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"Theorem 1 (Section V.A) establishes that finitely Ỹ-convex functions form a dense subset of all Ỹ-convex functions, giving a universal approximation property for generalized convex functions.",
"Theorem 2 (Section V.A) shows that, under semiconvexity conditions on Φ, gradients of finitely Ỹ-convex functions densely approximate gradients of all Ỹ-convex functions.",
"Theorem 4 (Section V.C) proves that the lean subset of finitely Ỹ-convex functions forms a convex parameter space, which is what allows bilevel objectives to be rewritten as single-level problems solvable with standard first-order optimization (Section I).",
"Table I reports multi-item auction experiments for n in {1,2,5,10,20} goods in which, for n up to 10, the learned mechanism's revenue matches the Straight-Jacket auction benchmark exactly.",
"Figures 2-3 show the learned parametrization recovers the known optimal posted price of 0.5 in the single-item auction case and finds a mixed-bundling pricing scheme matching theoretical benchmarks in the two-item case.",
"Kantorovich dual solutions for optimal transport are characterized as Ỹ-convex functions in Section IV, so gradients of the learned parametrization directly yield optimal transport maps via a diffeomorphism condition."
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