Learning Decision-Sufficient Representations for Linear Optimization

Abstract

We study how to construct compressed datasets that suffice to recover optimal decisions in linear programs with an unknown cost vector c lying in a prior set C. Recent work by Bennouna et al. 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 showing that computing d^ is NP-hard and deciding whether a dataset is globally sufficient is coNP-hard, thereby resolving a recent open problem posed by Bennouna et al. To address this worst-case intractability, we introduce pointwise sufficiency, a relaxation that requires sufficiency for an individual cost vector. Under nondegeneracy, we provide a polynomial-time cutting-plane algorithm for constructing pointwise-sufficient decision datasets. In a data-driven regime with i.i.d.\ costs, we further 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 O(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 O(d^/n) rather than O(d/n), where d is the ambient cost dimension.

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