  The classical analysis of Stochastic Gradient Descent (SGD) with polynomially
decaying stepsize $\eta_t = \eta/\sqrt{t}$ relies on well-tuned $\eta$
depending on problem parameters such as Lipschitz smoothness constant, which is
often unknown in practice. In this work, we prove that SGD with arbitrary $\eta
> 0$, referred to as untuned SGD, still attains an order-optimal convergence
rate $\widetilde{O}(T^{-1/4})$ in terms of gradient norm for minimizing smooth
objectives. Unfortunately, it comes at the expense of a catastrophic
exponential dependence on the smoothness constant, which we show is unavoidable
for this scheme even in the noiseless setting. We then examine three families
of adaptive methods $\unicode{x2013}$ Normalized SGD (NSGD), AMSGrad, and
AdaGrad $\unicode{x2013}$ unveiling their power in preventing such exponential
dependency in the absence of information about the smoothness parameter and
boundedness of stochastic gradients. Our results provide theoretical
justification for the advantage of adaptive methods over untuned SGD in
alleviating the issue with large gradients.
