Distributionally Robust Learning with Weakly Convex Losses: Convergence Rates and Finite-Sample Guarantees

We consider a distributionally robust stochastic optimization problem and formulate it as a stochastic two-level composition optimization problem with the use of the mean--semideviation risk measure. In this setting, we consider a single time-scale algorithm, involving two versions of the inner function value tracking: linearized tracking of a continuously differentiable loss function, and SPIDER tracking of a weakly convex loss function. We adopt the norm of the gradient of the Moreau envelope as our measure of stationarity and show that the sample complexity of $\mathcal{O}(\varepsilon^{-3})$ is possible in both cases, with only the constant larger in the second case. Finally, we demonstrate the performance of our algorithm with a robust learning example and a weakly convex, non-smooth regression example

Convexification of stochastic ordering

We consider sets defined by the usual stochastic ordering relation and by the second order stochastic dominance relation. Under fairy general assumptions we prove that in the space of integrable random variables the closed convex hull of the first set is equal to the second set. Keywords: Stochastic Dominance, Convexity

Risk-Averse Two-Stage Stochastic Linear Programming: Modeling and Decomposition

We formulate a risk-averse two-stage stochastic linear programming problem in which unresolved uncertainty remains after the second stage. The objective function is formulated as a composition of conditional risk measures. We analyze properties of the problem and derive necessary and sufficient optimality conditions. Next, we construct two decomposition methods for solving the problem. The first method is based on the generic cutting plane approach, while the second method exploits the composite structure of the objective function. We illustrate their performance on a portfolio optimization problem. 1

Optimization of Convex Risk Functions

We consider optimization problems involving convex risk functions. By employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models, and optimality and duality theory for problems involving risk functions