# The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions

arXiv: https://arxiv.org/abs/2104.05468 · HTML: https://mlsanity.com/paper/2104.05468/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2021.

## Built on by (most cited first, 4 of 4)

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- [PEPit: computer-assisted worst-case analyses of first-order optimization methods in Python](https://mlsanity.com/paper/2201.04040/) (2022)
- [A lower bound for stepsize-based acceleration of gradient descent](https://mlsanity.com/paper/2608.10418/) (2026)
- [Differentiable Arbitrating in Zero-sum Markov Games](https://mlsanity.com/paper/2302.10058/) (2023)

## Abstract (arXiv)

In this paper, we study the convergence rate of the gradient (or steepest descent) method with fixed step lengths for finding a stationary point of an L-smooth function. We establish a new convergence rate, and show that the bound may be exact in some cases, in particular when all step lengths lie in the interval (0,1/L]. In addition, we derive an optimal step length with respect to the new bound.