  We study convergence rates of AdaGrad-Norm as an exemplar of adaptive
stochastic gradient methods (SGD), where the step sizes change based on
observed stochastic gradients, for minimizing non-convex, smooth objectives.
Despite their popularity, the analysis of adaptive SGD lags behind that of non
adaptive methods in this setting. Specifically, all prior works rely on some
subset of the following assumptions: (i) uniformly-bounded gradient norms, (ii)
uniformly-bounded stochastic gradient variance (or even noise support), (iii)
conditional independence between the step size and stochastic gradient. In this
work, we show that AdaGrad-Norm exhibits an order optimal convergence rate of
$\mathcal{O}\left(\frac{\mathrm{poly}\log(T)}{\sqrt{T}}\right)$ after $T$
iterations under the same assumptions as optimally-tuned non adaptive SGD
(unbounded gradient norms and affine noise variance scaling), and crucially,
without needing any tuning parameters. We thus establish that adaptive gradient
methods exhibit order-optimal convergence in much broader regimes than
previously understood.
