Continuous Optimization for p-adic Models
Julian Salazar, Dimitri Kanevsky, Matt Harvey, Pascal Getreuer, Lucas Dixon
cs.LG · math.MG · math.NT · math.OC · stat.ML
Quick verdict
Claims first continuous gradient descent for p-adic models via Berkovich hull Γp; careful math and controlled small experiments, but no convergence guarantees.
Summary
The paper proposes the first native, continuous gradient-based optimization framework for machine-learning models with p-adic parameters. Because Qp is totally disconnected and standard p-adic losses are locally constant away from their minima, prior optimizers were discrete searches. The authors instead optimize over Γp, the convex hull of Qp inside the Berkovich affine line — a path-connected metric tree whose points pair a p-adic center with a real radius. They prove this hull with half the arc metric is the tight span (metric envelope) of (Qp, dp), then build forward propagation, a direct loss, directional derivatives, backpropagation, and grouped coordinate-descent approximations of joint gradient descent, Momentum and Adam. On synthetic 3-adic linear regression, their methods reach depth-5 coefficient recovery using 0.95–0.98 training-example equivalents versus 35.5 for Martins-style beam search of width five; under an adversarial initialization, Momentum/Adam recover all five seeds while plain GD recovers three of five. A valuation-logit softmax yields multiclass p-adic classifiers: perfect accuracy on matched-prime modulo-4/9 tasks (permuted-prime and Euclidean-raw controls sit at chance; Euclidean one-hot is also perfect), and on the Quillian semantic-network benchmark F1 of 0.804 versus 0.845 for batch beam search and 0.844 for a 703-parameter Euclidean MLP. Main limitations: no convergence guarantees, experiments limited to affine models plus one structurally favorable two-layer model, small scale, and radius propagation is only an upper bound when parameters interact.
Review complete
Assessment at a glance
4
risk flags identified
● high 0
● medium 2
● low 2
5 strengths · 4 weaknesses · 5 author questions
AI text signal = Binoculars zero-shot detector score over 20 sampled passages (Fast-DetectGPT co-reported) — absence does not prove human authorship, and a flag does not imply misconduct.
Levels reflect verification complexity — not a quality score. Evolve never rates your paper.
Key claims under review
- The paper presents the first native, continuous gradient descent (with Momentum and Adam variants) for machine-learning models with p-adic parameters, operating on the Berkovich hull Γp.
- (Γp, ½darc) is canonically the tight span / metric envelope of (Qp, |·|p), a new identification stated in Theorem C.6.
- Grouped-approximate gradient methods recover depth-5 Q3 linear coefficients with roughly 36x fewer training-example equivalents than width-five beam search (0.95–0.98 vs 35.5, Table 1).
- A valuation-logit softmax yields linear multiclass p-adic classifiers, resolving Martins (2025)'s Open Problem 6.1 and addressing 6.2.
- Radius propagation through composed stages is exact only when no parameter reaches a stage along two paths; in general it is an upper bound (the dependency problem), which currently limits deep p-adic networks.
Strengths
Canonical geometric foundation
Theorem C.6 proves (Γp, ½darc) is the tight span/metric envelope of (Qp, dp); the authors state this identification is new. This turns 'where can a p-adic parameter move continuously' from an ad hoc modeling choice into a canonical, isometry-invariant answer, with representative-independence proved (Lemma C.2, Corollary D.1).
Complete, internally careful optimization stack
Forward propagation via the multivariable Gauss seminorm with explicit active-term tracking (eq. 7–8), direct loss (eq. 9), directional chain rule (Proposition E.1), per-direction Momentum/Adam with equivariance proofs (Propositions C.8–C.9), and a cost model (Table 4, Appendix G.2). The framework is engineered, not gestural, and ships as an OSS library.
Honest, well-controlled empirics
Large, concrete efficiency gain over discrete search
Positions contributions against explicit open problems
Weaknesses
No convergence guarantees
Deep-model story untested on coupled architectures
Accuracy advantage over Euclidean baselines is absent
Small scale and protocol wrinkles
Full Review Report
$1.99 · one-timeUnlock the complete preliminary analysis Evolve produced for this paper.
