Continuous p-adic Optimization

We present the first method for intrinsic, continuous gradient-based optimization for models with $p$-adic parameters. To overcome the vanishing gradients caused by the discrete valuation of the $p$-adics, we lift parameters to the Berkovich affine line, a canonical analytification of the $p$-adic numbers that provides a path-connected space. This enables gradient descent that respects the non-Archimedean metric with a local transition cost that is linear. By extending losses to the Berkovich line, we get piecewise linear functions, corresponding to the cell decomposition of a tropical complex; gradients are efficiently computed via max-plus operations. We prove local convergence within tropical cells for linear models on the ${\mathbb{Q}}_p$ subtree, and demonstrate the success of our method on synthetic tasks and on the $p$-adic Quillian semantic benchmark [Martins, 2025], where we achieve performance equal or better than $\mathbb{R}$-valued counterparts while training in comparable time.