  We consider non-convex stochastic optimization using first-order algorithms
for which the gradient estimates may have heavy tails. We show that a
combination of gradient clipping, momentum, and normalized gradient descent
yields convergence to critical points in high-probability with best-known rates
for smooth losses when the gradients only have bounded $\mathfrak{p}$th moments
for some $\mathfrak{p}\in(1,2]$. We then consider the case of second-order
smooth losses, which to our knowledge have not been studied in this setting,
and again obtain high-probability bounds for any $\mathfrak{p}$. Moreover, our
results hold for arbitrary smooth norms, in contrast to the typical SGD
analysis which requires a Hilbert space norm. Further, we show that after a
suitable "burn-in" period, the objective value will monotonically decrease for
every iteration until a critical point is identified, which provides intuition
behind the popular practice of learning rate "warm-up" and also yields a
last-iterate guarantee.
