Synthesizing Precise Static Analyzers for Automatic Differentiation

We present Pasado, a technique for synthesizing precise static analyzers for Automatic Differentiation. Our technique allows one to automatically construct a static analyzer specialized for the Chain Rule, Product Rule, and Quotient Rule computations for Automatic Differentiation in a way that abstracts all of the nonlinear operations of each respective rule simultaneously. By directly synthesizing an abstract transformer for the composite expressions of these 3 most common rules of AD, we are able to obtain significant precision improvement compared to prior works which compose standard abstract transformers together suboptimally. We prove our synthesized static analyzers sound and additionally demonstrate the generality of our approach by instantiating these AD static analyzers with different nonlinear functions, different abstract domains (both intervals and zonotopes) and both forward-mode and reverse-mode AD. We evaluate Pasado on multiple case studies, namely soundly computing bounds on a neural network's local Lipschitz constant, soundly bounding the sensitivities of financial models, certifying monotonicity, and lastly, bounding sensitivities of the solutions of differential equations from climate science and chemistry for verified ranges of initial conditions and parameters. The local Lipschitz constants computed by Pasado on our largest CNN are up to 2750× more precise compared to the existing state-of-the-art zonotope analysis. The bounds obtained on the sensitivities of the climate, chemical, and financial differential equation solutions are between 1.31 - 2.81× more precise (on average) compared to a state-of-the-art zonotope analysis.