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

Forum topic · 小凯 · 2026-10-31

Summary

This arXiv paper (2510.25914) by Aron Brenner and Saurabh Amin proposes centering a Kullback-Leibler (KL) ambiguity ball on a density model fitted to the record—the learned nominal—instead of the empirical distribution. Unlike divergence balls around empirical distributions, which only allow reweighting of observed data, this learned nominal lets the adversary up-weight costly unobserved realizations in proportion to the model-assigned probabilities, while avoiding the over-hedging of transport balls that include distributions the data-generating process cannot produce. Because the resulting DRO problem can only accommodate a small number of scenarios, independent sampling would likely miss rare worst-case realizations. The authors therefore construct scenario sets adaptively, alternating between solving the finite problem and drawing new realizations from the learned nominal in directions guided by the current decision's worst case, correcting draws with importance weights. Experiments on a multi-product newsvendor problem and a capacity expansion problem with real grid data show that centering the ball on the learned nominal reduces cost under distribution shift when records are scarce and the model class captures structure enabling extrapolation beyond the record, with comparable in-distribution cost; advantages diminish as records grow or shifts deviate from learned structure.

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

Field: Machine Learning
Authors: Aron Brenner, Saurabh Amin
Published: 2025-10-28
arXiv: 2510.25914

Abstract

Distributionally robust optimization (DRO) hedges data-driven two-stage stochastic programs against error in the distribution estimated from a finite record, and standard ambiguity sets differ in how they treat realizations absent from that record. A divergence ball around the empirical distribution contains only reweightings of the record, whereas 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 as well.

The authors instead propose a Kullback-Leibler (KL) ball centered at a density model fitted to the record—the *learned nominal*—which lets the adversary up-weight costly unobserved realizations in proportion to the probability the model assigns them.

Key points

Ambiguity set choice: A learned nominal as the KL ball's center balances between empirical divergence balls (no new realizations) and transport balls (hedging against impossible realizations).
Scenario generation challenge: The resulting problem admits only a small number of scenarios, and independent samples at that scale likely miss rare worst-case realizations.
Adaptive method: The scenario set is built iteratively, alternating between solving the finite problem and sampling new realizations from the learned nominal; sampling directions are guided by the current decision's worst case, with draws corrected back to the model via importance weights.
Experiments: On a multi-product newsvendor problem and a capacity expansion problem based on real grid data, centering the ball on the learned nominal lowers cost under distribution shift when records are scarce and the model class captures structure enabling extrapolation beyond the record, while in-distribution costs remain comparable.
Limitations: The advantage diminishes as records grow or as the distribution shift deviates from the structure the model has learned.

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

Summary

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

Abstract

Key points

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