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We study Stochastic Gradient Descent with AdaGrad stepsizes: a popularadaptive (self-tuning) method for first-order stochastic optimization. Despitebeing well studied, existing analyses of this method suffer from variousshortcomings: they either assume some knowledge of the problem parameters,impose strong global Lipschitz conditions, or fail to give bounds that holdwith high probability. We provide a comprehensive analysis of this basic methodwithout any of these limitations, in both the convex and non-convex (smooth)cases, that additionally supports a general ``affine variance'' noise model andprovides sharp rates of convergence in both the low-noise andhigh-noise~regimes.

SGD with AdaGrad Stepsizes: Full Adaptivity with High Probability to Unknown Parameters, Unbounded Gradients and Affine Variance