Learning Decision-Sufficient Representations for Linear Optimization

摘要

We study how to construct compressed datasets that suffice to recover optimal decisions in linear programs with unknown cost vector c lying in a prior set 𝒞. Recent work by Bennouna et al. (2025a) provides an exact geometric characterization of sufficient decision datasets (SDDs) via an intrinsic decision-relevant dimension d^⋆. However, their algorithm for constructing minimum-size SDDs requires solving mixed-integer programs. In this paper, we establish hardness results: computing d^⋆ is NP-hard and deciding whether a dataset is globally sufficient is coNP-hard, thereby resolving the open problem posed by Bennouna et al. (2026). To circumvent worst-case intractability, we introduce pointwise sufficiency, a relaxation that requires sufficiency for an individual cost vector. We provide a polynomial-time cutting-plane algorithm to construct pointwise-sufficient decision datasets under nondegeneracy. In a data-driven regime with i.i.d. costs, we propose a cumulative algorithm that aggregates decision-relevant directions across samples, yielding a stable compression scheme of size at most d^⋆. This leads to a distribution-free PAC guarantee: with high probability over the training sample, the pointwise sufficiency failure probability on a fresh draw is at most Õ(d^⋆/n), and this rate is tight up to logarithmic factors. Finally, we apply decision-sufficient representations to contextual linear optimization, obtaining compressed predictors with generalization bounds scaling as Õ(√(d^⋆/n)) rather than Õ(√(d/n)), where d is the ambient cost dimension.