mlsanityData to 13 Oct 2026

Silver Rate Is (Almost) Optimal for Gradient Descent

arXiv 2609.09152 ↗First posted 8 Sep 2026math.OC cs.LGcited by 1

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Abstract (arXiv)

We study how far gradient descent (GD) can be accelerated by predetermined stepsizes in smooth convex optimization. Writing psil=log2(1+√2), we prove an Omegaleft(n^-psil-O(√loglog n/log n)right) non-anytime lower bound. In the anytime setting, every infinite schedule has infinitely many horizons with error Omegaleft(n^-frac2psil1+psil-O(√loglog 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.

Key facts. Cited by 1 paper in the map (data through 13 Oct 2026). Builds on 18 of its 31 references that are in the map. First posted on arXiv 8 Sep 2026.

Builds on

18 of 31 references.

References · 18

  1. 2015Smooth Strongly Convex Interpolation and Exact Worst-case Performance of First-order Methodscited by 37
  2. 2023Provably Faster Gradient Descent via Long Stepscited by 20
  3. 2023Acceleration by Stepsize Hedging II: Silver Stepsize Schedule for Smooth Convex Optimizationcited by 20
  4. 2023Acceleration by Stepsize Hedging I: Multi-Step Descent and the Silver Stepsize Schedulecited by 19
  5. 2022Branch-and-Bound Performance Estimation Programming: A Unified Methodology for Constructing Optimal Optimization Methodscited by 16
  6. 2024Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Stepscited by 11
  7. 2024Anytime Acceleration of Gradient Descentcited by 7
  8. 2024Acceleration by Random Stepsizes: Hedging, Equalization, and the Arcsine Stepsize Schedulecited by 6
  9. 2024Open Problem: Anytime Convergence Rate of Gradient Descentcited by 5
  10. 2024Accelerated Gradient Descent by Concatenation of Stepsize Schedulescited by 5
  11. 2024Composing Optimized Stepsize Schedules for Gradient Descentcited by 4
  12. 2026A lower bound for stepsize-based acceleration of gradient descentcited by 4
6 more
  1. 2026Lower Bounds for Anytime Acceleration of Gradient Descentcited by 3
  2. 2026Improved Gradient Descent Lower Bounds Beyond Nesterovcited by 2
  3. 2026Stronger Lower Bounds for (Non-)Anytime Acceleration of Gradient Descentcited by 2
  4. 2024Relaxed Proximal Point Algorithm: Tight Complexity Bounds and Acceleration without Momentumcited by 1
  5. 2025Optimized methods for composite optimization: a reduction perspectivecited by 1
  6. 2026Stepsize Hedging: an Alternative Mechanism for Accelerating Gradient Descentcited by 1

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Cited by year (2026 to 13 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026The Row Normalization Puzzle in Muoncited by 0