• Corpus ID: 198162011

Towards formalizing and extending differential programming using tangent categories

@inproceedings{Cruttwell2019TowardsFA,
  title={Towards formalizing and extending differential programming using tangent categories},
  author={Geoff S. H. Cruttwell and Jonathan Gallagher and Benjamin MacAdam},
  year={2019}
}
This paper gives an interpretation of a simple differential programming language (over finite dimensional R vector spaces), into a setting derived from synthetic differential geometry (SDG). The main theorem of this paper is Theorem 5.6, where we establish that there is always an interpretation of our simple differential programming language into a category of partial maps of a well-adapted smooth topos that preserves the derivative and all the control structures of the differential programming… 

Tables from this paper

Correctness of Automatic Differentiation via Diffeologies and Categorical Gluing

TLDR
It is shown that the characterisation of AD gives rise to an elegant semantic proof of its correctness based on a gluing construction on diffeological spaces, and how the analysis extends to other AD methods by considering a continuation-based method.

Higher Order Automatic Differentiation of Higher Order Functions

TLDR
This work characterises a forward-mode AD method on a higher order language with algebraic data types as the unique structure preserving macro given a choice of derivatives for basic operations, and describes a rich semantics for differentiable programming, based on diffeological spaces.

Dioptics: a Common Generalization of Open Games and Gradient-Based Learners

TLDR
It is shown that this category of gradient-based learners embeds naturally into the category of learners (with a choice of update rule and loss function), and that composing this embedding with reverse-mode automatic differentiation recovers the backpropagation functor L of [FST18].

Smart Choices and the Selection Monad

  • M. AbadiG. Plotkin
  • Computer Science
    2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2021
TLDR
A small language is defined that supports decision-making abstraction, rewards, and probabilities, and three denotational semantics with auxiliary monads for reward and probability are given, and the two kinds of semantics coincide by proving adequacy theorems.

Semantics for Automatic Differentiation Towards Probabilistic Programming Literature Review and Dissertation Proposal

TLDR
Stochastic Gradient Descent, Probabilistic Programming and Variational inference: Beyond Automatic Differentiation 3.2.3 Literature review: Beyond automatic Differentiation.

References

SHOWING 1-10 OF 37 REFERENCES

Differential Structure, Tangent Structure, and SDG

TLDR
It is shown that tangent structures appropriately span a very wide range of definitions, from the syntactic and structural differentials arising in computer science and combinatorics, through the concrete manifolds of algebraic and differential geometry, and finally to the abstract definitions of synthetic differential geometry.

Differential categories

TLDR
The notion of a categorical model of the differential calculus is introduced, and it is shown that it captures the not-necessarily-closed fragment of Ehrhard–Regnier differential $\lambda$-calculus.

A convenient differential category

TLDR
It is shown that the category of Mackey-complete, separated, topological convex bornological vector spaces and bornological linear maps is a differential category and will ultimately yield a wide variety of models of differential logics.

Finiteness spaces

  • T. Ehrhard
  • Mathematics, Computer Science
    Mathematical Structures in Computer Science
  • 2005
TLDR
A new denotational model of linear logic based on the purely relational model, where webs are equipped with a notion of ‘finitary’ subsets satisfying a closure condition and proofs are interpreted as finitary sets is investigated.

Tangent spaces and tangent bundles for diffeological spaces

We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and

Synthetic Differential Geometry

Preface to the second edition (2005) Preface to the first edition (1981) Part I. The Synthetic Ttheory: 1. Basic structure on the geometric line 2. Differential calculus 3. Taylor formulae - one

Restriction categories as enriched categories

CARTESIAN DIFFERENTIAL CATEGORIES

This paper revisits the authors' notion of a dierential category from a dierent perspective. A dierential category is an additive symmetric monoidal category with a comonad (a \coalgebra modality")

Restriction categories I: categories of partial maps