Kevin-Martin Aigner (Uni Erlangen) on "Online Learning for Optimization Problems with Unknown or Uncertain Cost Functions"

We consider the robust treatment of stochastic optimization problems involving random vectors with unknown discrete probability distributions. With this problem class, we demonstrate the basic concepts of data-driven optimization under uncertainty. Furthermore, we introduce a new iterative approach that uses scenario observations to learn more about the uncertainty over time. This means our solutions become less and less conservative, interpolating between distributionally robust and stochastic optimization. We achieve this by solving the distributionally robust optimization problem over time via an online-learning approach while iteratively updating the ambiguity sets. We provide a regret bound for the quality of the obtained solutions that converges at a rate of O(log(T)/T) and illustrate the effectiveness of our procedure by numerical experiments. Our proposed algorithm is able to solve the online learning problem significantly faster than equivalent reformulations. This is joint work with Kristin Braun, Frauke Liers, Sebastian Pokutta, Oskar Schneider, Kartikey Sharma and Sebastian Tschuppik.