Probabilistic Abstract Interpretation of Deep Neural Networks

“The extraction of (symbolic) rules which describe the operation of (deep) neural networks which have been trained to perform a certain task is central to explaining their inner workings in order to judge their correctness, bias-freeness, etc. Besides this it also appears to be important to short-cut the learning process by `injecting’ symbolic knowledge in an appropriate way. The aim of this research is to apply classical and probabilistic program analysis techniques in order to verify and `understand’ (deep) neural networks. This concerns in particular the use of Probabilistic Abstract Interpretation (PAI) in order to extract (not least symbolic but also statistical) information about the structure of the knowledge represented in trained networks as well as the development of methods to initialise DNNs. While there is work on (classical) Abstract Interpretation to verify DNNs (e.g. AI^2@ETH, cf http://safeai.ethz.ch/) the use of PAI adds a novel aspect. In particular, as PAI – in contrast to classical Abstract Interpretation – allows for (numerically) close rather than safe (upper/lower) abstractions or approximations. A starting point for research could be our paper on Probabilistic Abstract Interpretation https://link.springer.com/chapter/10.1007/978-3-642-13678-8_1 as well as a the lecture notes of the course Program Analysis https://www.doc.ic.ac.uk/~herbert/teaching.html (CO470). Technically, a central notions is the concept of the Moore-Penrose pseudo-inverse (closely related to so-called linear regression and least square approximation) and the use of tensor products for representing `compositional’ information.”