Open Collections

Automatic differentiation in PCF

Banff International Research Station for Mathematical Innovation and Discovery

We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration except for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of non-null polynomials.

Mathematics; Category Theory; Homological Algebra; Algebraic Topology; Algebraic Structures

video/mp4

Author affiliation: IRIF Paris France

2023-10-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86384

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/