## High Probability Bounds for a Class of Nonconvex Algorithms with AdaGrad Stepsize

### Ali Kavis · Kfir Y Levy · Volkan Cevher

Keywords: [ adaptive methods ] [ stochastic optimization ] [ nonconvex optimization ]

[ Abstract ]
Thu 28 Apr 2:30 a.m. PDT — 4:30 a.m. PDT

Abstract: 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.

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