Combinatory Adjoints and Differentiation

Martin Elsman
(University of Copenhagen)
Fritz Henglein
(University of Copenhagen)
Robin Kaarsgaard
(University of Edinburgh)
Mikkel Kragh Mathiesen
(University of Copenhagen)
Robert Schenck
(University of Copenhagen)

We develop a compositional approach for automatic and symbolic differentiation based on categorical constructions in functional analysis where derivatives are linear functions on abstract vectors rather than being limited to scalars, vectors, matrices or tensors represented as multi-dimensional arrays.

We show that both symbolic and automatic differentiation can be performed using a differential calculus for generating linear functions representing Fréchet derivatives based on rules for primitive, constant, linear and bilinear functions as well as their sequential and parallel composition. Linear functions are represented in a combinatory domain-specific language.

Finally, we provide a calculus for symbolically computing the adjoint of a derivative without using matrices, which are too inefficient to use on high-dimensional spaces. The resulting symbolic representation of a derivative retains the data-parallel operations from the input program. The combination of combinatory differentiation and computing formal adjoints turns out to be behaviorally equivalent to reverse-mode automatic differentiation. In particular, it provides opportunities for optimizations where matrices are too inefficient to represent linear functions.

In Jeremy Gibbons and Max S. New: Proceedings Ninth Workshop on Mathematically Structured Functional Programming (MSFP 2022), Munich, Germany, 2nd April 2022, Electronic Proceedings in Theoretical Computer Science 360, pp. 1–26.
Published: 30th June 2022.

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