  We study a scalable alternative to robust gradient descent (RGD) techniques
that can be used when the gradients can be heavy-tailed, though this will be
unknown to the learner. The core technique is simple: instead of trying to
robustly aggregate gradients at each step, which is costly and leads to
sub-optimal dimension dependence in risk bounds, we choose a candidate which
does not diverge too far from the majority of cheap stochastic sub-processes
run for a single pass over partitioned data. In addition to formal guarantees,
we also provide empirical analysis of robustness to perturbations to
experimental conditions, under both sub-Gaussian and heavy-tailed data. The
result is a procedure that is simple to implement, trivial to parallelize,
which keeps the formal strength of RGD methods but scales much better to large
learning problems.
