Differential Calculus with Imprecise Input and Its Logical Framework

Abstract

We develop a domain-theoretic Differential Calculus for locally Lipschitz functions on finite dimensional real spaces with imprecise input/output. The inputs to these functions are hyper-rectangles and the outputs are compact real intervals. This extends the domain of application of Interval Analysis and exact arithmetic to the derivative. A new notion of a tie for these functions is introduced, which in one dimension represents a modification of the notion previously used in the one-dimensional framework. A Scott continuous sub-differential for these functions is then constructed, which satisfies a weaker form of calculus compared to that of the Clarke sub-gradient. We then adopt a Program Logic viewpoint using the equivalence of the category of stably locally compact spaces with that of semi-strong proximity lattices. We show that given a localic approximable mapping representing a locally Lipschitz map with imprecise input/output, a localic approximable mapping for its sub-differential can be constructed, which provides a logical formulation of the sub-differential operator.

A simple differentiable programming language

“…For the former, one might use Clarke sub-gradients [Clarke 1990], following [Di Gianantonio and Edalat 2013] (the Clarke sub-gradient of ReLU at 0 is the interval [0, 1]); for the latter, one may use C k functions. Yet another possibility would be to work with approximate reals, rather than reals, and to seek numerical accuracy theorems; one might employ a domain-theoretic notion of sub-differentiation of functions over Scott's interval domain, generalizing the Clarke sub-gradient (see [Edalat and Lieutier 2004;Edalat and Maleki 2018]).…”