Divide and Slide: Layer-Wise Refinement for Output Range Analysis of Deep Neural Networks

Chao Huang*, Jiameng Fan, Xin Chen, Wenchao Li, Qi Zhu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


In this article, we present a layer-wise refinement method for neural network output range analysis. While approaches such as nonlinear programming (NLP) can directly model the high nonlinearity brought by neural networks in output range analysis, they are known to be difficult to solve in general. We propose to use a convex polygonal relaxation (overapproximation) of the activation functions to cope with the nonlinearity. This allows us to encode the relaxed problem into a mixed-integer linear program (MILP), and control the tightness of the relaxation by adjusting the number of segments in the polygon. Starting with a segment number of 1 for each neuron, which coincides with a linear programming (LP) relaxation, our approach selects neurons layer by layer to iteratively refine this relaxation. To tackle the increase of the number of integer variables with tighter refinement, we bridge the propagation-based method and the programming-based method by dividing and sliding the layer-wise constraints. Specifically, given a sliding number $s$ , for the neurons in layer $l$ , we only encode the constraints of the layers between $l-s$ and $l$. We show that our overall framework is sound and provides a valid overapproximation. Experiments on deep neural networks demonstrate significant improvement on output range analysis precision using our approach compared to the state-of-the-art.

Original languageEnglish (US)
Article number9211410
Pages (from-to)3323-3335
Number of pages13
JournalIEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Issue number11
StatePublished - Nov 2020


  • Linear programming (LP)
  • mixed-integer linear programming (MILP)
  • neural networks
  • output range analysis
  • refinement

ASJC Scopus subject areas

  • Software
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering


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