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

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 **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 **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 $O(1/\sqrt{T})$.