# Backprop as Functor: A compositional perspective on supervised learning

@article{Fong2019BackpropAF, title={Backprop as Functor: A compositional perspective on supervised learning}, author={Brendan Fong and David I. Spivak and R{\'e}my Tuy{\'e}ras}, journal={2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, year={2019}, pages={1-13} }

A supervised learning algorithm searches over a set of functions $A\rightarrow B$ parametrised by a space $P$ to find the best approximation to some ideal function $f:A\rightarrow B$. It does this by taking examples $(a, f(a))\in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent-with respect to a fixed step size and an error function satisfying a certain property-defines a monoidal functor…

## 59 Citations

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This paper shows both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functorEmbedding the categories of learners into a categories of symmetric lenses.

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## References

SHOWING 1-10 OF 21 REFERENCES

Lenses and Learners

- MathematicsBx@PLW
- 2019

This paper shows both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functorEmbedding the categories of learners into a categories of symmetric lenses.

Relational lenses: a language for updatable views

- Computer SciencePODS '06
- 2006

The approach is to define a bi-directional query language, in which every expression can be read bot(from left to right) as a view definition and (from right to left) as an update policy.

A compositional framework for Markov processes

- Mathematics, Computer Science
- 2016

It is proved that black boxing gives a symmetric monoidal dagger functor sending open detailed balanced Markov processes to open circuits made of linear resistors, and described how to “black box” an open Markov process.

The algebra of open and interconnected systems

- Computer Science
- 2016

This thesis develops the theory of hypergraph categories and introduces the tools of decorated cospans and corelations, a more powerful version that permits construction of all hyper graph categories and hypergraph functors.

Geometric Deep Learning: Going beyond Euclidean data

- Computer ScienceIEEE Signal Processing Magazine
- 2017

Deep neural networks are used for solving a broad range of problems from computer vision, natural-language processing, and audio analysis where the invariances of these structures are built into networks used to model them.

Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams

- Mathematics, Computer Science
- 2014

This paper uses the language of operads to study the algebraic nature of assembling complex dynamical systems from an interconnection of simpler ones, and defines two W-algebras, G and L, which associate semantic content to the structures in W.

From open learners to open games

- Computer ScienceArXiv
- 2019

It is proved that there is a faithful symmetric monoidal functor from the former to the latter, which means that any supervised neural network can be seen as an open game in a canonical way.

Understanding deep image representations by inverting them

- Computer Science2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
- 2015

Image representations, from SIFT and Bag of Visual Words to Convolutional Neural Networks (CNNs), are a crucial component of almost any image understanding system. Nevertheless, our understanding of…

Categories for the Working Mathematician

- Mathematics
- 1971

I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large…

Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning

- Physics
- 2017

This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations.