2020/06/14 by Othmane Sebbouh, Robert M. Gower, Sebbouh, Othmane +3
Computer Science · Mathematics · #Complexity and Algorithms in Graphs #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Mathematical Approximation and Integration #Optimization and Control (math.OC) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2006.07867
openalex publication_date 2020/06/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.