  We study scalable alternatives to robust gradient descent (RGD) techniques
that can be used when the losses and/or 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 instead focus
computational effort on robustly choosing (or newly constructing) a strong
candidate based on a collection of cheap stochastic sub-processes which can be
run in parallel. The exact selection process depends on the convexity of the
underlying objective, but in all cases, our selection technique amounts to a
robust form of boosting the confidence of weak learners. 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,
along with applications to a variety of benchmark datasets. The overall
take-away is an extensible procedure that is simple to implement, trivial to
parallelize, which keeps the formal merits of RGD methods but scales much
better to large learning problems.
