In high-stakes machine learning applications, it is crucial to not only perform well on average, but also when restricted to difficult examples. To address this, we consider the problem of training models in a risk averse manner. We propose an adaptive sampling algorithm for stochastically optimizing the Conditional Value-at-Risk (CVaR) of a loss distribution. We use a distributionally robust formulation of the CVaR to phrase the problem as a zero-sum game between two players, and solve it efficiently using regret minimization. Our approach relies on sampling from structured Determinantal Point Processes (DPPs), which allows scaling it to large data sets. Finally, we empirically demonstrate its effectiveness on large-scale convex and non-convex learning tasks.