For part (2), apply Lemma 43, which gives a uniform generalization bound over the larger class HU⋆,d⋆ (and therefore also over its subsetH U⋆,d⋆(C))

Part (1) follows from Lemma 44 by taking f=f ⋆ ∈ H(C) : the compressed predictor ˆf⋆(ξ) = liftU⋆(U ⊤ ⋆ (f⋆(ξ)−c 0)) induces the same decisions as f⋆, hence has the same population SPO risk, is a risk minimizer inH U⋆,d⋆(C) · 2012

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Learning Decision-Sufficient Representations for Linear Optimization

math.OC · 2026-03-19 · unverdicted · novelty 7.0

Proves NP-hardness of computing decision-relevant dimension d* and coNP-hardness of global sufficiency in linear optimization, then gives poly-time pointwise algorithms, a cumulative compression scheme of size at most d*, and PAC bounds scaling with d* for contextual linear optimization.

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Learning Decision-Sufficient Representations for Linear Optimization math.OC · 2026-03-19 · unverdicted · none · ref 15
Proves NP-hardness of computing decision-relevant dimension d* and coNP-hardness of global sufficiency in linear optimization, then gives poly-time pointwise algorithms, a cumulative compression scheme of size at most d*, and PAC bounds scaling with d* for contextual linear optimization.

For part (2), apply Lemma 43, which gives a uniform generalization bound over the larger class HU⋆,d⋆ (and therefore also over its subsetH U⋆,d⋆(C))

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