# New Bounds for the Last Iterate of the Stochastic subGradient Method

arXiv: https://arxiv.org/abs/2606.24879 · HTML: https://mlsanity.com/paper/2606.24879/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 19 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Tight Analyses for Non-Smooth Stochastic Gradient Descent](https://mlsanity.com/paper/1812.05217/) (2018)
- [Revisiting the Last-Iterate Convergence of Stochastic Gradient Methods](https://mlsanity.com/paper/2312.08531/) (2023)
- [Exact convergence rate of the last iterate in subgradient methods](https://mlsanity.com/paper/2307.11134/) (2023)
- [Stochastic Nonsmooth Convex Optimization with Heavy-Tailed Noises: High-Probability Bound, In-Expectation Rate and Initial Distance Adaptation](https://mlsanity.com/paper/2303.12277/) (2023)
- [General Tail Bounds for Non-Smooth Stochastic Mirror Descent](https://mlsanity.com/paper/2312.07142/) (2023)
- [Gradient Descent's Last Iterate is Often (slightly) Suboptimal](https://mlsanity.com/paper/2604.13870/) (2026)
- [An Improved Analysis of the Clipped Stochastic subGradient Method under Heavy-Tailed Noise](https://mlsanity.com/paper/2410.00573/) (2024)
- [The Convergence Rate of SGD's Final Iterate: Analysis on Dimension Dependence](https://mlsanity.com/paper/2106.14588/) (2021)

## Abstract (arXiv)

We study the last iterate of the stochastic subgradient method for one-dimensional convex Lipschitz objectives. For a fixed horizon n, we consider the standard fixed stepsizes η =Theta(1/√ n). We prove that, for such stepsize policies, under additive i.i.d. subgradient noise with uniformly bounded variance, the last iterate features an optimization error of order 1/√ n, thereby removing the extra (log n) factor present in existing generic bounds. On the other hand, we show that without the i.i.d. assumption, the optimization error can be of order (log n)/√ n. Thus, under the uniformly bounded variance assumption alone, the last iterate of SsGM is suboptimal even in dimension one, resolving negatively an open problem posed in Koren and Segal, COLT, 2020.