We prove that the iterates produced by, either the scalar step size variant, or the coordinatewise variant of AdaGrad algorithm, are convergent sequences when applied to convex objective functions with Lipschitz gradient. The key insight is to remark that such AdaGrad sequences satisfy a variable metric quasi-Fej\'er monotonicity property, which allows to prove convergence.

翻译：我们证明,使用利普施茨梯度对曲线客观函数应用时,由卡路里级阶梯大小变异或AdaGrad算法协调变异产生的迭代是趋同序列。 关键的观点是,AdaGrad序列满足了可变的准Fej\'er单音特性,从而可以证明趋同。