  We develop a variant of the stochastic prox-linear method for minimizing the
Conditional Value-at-Risk (CVaR) objective. CVaR is a risk measure focused on
minimizing worst-case performance, defined as the average of the top quantile
of the losses. In machine learning, such a risk measure is useful to train more
robust models. Although the stochastic subgradient method (SGM) is a natural
choice for minimizing the CVaR objective, we show that our stochastic
prox-linear (SPL+) algorithm can better exploit the structure of the objective,
while still providing a convenient closed form update. Our SPL+ method also
adapts to the scaling of the loss function, which allows for easier tuning. We
then specialize a general convergence theorem for SPL+ to our setting, and show
that it allows for a wider selection of step sizes compared to SGM. We support
this theoretical finding experimentally.
