Stochastic Gradient Descent with Momentum is Algorithmically Stable

Variance reduction (VR) methods employ stochastic gradients with decreasing variance, and they have been widely applied to solve large-scale optimization problems in machine learning because of their efficiency. Existing theoretical studies of VR methods are mainly focused on the convergence analysis, leaving the generalization behavior largely unexplored. In this paper, we bridge this gap by developing the first non-vacuous generalization analysis of the representative VR method: Stochastic Variance Reduced Gradient (SVRG), through the lens of algorithmic stability. In particular, we establish sharp stability bounds of the SVRG in both convex and strongly convex settings by exploiting its algorithmic structure. The obtained bounds are data-dependent, because the training errors are incorporated along the trajectory. Our analysis clarifies the interplay between optimization and generalization, leading to optimal excess population risk bounds in both settings. Our approach differs substantially from existing analyses of stochastic algorithms in the sense that we decompose the SVRG update as an SGD-like step plus a zero-mean correction term and then introduce novel Lyapunov functions to absorb the additional gradient terms induced by the reference points. Our analytical framework can be generalized to other VR methods, and we demonstrate the generalization by the well-known Stochastic Average Gradient Accelerated (SAGA) method.

We analyze generalization error, uniform stability, and uniform argument stability of gradient descent (GD) and stochastic gradient descent (SGD) over discrete parameter spaces, where each update involves deterministic or stochastic rounding. We show that deterministic rounding degrades the generalization error of GD on convex, Lipschitz, and smooth loss functions, increasing the rate from $O(T/n)$ to $O(T/\sqrt{n})$, and establish matching lower bounds. We further prove that uniform stability of GD becomes $Ω(T)$, showing that stability-based generalization bounds are vacuous in this setting. In contrast, for the same losses, stochastic gradient descent with deterministic rounding admits nontrivial uniform stability guarantees, which differ qualitatively from the real-valued case and exhibit distinct dependencies on the number of iterations and the dimension: we prove tight bounds $O(T/n)$ for one dimension and $O(T^2/n)$ for higher dimensions. We also show that stochastic rounding can introduce generalization error that increases with the dimension; such a phenomenon is absent in standard real-valued optimization and in the deterministic rounding case. Finally, we provide upper bounds on uniform argument stability for stochastic rounding schemes and show that these bounds are tight when the loss can be represented as a sum of coordinate-wise functions.

Many stochastic gradient methods are believed not to converge when the noise in stochastic gradients has only a finite $p$-th moment for $p\in\left(1,2\right)$, a setting known as the heavy-tailed noise assumption. However, some recent studies have found that Stochastic Gradient Descent ($\textsf{SGD}$), without any modification to its update rule, can surprisingly converge in expectation for convex problems with bounded domains, highlighting the potential of classical stochastic gradient methods. Inspired by this recent progress, we provide a comprehensive study of stochastic optimization under heavy-tailed noise and establish new in-expectation convergence results for Stochastic Mirror Descent ($\textsf{SMD}$) and Accelerated Stochastic Mirror Descent ($\textsf{ASMD}$) in convex optimization, and for $\textsf{SGD}$ and Stochastic Gradient Descent with Momentum ($\textsf{SGDM}$) in nonconvex optimization. Notably, our results not only hold without algorithmic changes but also avoid restrictive assumptions, such as bounded domains, imposed in prior work. More importantly, our analysis provides a new, elegant, and powerful framework for studying heavy-tailed stochastic optimization, opening a new route to understanding first-order stochastic gradient methods.