Revisiting AdaGrad in Stochastic Convex Optimization: Last Iterates, High Probability, and Lower Bounds

Abstract:AdaGrad and AdaGrad-Norm are widely used adaptive methods, but their precise behavior in stochastic convex optimization remains less understood. We first show that AdaGrad-Norm and AdaGrad do not admit any universal \textbf{last-iterate} rate, even under sub-Gaussian noise and bounded iterates. We then prove that bounded variance alone is too weak: even with bounded iterates, it cannot yield \textbf{high-probability average-iterate} rates, and without bounded iterates it may not even guarantee convergence in expectation. Besides, we construct tight $\Omega(\log T/\sqrt T)$ lower bounds for both AdaGrad-Norm and AdaGrad under sub-Gaussian noise, showing that $\log T$ in existing average-iterate upper bounds is unavoidable. Finally, we show that this $\log T$ loss disappears once bounded-iterate condition is imposed: under a general ABC condition, both methods achieve rates ${O}(1/\sqrt{T})$.

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

arXiv:2609.32729 [math.OC]

(or arXiv:2609.32729v1 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2609.32729 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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