mlsanityData to 14 Oct 2026

Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimization

arXiv 2309.16530 ↗First posted 28 Sep 2023math.OCcited by 20

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Abstract (arXiv)

We provide a concise, self-contained proof that the Silver Stepsize Schedule proposed in Part I directly applies to smooth (non-strongly) convex optimization. Specifically, we show that with these stepsizes, gradient descent computes an ε-minimizer in O(ε^-logρ 2) = O(ε-0.7864) iterations, where ρ = 1+√2 is the silver ratio. This is intermediate between the textbook unaccelerated rate O(ε-1) and the accelerated rate O(ε-1/2) due to Nesterov in 1983. The Silver Stepsize Schedule is a simple explicit fractal: the i-th stepsize is 1+ρv(i)-1 where v(i) is the 2-adic valuation of i. The design and analysis are conceptually identical to the strongly convex setting in Part I, but simplify remarkably in this specific setting.

Key facts. Cited by 20 papers in the map (data through 14 Oct 2026). First posted on arXiv 28 Sep 2023.

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Citing papers per year: 2023: 1; 2024: 3; 2025: 6; 2026: 10.

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  1. 2024Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiencycited by 35
  2. 2025Negative Stepsizes Make Gradient-Descent-Ascent Convergecited by 11
  3. 2024Anytime Acceleration of Gradient Descentcited by 7
  4. 2025Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizescited by 7
  5. 2024Open Problem: Anytime Convergence Rate of Gradient Descentcited by 5
  6. 2025CyclicReflex: Improving Reasoning Models via Cyclical Reflection Token Schedulingcited by 5
  7. 2025Convergence Rates for Gradient Descent on the Edge of Stability in Overparametrised Least Squarescited by 4
  8. 2026Gradient Descent's Last Iterate is Often (slightly) Suboptimalcited by 4
  9. 2026A lower bound for stepsize-based acceleration of gradient descentcited by 4
  10. 2025Gradient Methods with Online Scaling Part I. Theoretical Foundationscited by 3
  11. 2023On the convergence of adaptive first order methods: proximal gradient and alternating minimization algorithmscited by 2
  12. 2026Improved Gradient Descent Lower Bounds Beyond Nesterovcited by 2
8 more
  1. 2026Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimensioncited by 1
  2. 2026Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minimacited by 1
  3. 2026Silver Rate Is (Almost) Optimal for Gradient Descentcited by 1
  4. 2025Empirical and computer-aided robustness analysis of long-step and accelerated methods in smooth convex optimizationcited by 0
  5. 2026Exponential Convergence of (Stochastic) Gradient Descent for Separable Logistic Regressioncited by 0
  6. 2026A short proof of near-linear convergence of adaptive gradient descent under fourth-order growth and convexitycited by 0
  7. 2026A Lower Bound for the Heavy-Ball Method on Smooth Convex Functionscited by 0
  8. 2026The Row Normalization Puzzle in Muoncited by 0