Confusion of Tagged Perturbations in Forward Automatic Differentiation of Higher-Order Functions

Forward Automatic Differentiation (AD) is a technique for augmenting programs to both perform their original calculation and also compute its directional derivative. The essence of Forward AD is to attach a derivative value to each number, and propagate these through the computation. When derivatives are nested, the distinct derivative calculations, and their associated attached values, must be distinguished. In dynamic languages this is typically accomplished by creating a unique tag for each application of the derivative operator, tagging the attached values, and overloading the arithmetic operators. We exhibit a subtle bug, present in fielded implementations, in which perturbations are confused despite the tagging machinery. 1 Forward AD using Tagged Tangents Forward AD (Wengert, 1964) computes the derivative of a function f : R → α at a point c by evaluating f(c+ ε) under a nonstandard interpretation that associates a conceptually infinitesimal perturbation with each real number, propagates these augmented values according to the rules of calculus (Leibniz, 1664), and extracts the perturbation of the result. When x is a number, we use x+ x̄ε to denote a tangent-vector bundle: the primal value x bundled with the tangent value x̄, where x̄ has the same type as x. We consider this tangent-vector bundle to also be a number, with arithmetic defined by regarding it as a truncated power series, or equivalently, by taking ε = 0 but ε 6= 0. This implies that f(x + x̄ε) = f(x) + x̄f (x)ε where f (x) is the first derivative of f at x (Newton, 1704). Hamilton Inst & Dept Comp Sci, NUI Maynooth, Co. Kildare, Ireland School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907-2035, USA