# Continuous Optimization for p-adic Models

arXiv: https://arxiv.org/abs/2609.25501 · HTML: https://mlsanity.com/paper/2609.25501/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Riemannian approach to batch normalization](https://mlsanity.com/paper/1709.09603/) (2017)
- [An Asynchronous Parallel Stochastic Coordinate Descent Algorithm](https://mlsanity.com/paper/1311.1873/) (2013)
- [AI co-mathematician: Accelerating mathematicians with agentic AI](https://mlsanity.com/paper/2605.06651/) (2026)
- [Non-ergodic Convergence Analysis of Heavy-Ball Algorithms](https://mlsanity.com/paper/1811.01777/) (2018)
- [Ultrametric Fitting by Gradient Descent](https://mlsanity.com/paper/1905.10566/) (2019)
- [Random Permutations Fix a Worst Case for Cyclic Coordinate Descent](https://mlsanity.com/paper/1607.08320/) (2016)
- [v-PuNNs: van der Put Neural Networks for Transparent Ultrametric Representation Learning](https://mlsanity.com/paper/2508.01010/) (2025)
- [Learning with the p-adics](https://mlsanity.com/paper/2512.22692/) (2025)
- [Number Theory Meets Linguistics: Modelling Noun Pluralisation Across 1497 Languages Using 2-adic Metrics](https://mlsanity.com/paper/2211.13124/) (2022)
- [Stochastic and incremental subgradient methods for convex optimization on Hadamard spaces](https://mlsanity.com/paper/2412.06730/) (2024)
- [p-adic Character Neural Network](https://mlsanity.com/paper/2603.29905/) (2026)
- [Tropical Gradient Descent](https://mlsanity.com/paper/2405.19551/) (2024)

## Abstract (arXiv)

We present the first method for native, continuous gradient descent for machine learning models with p-adic parameters. Existing native optimizers are discrete, mostly combinatorial searches, as the p-adic numbers mathbbQp are totally disconnected, with standard losses that are flat away from their minima. To enable continuous optimization, we propose working with mathbbQp via its Berkovich affine line: a canonical, path-connected expansion of mathbbQp that preserves its isometries and uniquely extends its analytic maps. This hull is a metric tree with interpretable points and local derivatives, which we show enables effective optimizers and backpropagation. We formulate gradient descent and show that its approximations efficiently learn linear models with coefficients in mathbbQp to do modular arithmetic, an XOR-like task not expressible by linear models in ℝ. We also demonstrate momentum and Adam variants, linear regression, and classification on binary-encoded hierarchies (Quillian semantic networks), addressing open problems posed by Martins (2025). Library at https://github.com/google-deepmind/padic-ml