Rencontre, Feb. 6, 2020


10:30 – 12:00
Damiano Mazza (CNRS, Univ. Paris 13)

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.

14:00 – 15:00
Cyrille Chenavier (Johannes Kepler University (Linz, Austria))

We introduce topological rewriting systems as a generalisation of abstract rewriting systems, where we replace the set of terms by a topological space. Abstract rewriting systems correspond to topological rewriting systems for the discrete topology. We introduce the topological confluence property as an approximation of the confluence property, and use it to characterise standard bases in terms of rewriting theory. Using a representation of linear topological rewriting systems with continuous reduction operators, we also interpret the topological confluence property in terms of lattice operations. Finally, we investigate duality for reduction operators that we relate to series representations and syntactic algebras, from which we deduce a duality for proving that an algebra is syntactic or not.

15:30 – 16:30
Luc Pellissier (INRIA and École Polytechnique)