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Bounding and Counting Linear Regions of Deep Neural Networks

Thiago Serra (tserra***at***gmail.com)
Christian Tjandraatmadja (ctjandra***at***andrew.cmu.edu)
Srikumar Ramalingam (srikumar***at***cs.utah.edu)

Abstract: We investigate the complexity of deep neural networks (DNN) that represent piecewise linear (PWL) functions. In particular, we study the number of linear regions, i.e. pieces, that a PWL function represented by a DNN can attain, both theoretically and empirically. We present (i) tighter upper and lower bounds for the maximum number of linear regions on rectifier networks, which are exact for inputs of dimension one; (ii) a first upper bound for multi-layer maxout networks; and (iii) a first method to perform exact enumeration or counting of the number of regions by modeling the DNN with a mixed-integer linear formulation. These bounds come from leveraging the dimension of the space defining each linear region. The results also indicate that a deep rectifier network can only have more linear regions than every shallow counterpart with same number of neurons if that number exceeds the dimension of the input.

Keywords: deep learning, linear regions, piecewise-linear activations, mixed-integer linear programming, solution couting

Category 1: Integer Programming ((Mixed) Integer Linear Programming )

Category 2: Applications -- Science and Engineering (Statistics )

Citation: ICML 2018

Download: [PDF]

Entry Submitted: 01/08/2018
Entry Accepted: 01/09/2018
Entry Last Modified: 06/09/2018

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