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Backpropagation is a classic automatic differentiation algorithm computingthe gradient of functions specified by a certain class of simple, first-orderprograms, called computational graphs. It is a fundamental tool in severalfields, most notably machine learning, where it is the key for efficientlytraining (deep) neural networks. Recent years have witnessed the quick growthof a research field called differentiable programming, the aim of which is toexpress computational graphs more synthetically and modularly by resorting toactual programming languages endowed with control flow operators andhigher-order combinators, such as map and fold. In this paper, we extend thebackpropagation algorithm to a paradigmatic example of such a programminglanguage: we define a compositional program transformation from thesimply-typed lambda-calculus to itself augmented with a notion of linearnegation, and prove that this computes the gradient of the source program withthe same efficiency as first-order backpropagation. The transformation iscompletely effect-free and thus provides a purely logical understanding of thedynamics of backpropagation.

Backpropagation in the Simply Typed Lambda-calculus with Linear Negation