Fast Last-Iterate Convergence of SGD in the Smooth Interpolation Regime
This provides theoretical guarantees for training over-parameterized models and analyzing forgetting in continual learning, representing an incremental extension of prior work beyond least squares regression.
The paper tackles the problem of analyzing the last iterate convergence of stochastic gradient descent (SGD) for smooth convex objectives in the interpolation regime, establishing a near-optimal rate of O(1/T + σ_*/√T) for well-tuned stepsizes and improving the rate to O(1/√T) when noise at optimum is zero.
We study population convergence guarantees of stochastic gradient descent (SGD) for smooth convex objectives in the interpolation regime, where the noise at optimum is zero or near zero. The behavior of the last iterate of SGD in this setting -- particularly with large (constant) stepsizes -- has received growing attention in recent years due to implications for the training of over-parameterized models, as well as to analyzing forgetting in continual learning and to understanding the convergence of the randomized Kaczmarz method for solving linear systems. We establish that after $T$ steps of SGD on $β$-smooth convex loss functions with stepsize $0 < η< 2/β$, the last iterate exhibits expected excess risk $\widetilde{O}(\frac{1}{η(2-βη) T^{1-βη/2}} + \fracη{(2-βη)^2} T^{βη/2} σ_\star^2)$, where $σ_\star^2$ denotes the variance of the stochastic gradients at the optimum. In particular, for a well-tuned stepsize we obtain a near optimal $\widetilde{O}(1/T + σ_\star/\sqrt{T})$ rate for the last iterate, extending the results of Varre et al. (2021) beyond least squares regression; and when $σ_\star=0$ we obtain a rate of $\smash{O(1/\sqrt T)}$ with $η=1/β$, improving upon the best-known $\smash{O(T^{-1/4})}$ rate recently established by Evron et al. (2025) in the special case of realizable linear regression.