Continuous Optimization for p-adic Models
Abstract
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 ℚp are totally disconnected, with standard losses that are flat away from their minima. To enable continuous optimization, we propose working with ℚp via its Berkovich affine line: a canonical, path-connected expansion of ℚp 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 ℚp 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
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BibTeX
@misc{salazar2026continuous,
title = {{Continuous Optimization for p-adic Models}},
author = {Julian Salazar and Dimitri Kanevsky and Matt Harvey and Pascal Getreuer and Lucas Dixon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25501},
eprint = {2609.25501},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25501},
}
RIS
TY - UNPB
TI - Continuous Optimization for p-adic Models
AU - Julian Salazar
AU - Dimitri Kanevsky
AU - Matt Harvey
AU - Pascal Getreuer
AU - Lucas Dixon
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25501
UR - https://arxiv.org/abs/2609.25501
AB - 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 ℚ_p are totally disconnected, with standard losses that are flat away from their minima. To enable continuous optimization, we propose working with ℚ_p via its Berkovich affine line: a canonical, path-connected expansion of ℚ_p 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 ℚ_p 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
ER -