# Accelerated Objective Gap and Gradient Norm Convergence for Gradient Descent via Long Steps

arXiv: https://arxiv.org/abs/2403.14045 · HTML: https://mlsanity.com/paper/2403.14045/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2024.

## Built on by (most cited first, 11 of 11)

- [Anytime Acceleration of Gradient Descent](https://mlsanity.com/paper/2411.17668/) (2024)
- [Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes](https://mlsanity.com/paper/2504.04105/) (2025)
- [Projected gradient methods for nonconvex and stochastic smooth optimization: new complexities and auto-conditioned stepsizes](https://mlsanity.com/paper/2412.14291/) (2024)
- [Convergence Rates for Gradient Descent on the Edge of Stability in Overparametrised Least Squares](https://mlsanity.com/paper/2510.17506/) (2025)
- [Gradient Descent's Last Iterate is Often (slightly) Suboptimal](https://mlsanity.com/paper/2604.13870/) (2026)
- [A lower bound for stepsize-based acceleration of gradient descent](https://mlsanity.com/paper/2608.10418/) (2026)
- [Improved Gradient Descent Lower Bounds Beyond Nesterov](https://mlsanity.com/paper/2609.02855/) (2026)
- [Tight Bounds for Logistic Regression with Large Stepsize Gradient Descent in Low Dimension](https://mlsanity.com/paper/2602.12471/) (2026)
- [Dynamics of Gradient Descent with Large Step Size Near a Manifold of Flat Minima](https://mlsanity.com/paper/2607.08380/) (2026)
- [Silver Rate Is (Almost) Optimal for Gradient Descent](https://mlsanity.com/paper/2609.09152/) (2026)
- [A Lower Bound for the Heavy-Ball Method on Smooth Convex Functions](https://mlsanity.com/paper/2609.08656/) (2026)

## Abstract (arXiv)

This work considers gradient descent for L-smooth convex optimization with stepsizes larger than the classic regime where descent can be ensured. The stepsize schedules considered are similar to but differ slightly from the recent silver stepsizes of Altschuler and Parrilo. For one of our stepsize sequences, we prove a Oleft(N- 1.2716dotsright) convergence rate in terms of objective gap decrease and for the other, we show the same rate of decrease for squared-gradient-norm decrease. This first result improves on the recent result of Altschuler and Parrilo by a constant factor, while the second results improve on the exponent of the prior best squared-gradient-norm convergence guarantee of Oleft(N-1right).