Report preview
Scope & Claims
This is a method paper at the intersection of machine learning and non-Archimedean geometry. The authors claim the first native, continuous gradient-based optimization for models with p-adic parameters, resolving Martins (2025)'s Open Problem 6.1 (linear multiclass p-adic classifiers) and addressing 6.2 (gradient-based optimization). The 'first' claim is scoped carefully in Section 2 and Appendix B: prior native p-adic optimizers (Khrennikov–Tirozzi 2000; Bradley 2009; Baker et al. 2025; Zubarev 2025; Martins 2025; Mihara 2026a–c) are discrete searches, while Zúñiga-Galindo (2024) and N'guessan (2025) optimize real surrogates. Within that taxonomy the claim is plausible, though 'native' and 'continuous' do the definitional work and priority rests on this scoping.
The central object is Γp, the convex hull of Qp in the Berkovich affine line (Section 3.1): a path-connected metric tree whose points ζ(x,r) pair a p-adic center with a real radius. Theorem C.6 proves (Γp, ½darc) is the tight span/metric envelope of (Qp, dp) — the authors state this identification is new — and it is the paper's most distinctive contribution because it converts 'where can a p-adic parameter move continuously' from a modeling choice into a canonical answer.
The remaining sections cover Methodology Assessment, Statistical Evidence, Results & Reproducibility, Limitations & Risks, and Verdict & Recommendations.
- All 5 strengths & 4 weaknesses, analyzed in detail
- 4 risk flags with severity, section & reviewer questions
- Methods & statistical evidence assessment
- 5 questions a peer reviewer would ask
- Full six-dimension evaluation report & recommendations
One-time purchase · this paper only
One-time purchase, this paper only. AI-generated preliminary analysis — not peer review.
Frequently asked questions
Why can't gradient descent be applied directly to p-adic numbers?
The field Qp is totally path-disconnected — every continuous path in it is constant — and standard losses such as p-adic absolute error are locally constant away from exact fits, so there is no usable gradient information. This paper works instead in the Berkovich hull Γp, where an added real radius coordinate supplies the missing continuous direction for optimization.
What is Γp in this paper?
Γp is the convex hull of Qp inside the Berkovich affine line: an infinite metric tree whose points pair a p-adic center x with a real radius r ≥ 0, with Qp embedded as the radius-zero leaves. The authors prove that Γp with half the arc metric is the tight span (metric envelope) of (Qp, its p-adic distance), making it a canonical continuous enlargement of Qp.
Do p-adic models beat neural networks in this paper?
No. On modulo classification a Euclidean one-hot linear model also reaches perfect accuracy, and on the Quillian benchmark a 703-parameter Euclidean MLP achieves F1 0.844 versus 0.804 for the p-adic Adam method (batch beam search: 0.845). The demonstrated advantage is training efficiency relative to discrete p-adic optimizers such as beam search, plus native p-adic parametrization with permuted-code controls showing the ultrametric encoding matters.
Is code available for p-adic gradient descent?
Yes. The authors released an open-source library at github.com/google-deepmind/padic-ml implementing the hull arithmetic, the gradient descent/Momentum/Adam optimizers, and the experiments described in the paper; all experimental data is synthetic or public, so results are CPU-scale reproducible.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Figures 1–2 are described only via captions; the tree geometry and Figure 2's update path cannot be visually verified from extracted text.
- The formal proofs (Theorems C.6 and H.1, Propositions D.3 and E.1) were assessed from their statements, not re-derived line-by-line in this pass.
- Appendix G.4's inline tie-breaking table appears garbled by text extraction; exact per-candidate losses were not re-verified.
- The external repository github.com/google-deepmind/padic-ml was not inspected; code-availability is taken from the paper's claims.
Go deeper on this paper
Wrote a paper? Stress-test it before reviewers do.
Upload your manuscript for the same structured scrutiny — reasoning, statistics, evidence, figures, and reproducibility.