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

Abstract: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.

Optimization and Control (math.OC); Machine Learning (cs.LG)

arXiv:2604.13870 [math.OC]

(or arXiv:2604.13870v2 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2604.13870 Focus to learn more arXiv-issued DOI via DataCite

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