Nearly Optimal $L_p$ Risk Minimization

Convex risk measures play a foundational role in the area of stochastic optimization. However, in contrast to risk neutral models, their applications are still limited due to the lack of efficient solution methods. In particular, the mean $L_p$ semi-deviation is a classic risk minimization model, but its solution is highly challenging due to the composition of concave-convex functions and the lack of uniform Lipschitz continuity. In this paper, we discuss some progresses on the design of efficient algorithms for $L_p$ risk minimization, including a novel lifting reformulation to handle the concave-convex composition, and a new stochastic approximation method to handle the non-Lipschitz continuity. We establish an upper bound on the sample complexity associated with this approach and show that this bound is not improvable for $L_p$ risk minimization in general through the construction of a nearly matching lower complexity bound.