# Deep Learning with Parametric Lenses

arXiv: https://arxiv.org/abs/2404.00408 · HTML: https://mlsanity.com/paper/2404.00408/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 61 references that are in the map.
- First posted on arXiv 30 Mar 2024.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [BinaryConnect: Training Deep Neural Networks with binary weights during propagations](https://mlsanity.com/paper/1511.00363/) (2015)
- [Understanding Deep Image Representations by Inverting Them](https://mlsanity.com/paper/1412.0035/) (2014)
- [Backprop as Functor: A compositional perspective on supervised learning](https://mlsanity.com/paper/1711.10455/) (2017)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Towards Verified Artificial Intelligence](https://mlsanity.com/paper/1606.08514/) (2016)
- [The simple essence of automatic differentiation](https://mlsanity.com/paper/1804.00746/) (2018)
- [Reverse derivative categories](https://mlsanity.com/paper/1910.07065/) (2019)
- [Compositional game theory](https://mlsanity.com/paper/1603.04641/) (2016)
- [Lenses and Learners](https://mlsanity.com/paper/1903.03671/) (2019)
- [Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits](https://mlsanity.com/paper/2101.10488/) (2021)
- [An Introduction to String Diagrams for Computer Scientists](https://mlsanity.com/paper/2305.08768/) (2023)
- [Compositional Deep Learning](https://mlsanity.com/paper/1907.08292/) (2019)
- [Profunctor Optics, a Categorical Update](https://mlsanity.com/paper/2001.07488/) (2020)
- [Categorical Stochastic Processes and Likelihood](https://mlsanity.com/paper/2005.04735/) (2020)
- [Differentiable Causal Computations via Delayed Trace](https://mlsanity.com/paper/1903.01093/) (2019)
- [Limits of bimorphic lenses](https://mlsanity.com/paper/1808.05545/) (2018)
- [Categories of Differentiable Polynomial Circuits for Machine Learning](https://mlsanity.com/paper/2203.06430/) (2022)
- [Diagrammatic Semantics for Digital Circuits](https://mlsanity.com/paper/1703.10247/) (2017)
- [Translating Extensive Form Games to Open Games with Agency](https://mlsanity.com/paper/2105.06763/) (2021)
- [Generalized Lens Categories via functors Crm op→mathsfCat](https://mlsanity.com/paper/1908.02202/) (2019)

## Built on by (most cited first, 3 of 3)

- [Self-Revising Discovery Systems for Science: A Categorical Framework for Agentic Artificial Intelligence](https://mlsanity.com/paper/2606.01444/) (2026)
- [Game-Theoretic Gradient Control for Robust Neural Network Training](https://mlsanity.com/paper/2507.19143/) (2025)
- [Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning](https://mlsanity.com/paper/2604.07242/) (2026)

## Abstract (arXiv)

We propose a categorical semantics for machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, and different architectures, shedding new light on their similarities and differences. Furthermore, our approach to learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realised in the discrete setting of Boolean and polynomial circuits. We demonstrate the practical significance of our framework with an implementation in Python.