Last-Iterate Convergence of Randomized Kaczmarz and SGD with Greedy Step Size

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.

Random reshuffling stochastic gradient descent (RR-SGD) is widely used for finite-sum optimization in machine learning. For convex $L$-smooth components, recent work establishes convergence guarantees for an averaged iterate with component stepsizes as large as order $1/L$. However, the final iterate is a natural output in practice, and existing last-iterate guarantees in this setting require stepsizes that scale inversely with the number of components $n$. We establish an expected objective-gap bound for the last iterate of standard RR-SGD on unconstrained problems with a constant component stepsize satisfying $0<η\lesssim 1/(L(1 + \log(nK)))$, where $K$ is the number of epochs. By retaining the sampling bias at random comparison points, we remove the explicit $1/n$ factor from the stepsize restriction. Combined with existing small-step bounds, our result matches the corresponding averaged-iterate convergence rate up to logarithmic factors.

We consider the well-studied setting of minimizing a convex Lipschitz function using either gradient descent (GD) or its stochastic variant (SGD), and examine the last iterate convergence. By now, it is known that standard stepsize choices lead to a last iterate convergence rate of $\log T/\sqrt{T}$ after $T$ steps. A breakthrough result of Jain et al. [2019] recovered the optimal $1/\sqrt{T}$ rate by constructing a non-standard stepsize sequence. However, this sequence requires choosing $T$ in advance, as opposed to common stepsize schedules which apply for any time horizon. Moreover, Jain et al. conjectured that without prior knowledge of $T$, no stepsize sequence can ensure the optimal error for SGD's last iterate, a claim which so far remained unproven. We prove this conjecture, and in fact show that even in the noiseless case of GD, it is impossible to avoid an excess poly-log factor in $T$ when considering an anytime last iterate guarantee.