  In this paper, we propose a new, simplified high probability analysis of
AdaGrad for smooth, non-convex problems. More specifically, we focus on a
particular accelerated gradient (AGD) template (Lan, 2020), through which we
recover the original AdaGrad and its variant with averaging, and prove a
convergence rate of $\mathcal O (1/ \sqrt{T})$ with high probability without
the knowledge of smoothness and variance. We use a particular version of
Freedman's concentration bound for martingale difference sequences (Kakade &
Tewari, 2008) which enables us to achieve the best-known dependence of $\log (1
/ \delta )$ on the probability margin $\delta$. We present our analysis in a
modular way and obtain a complementary $\mathcal O (1 / T)$ convergence rate in
the deterministic setting. To the best of our knowledge, this is the first high
probability result for AdaGrad with a truly adaptive scheme, i.e., completely
oblivious to the knowledge of smoothness and uniform variance bound, which
simultaneously has best-known dependence of $\log( 1/ \delta)$. We further
prove noise adaptation property of AdaGrad under additional noise assumptions.
