Life sciences · Preprint

Silver Rate Is (Almost) Optimal for Gradient Descent Acceleration

arXiv · September 8, 2026

Insufficient Information Why this matters to you Expert discussion

The material analysed did not support any firm read.

AI Summary

This is a preprint establishing theoretical convergence bounds for gradient descent optimization using predetermined step sizes. The work is a pure mathematics contribution to optimization theory and has no direct clinical, biological, or medical application.

What was studied

Preprint.

Key findings

Non-anytime lower bound of Ω(n^(−p_sil−O(√(log log n/log n)))) where p_sil = log₂(1+√2) Anytime setting lower bound of Ω(n^(−2p_sil/(1+p_sil)−O(√(log log n/log n)))) Results determine optimal polynomial convergence exponents in both non-anytime and anytime settings

Safety

Safety was not reported in the material analysed. Check the source before drawing any conclusion about harm.

Who this may help

The source did not state who this applies to in practice.

Plain-language summary

This is a theoretical mathematics paper providing convergence proofs for gradient descent optimization; it does not involve clinical, biological, or medical evidence and is therefore not applicable to PeerCurrent's clinical and life-science audience.

Reported figures

Quoted from the source exactly as published.

Silver ratio exponentp_sil = log₂(1+√2)

Non-anytime lower bound exponentΩ(n^(−p_sil−O(√(log log n/log n))))

Anytime lower bound exponentΩ(n^(−2p_sil/(1+p_sil)−O(√(log log n/log n))))

Caveats

This is a preprint and has not been peer reviewed.
Content is pure mathematical theory with no empirical validation or clinical relevance.
No biological, clinical, or medical application is described or claimed.

Evidence assessment

Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.

Study design

insufficient

Mathematical proof; not an empirical study or clinical investigation.

Directness

insufficient

Theoretical optimization analysis with no clinical or biological endpoint.

Consistency

insufficient

Single theoretical result; no comparison to prior empirical findings reported.

Risk of bias

insufficient

Proof-based work; bias assessment not applicable to mathematical derivations.

Sample adequacy

insufficient

No empirical sample; theoretical analysis only.

Source abstract

We study how far gradient descent (GD) can be accelerated by predetermined nonnegative stepsizes in smooth convex optimization. Writing $p_{\mathrm{sil}}=\log_2(1+\sqrt{2})$, we prove an $Ω\left(n^{-p_{\mathrm{sil}}-O(\sqrt{\log\log n/\log n})}\right)$ non-anytime lower bound. In the anytime setting, every infinite nonnegative schedule has infinitely many horizons with error $Ω\left(n^{-\frac{2p_{\mathrm{sil}}}{1+p_{\mathrm{sil}}}-O(\sqrt{\log\log n/\log n})}\right)$. Together with the silver-schedule upper bound [Altschuler and Parrilo, 2025] and the anytime upper bound [Zhang et al., 2025], our results determine the optimal polynomial convergence exponents in both settings.

Follow from this

Taken from the source record, never inferred. Follow any of these and new work involving them reaches your briefing.

Yuhan YeKaizhao Liu