Backpropagation in the Simply Typed Lambda-calculus with Linear Negation

TL;DR

This paper extends backpropagation to a lambda-calculus with linear negation, providing a logical, effect-free framework for automatic differentiation in differentiable programming languages.

Contribution

It introduces a compositional program transformation for the simply-typed lambda-calculus with linear negation that computes gradients efficiently and effect-free, offering a logical perspective on backpropagation.

Findings

The transformation computes gradients with the same efficiency as traditional backpropagation.

It is effect-free and purely logical, enhancing theoretical understanding.

The approach bridges differentiable programming with formal logical systems.

Abstract

Backpropagation is a classic automatic differentiation algorithm computing the gradient of functions specified by a certain class of simple, first-order programs, called computational graphs. It is a fundamental tool in several fields, most notably machine learning, where it is the key for efficiently training (deep) neural networks. Recent years have witnessed the quick growth of a research field called differentiable programming, the aim of which is to express computational graphs more synthetically and modularly by resorting to actual programming languages endowed with control flow operators and higher-order combinators, such as map and fold. In this paper, we extend the backpropagation algorithm to a paradigmatic example of such a programming language: we define a compositional program transformation from the simply-typed lambda-calculus to itself augmented with a notion of linear negation, and prove that this computes the gradient of the source program with the same efficiency as first-order backpropagation. The transformation is completely effect-free and thus provides a purely logical understanding of the dynamics of backpropagation.