# Last-Iterate Complexity of SGD for Convex and Smooth Stochastic Problems

arXiv: https://arxiv.org/abs/2507.14122 · HTML: https://mlsanity.com/paper/2507.14122/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jul 2025.

## Built on by (most cited first, 5 of 5)

- [Gradient Descent's Last Iterate is Often (slightly) Suboptimal](https://mlsanity.com/paper/2604.13870/) (2026)
- [Small Gradient Norm Regret for Online Convex Optimization](https://mlsanity.com/paper/2601.13519/) (2026)
- [Convergence Rate of the Last Iterate of Stochastic Proximal Algorithms](https://mlsanity.com/paper/2602.05489/) (2026)
- [Stochastic Nonconvex Bilevel Optimization: Improved Rates Without Rare-Visit Assumption](https://mlsanity.com/paper/2609.06580/) (2026)
- [Revisiting AdaGrad in Stochastic Convex Optimization: Last Iterates, High Probability, and Lower Bounds](https://mlsanity.com/paper/2609.32729/) (2026)

## Abstract (arXiv)

Most results on Stochastic Gradient Descent (SGD) in the convex and smooth setting are presented under the form of bounds on the ergodic function value gap. It is an open question whether bounds can be derived directly on the last iterate of SGD in this context. Recent advances suggest that it should be possible. For instance, it can be achieved by making the additional, yet unverifiable, assumption that the variance of the stochastic gradients is uniformly bounded. In this paper, we show that there is no need of such an assumption, and that SGD enjoys a tilde O left( T-1/2 right) last-iterate complexity rate for convex smooth stochastic problems.