Adaptive Scenario Generation for KL-DRO with a Learned Density Model

Distributionally robust optimization (DRO) hedges data-driven two-stage stochastic programs against error in the distribution estimated from a finite record. Standard ambiguity sets treat realizations absent from the record in one of two ways. A divergence ball around the empirical distribution contains only reweightings of the record. A transport ball also contains distributions supported off the record, including on realizations the data-generating process cannot produce, against which the decision is then hedged. We propose a Kullback–Leibler (KL) ball centered at a density model fitted to the record, the learned nominal, so that the adversary can up-weight costly unobserved realizations in proportion to the probability the model assigns them. Because an independent sample of the few scenarios the problem can accommodate is likely to omit the rare realizations on which the worst case concentrates, we build the scenario set adaptively. We alternate between solving the finite problem and drawing new realizations from the learned nominal, steered toward the worst case at the current decision, with importance weights that keep the ball centered at the model. On a multi-item newsvendor problem and a capacity expansion problem on realistic grid data, centering the ball at the learned nominal rather than at the record lowers cost under distribution shift, at comparable in-distribution cost, when the record is scarce and the model class captures structure that lets it extrapolate beyond the record. The advantage fades as the record grows or as the shift departs from the structure the model has learned.