# Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball

arXiv: https://arxiv.org/abs/2006.07867 · HTML: https://mlsanity.com/paper/2006.07867/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2020.

## Built on by (most cited first, 7 of 7)

- [DecentLaM: Decentralized Momentum SGD for Large-batch Deep Training](https://mlsanity.com/paper/2104.11981/) (2021)
- [Adaptivity without Compromise: A Momentumized, Adaptive, Dual Averaged Gradient Method for Stochastic Optimization](https://mlsanity.com/paper/2101.11075/) (2021)
- [On the Convergence of Stochastic Extragradient for Bilinear Games using Restarted Iteration Averaging](https://mlsanity.com/paper/2107.00464/) (2021)
- [Minibatch and Momentum Model-based Methods for Stochastic Weakly Convex Optimization](https://mlsanity.com/paper/2106.03034/) (2021)
- [The Role of Momentum Parameters in the Optimal Convergence of Adaptive Polyak's Heavy-ball Methods](https://mlsanity.com/paper/2102.07314/) (2021)
- [(Accelerated) Noise-adaptive Stochastic Heavy-Ball Momentum](https://mlsanity.com/paper/2401.06738/) (2024)
- [Dynamics of Stochastic Momentum with Sparse Updates in High Dimensions](https://mlsanity.com/paper/2605.28961/) (2026)

## Abstract (arXiv)

We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem. For SGD, in the convex and smooth setting, we provide the first almost sure asymptotic convergence rates for a weighted average of the iterates . More precisely, we show that the convergence rate of the function values is arbitrarily close to o(1/√k), and is exactly o(1/k) in the so-called overparametrized case. We show that these results still hold when using stochastic line search and stochastic Polyak stepsizes, thereby giving the first proof of convergence of these methods in the non-overparametrized regime. Using a substantially different analysis, we show that these rates hold for SHB as well, but at the last iterate. This distinction is important because it is the last iterate of SGD and SHB which is used in practice. We also show that the last iterate of SHB converges to a minimizer almost surely. Additionally, we prove that the function values of the deterministic HB converge at a o(1/k) rate, which is faster than the previously known O(1/k). Finally, in the nonconvex setting, we prove similar rates on the lowest gradient norm along the trajectory of SGD.