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Last-Iterate Complexity of SGD for Convex and Smooth Stochastic Problems

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arxiv 2507.14122 v1 pith:R2H5UL7O submitted 2025-07-18 math.OC

Last-Iterate Complexity of SGD for Convex and Smooth Stochastic Problems

classification math.OC
keywords stochasticconvexsmoothassumptionboundscomplexitylast-iterateproblems
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Most results on Stochastic Gradient Descent (SGD) in the convex and smooth setting are presented under the form of bounds on the ergodic function value gap. It is an open question whether bounds can be derived directly on the last iterate of SGD in this context. Recent advances suggest that it should be possible. For instance, it can be achieved by making the additional, yet unverifiable, assumption that the variance of the stochastic gradients is uniformly bounded. In this paper, we show that there is no need of such an assumption, and that SGD enjoys a $\tilde O \left( T^{-1/2} \right)$ last-iterate complexity rate for convex smooth stochastic problems.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Gradient Descent's Last Iterate is Often (slightly) Suboptimal

    math.OC 2026-04 unverdicted novelty 8.0

    Proves it is impossible to achieve optimal last-iterate rates for GD and SGD without knowing the horizon T in advance, incurring an unavoidable poly-log factor penalty even in the deterministic case.

  2. Stochastic Krasnoselskii-Mann Iterations: Convergence without Uniformly Bounded Variance

    math.OC 2026-04 unverdicted novelty 7.0

    Stochastic Krasnoselskii-Mann iterations converge almost surely weakly with finite variance only at a single fixed point, recovering best-known rates without uniform variance bounds.

  3. Stochastic Krasnoselskii-Mann Iterations: Convergence without Uniformly Bounded Variance

    math.OC 2026-04 unverdicted novelty 7.0

    Stochastic Krasnoselskii-Mann iterations converge almost surely and with rates under finite variance at a single fixed point rather than uniform variance bounds, recovering optimal complexity and providing first such ...