Robust Learning via Nested Distributionally Robust Optimization

Abstract

Distributionally Robust Optimization (DRO) is widely used to improve model robustness, with existing methods addressing either geometric perturbations (e.g., input shifts) or statistical contamination (e.g., heavy-tailed noise and outliers) effectively. However, these uncertainty sources often co-exist. Coupling them through a single divergence or optimal transport constraint conflates geometric displacement with loss-based outlierness, which often leads to the discarding of informative high-leverage samples. We introduce nested DRO, a bilevel formulation that combines optimal transport with an outer $\\phi$-divergence constraint to decouple geometric smoothing from statistical robustness. We prove that this structure naturally induces a geometry-invariant, loss-based reweighting mechanism that separates outlier suppression from transport-induced regularization. We derive a tractable strong dual for the resulting non-convex problem and show its equivalence to variance-regularized risk minimization, providing a rigorous theoretical justification for reweighting gains as a natural consequence of dualization. Empirical results on synthetic and real datasets demonstrate that nested DRO consistently outperforms geometry-coupled DRO baselines, particularly under heavy-tailed contamination where preserving high-leverage structure is crucial.