practitioners use heuristics andad hocreasoning to identify imprecision issues and improve approximations, but there is a lack of computability-theoretic foundations about whether those efforts can succeed.

This paper shows a surprising interplay between undecidability and decidable approximations: there exists a class of undecidable problems, such that it is computable to transform any decidable approximation to a witness input demonstrating its imprecision. We call those undecidable problemswitnessable problems. For example, if a program propertyPis witnessable, then there exists a computable functionf_{P}, such thatf_{P}takes as input the code of any program analyzer targetingPand produces an input programwon which the program analyzer is imprecise. An even more surprising fact is that the class of witnessable problems includes almost all undecidable problems in programming language theory and formal methods. Specifically, we prove the diagonal halting problemKis witnessable, and the class of witnessable problems is closed under complements and many-one reductions. In particular, all “non-trivial semantic properties of programs” mentioned in Rice’s theorem are witnessable. We also explicitly construct a problem in the non-witnessable (and undecidable) class and show that both classes have cardinality 2^{ℵ0}.

Our results offer a new perspective on the understanding of undecidability: for witnessable problems, although it is impossible to solve them precisely, it is always possible to improve any decidable approximation to make it closer to the precise solution. This fact formally demonstrates that research efforts on such approximations are promising and shows there exist universal ways to identify precision issues of program analyzers, program verifiers, SMT solvers, etc., because their essences are decidable approximations of witnessable problems.

We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of k ReLUs minimizing the squared error (for k>1) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error ϵ. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest k-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in (1/epsilon)^2. Together with a previous work regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on epsilon.

Allender, Eric; Ilango, Rahul; Vafa, Neekon(
, Theory of computing systems)

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions, and is provably not hard under "local" reductions computable in TIME(n^0.49) . The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC^0) is closely related to many of the longstanding open questions in complexity theory.
All prior hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function. Some of these results were proved by exploiting
a connection to a notion of time-bounded Kolmogorov complexity (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed, but this approach establishes only hardness of extremely good approximations of the form 1+o(1), and these improved hardness results are not yet known to hold for MCSP. In particular, it is known
that MKTP is hard for the complexity class DET under nonuniform AC^0 m-reductions, implying MKTP is not in AC^0[p] for any prime p. It was still open if similar circuit lower bounds hold for MCSP. One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond 1 + o(1) to omega(1), as KT-complexity and circuit size are polynomially-related. In this paper, we show that this approach cannot succeed.
More specically, we prove that PARITY does not reduce to the problem of computing superlinear approximations to KT-complexity or circuit size via AC^0-Turing reductions that make O(1) queries. This is signicant, since approximating any set in P/poly AC^0-reduces to
just one query of a much worse approximation of circuit size or KT-complexity. For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in earlier work (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC^0 reductions. It may also be a step toward conrming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform AC0 m-reductions.

Bui-Thanh, Tan(
, IMA Journal of Applied Mathematics)

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.

Sallam, Gamal; Gupta, Gagan R.; Li, Bin; Ji, Bo(
, IEEE INFOCOM 2018 - IEEE Conference on Computer Communications)

With the advent of Network Function Virtualization (NFV), Physical Network Functions (PNFs) are gradually being replaced by Virtual Network Functions (VNFs) that are hosted on general purpose servers. Depending on the call flows for specific services, the packets need to pass through an ordered set of network functions (physical or virtual) called Service Function Chains (SFC) before reaching the destination. Conceivably for the next few years during this transition, these networks would have a mix of PNFs and VNFs, which brings an interesting mix of network problems that are studied in this paper: (1) How to find an SFC-constrained shortest path between any pair of nodes? (2) What is the achievable SFC-constrained maximum flow? (3) How to place the VNFs such that the cost (the number of nodes to be virtualized) is minimized, while the maximum flow of the original network can still be achieved even under the SFC constraint? In this work, we will try to address such emerging questions. First, for the SFC-constrained shortest path problem, we propose a transformation of the network graph to minimize the computational
complexity of subsequent applications of any shortest path algorithm. Second, we formulate the SFC-constrained maximum flow problem as a fractional multicommodity flow problem, and develop a combinatorial algorithm for a special case of practical
interest. Third, we prove that the VNFs placement problem is NP-hard and present an alternative Integer Linear Programming (ILP) formulation. Finally, we conduct simulations to elucidate our theoretical results.

Wang, Zi, Albarghouthi, Aws, Prakriya, Gautam, and Jha, Somesh. Interval universal approximation for neural networks. Retrieved from https://par.nsf.gov/biblio/10328257. Proceedings of the ACM on Programming Languages 6.POPL Web. doi:10.1145/3498675.

Wang, Zi, Albarghouthi, Aws, Prakriya, Gautam, and Jha, Somesh.
"Interval universal approximation for neural networks". Proceedings of the ACM on Programming Languages 6 (POPL). Country unknown/Code not available. https://doi.org/10.1145/3498675.https://par.nsf.gov/biblio/10328257.

@article{osti_10328257,
place = {Country unknown/Code not available},
title = {Interval universal approximation for neural networks},
url = {https://par.nsf.gov/biblio/10328257},
DOI = {10.1145/3498675},
abstractNote = {To verify safety and robustness of neural networks, researchers have successfully applied abstract 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 the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (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 of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable 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 Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ 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.},
journal = {Proceedings of the ACM on Programming Languages},
volume = {6},
number = {POPL},
author = {Wang, Zi and Albarghouthi, Aws and Prakriya, Gautam and Jha, Somesh},
}

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