Near-Optimal Last-Iterate Convergence of SGD in the Smooth Interpolation Regime

We study the last iterate of stochastic gradient descent (SGD) with a constant stepsize on smooth convex objectives in the low-noise and interpolation regimes, where the variance $σ_\star^2$ of the stochastic gradients at the optimum is small or zero. We establish that after $T$ steps of SGD on $β$-smooth convex losses with any constant stepsize $0<η<2/β$, the last iterate attains expected excess risk $\widetilde O\bigl(\tfrac{1}{η(2-βη)^3T}+\tfracη{(2-βη)^4}σ_\star^2\bigr)$, which gives the rate $\smash{\widetilde O(1/T)}$ for every constant stepsize in the interpolation regime ($σ_\star=0$), matching the $Ω(1/T)$ lower bound up to logarithmic factors and improving upon the $O(1/\sqrt T)$ rate of Attia et al. (2026) for $η=1/β$. Our analysis is based on a new technique, which compares the second moments of the stochastic gradients along the SGD trajectory across only logarithmically many time scales. For quadratic losses in the interpolation regime, an adaptation of this technique gives an $O(1/T)$ rate for every constant stepsize, extending the recent optimal rate for the randomized Kaczmarz method of Derezinski and Dong (2026) to stepsizes above the greedy stepsize with an arguably simpler proof.