A general construction for abstract interpretation of higher-order automatic differentiation

We present a novel, general construction to abstractly interpret higher-order automatic differentiation (AD). Our construction allows one to instantiate an abstract interpreter for computing derivatives up to a chosen order. Furthermore, since our construction reduces the problem of abstractly reasoning about derivatives to abstractly reasoning about real-valued straight-line programs, it can be instantiated with almost any numerical abstract domain, both relational and non-relational. We formally establish the soundness of this construction.

We implement our technique by instantiating our construction with both the non-relational interval domain and the relational zonotope domain to compute both first and higher-order derivatives. In the latter case, we are the first to apply a relational domain to automatic differentiation for abstracting higher-order derivatives, and hence we are also the first abstract interpretation work to track correlations across not only different variables, but different orders of derivatives.

We evaluate these instantiations on multiple case studies, namely robustly explaining a neural network and more precisely computing a neural network's Lipschitz constant. For robust interpretation, first and second derivatives computed via zonotope AD are up to 4.76× and 6.98× more precise, respectively, compared to interval AD. For Lipschitz certification, we obtain bounds that are up to 11,850× more precise with zonotopes, compared to the state-of-the-art interval-based tool.