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The hexatope and octatope abstract domains for neural network verification

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Abstract

Efficient verification algorithms for neural networks often depend on various abstract domains such as intervals, zonotopes, and linear star sets. The choice of the abstract domain presents an expressiveness vs. scalability trade-off: simpler domains are less precise but yield faster algorithms. This paper investigates the hexatope and octatope abstract domains in the context of neural net verification. Hexatopes are affine transformations of higher-dimensional hexagons, defined by difference constraint systems, and octatopes are affine transformations of higher-dimensional octagons, defined by unit-two-variable-per-inequality constraint systems. These domains generalize the idea of zonotopes which can be viewed as affine transformations of hypercubes. On the other hand, they can be considered as a restriction of linear star sets, which are affine transformations of arbitrary \(\mathcal {H}\)-Polytopes. This distinction places hexatopes and octatopes firmly between zonotopes and linear star sets in their expressive power, but what about the efficiency of decision procedures? An important analysis problem for neural networks is the exact range computation problem that asks to compute the exact set of possible outputs given a set of possible inputs. For this, three computational procedures are needed: (1) optimization of a linear cost function; (2) affine mapping; and (3) over-approximating the intersection with a half-space. While zonotopes allow an efficient solution for these approaches, star sets solves these procedures via linear programming. We show that these operations are faster for hexatopes and octatopes than they are for the more expressive linear star sets by reducing the linear optimization problem over these domains to the minimum cost network flow, which can be solved in strongly polynomial time using the Out-of-Kilter algorithm. Evaluating exact range computation on several ACAS Xu neural network benchmarks, we find that hexatopes and octatopes show promise as a practical abstract domain for neural network verification.

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Acknowledgement

This material is based upon work supported by the Air Force Office of Scientific Research and the Office of Naval Research under award numbers FA9550-19-1-0288, FA9550-21-1-0121, FA9550-22-1-0450, FA9550-22-1-0029, N00014-22-1-2156, and FA9550-23-1-0066. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Air Force or the United States Navy. This research was supported in part by the Air Force Research Laboratory Information Directorate through the Air Force Office of Scientific Research Summer Faculty Fellowship Program, contract numbers FA8750-15-3-6003, FA9550-15-0001, and FA9550-20-F-0005. This work is also supported by the National Science Foundation (NSF) grant CCF-2009022 and by NSF CAREER awards CCF-2146563 and CNF-2237229.

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, USA

    Stanley Bak

  2. University of Colorado Boulder, Boulder, USA

    Taylor Dohmen, Ashutosh Trivedi & Alvaro Velasquez

  3. West Virginia University, Morgantown, USA

    K. Subramani & Piotr Wojciechowski

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  1. Stanley Bak
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  3. K. Subramani
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Correspondence to K. Subramani.

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Bak, S., Dohmen, T., Subramani, K. et al. The hexatope and octatope abstract domains for neural network verification. Form Methods Syst Des 64, 178–199 (2024). https://doi.org/10.1007/s10703-024-00457-y

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