Probabilistic Abstract Interpretation on Neural Networks via Grids Approximation

Abstract

Probabilistic abstract interpretation is a theory used to extract particular properties of a computer program when it is infeasible to test every single inputs. In this paper we apply the theory on neural networks for the same purpose: to analyse density distribution flow of all possible inputs of a neural network when a network has uncountably many or countable but infinitely many inputs. We show how this theoretical framework works in neural networks and then discuss different abstract domains and corresponding Moore-Penrose pseudo-inverses together with abstract transformers used in the framework. We also present experimental examples to show how this framework helps to analyse real world problems.

Figures (5)

• Figure 1: Probabilistic abstract interpretation with Moore-Penrose pseudo-inverse pairs $A$ and $G$. The left hand side $C \xrightarrow{\text{$f$}} D$ is the concrete domain and the right hand side $C^{\#} \xrightarrow{\text{$f^{\#}$}} D^{\#}$ is the abstract domain.
• Figure 2: lattice of abstract domain
• Figure 3: abstract elements through layer
• Figure 4: ReLU transformation
• Figure 5: (a) A plot of model architecture of digit classifier. (b) A table of model architecture showing total number of weights and number of weights in each layer.

Theorems & Definitions (2)

• Definition 1
• Proposition 1