{"ID":5937097,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-09T12:56:29.822227725Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05024","arxiv_id":"2607.05024","title":"Post-Learning Inference for Combinatorial Optimizers with High-Dimensional Sparse Contextual Information via Minimal Directional Perturbation","abstract":"We study post-learning inference for structural properties of data-dependent combinatorial optimizers. The target is whether an oracle optimizer, rather than a latent parameter or smooth functional, belongs to a prescribed class, such as a category-mix, inventory, or resource-feasibility class. We focus on a high-dimensional contextual multinomial logit model with sequentially adaptive data collection, where the parameter-to-optimizer map is discontinuous and the policy induces temporal dependence. We propose a novel perturbation test based on a nonsmooth max-difference revenue statistic comparing the best null assortment with the best alternative assortment. The test perturbs the estimated terminal revenue surface on the selected support: random unit directions capture directional uncertainty, while the minimal perturbation radius captures magnitude uncertainty and yields a p-value. This localizes inference near the null--alternative boundary and avoids uniform error control over the full candidate class. The data are collected by an \\(\\ell_1\\)-penalized online likelihood policy that performs variable selection while controlling regret. Using a new anti-concentration argument for Gaussian maxima differences and martingale Gaussian coupling, we establish uniform estimation rates, effective support recovery, and asymptotic validity of the proposed p-value under adaptive assortment selection. We prove asymptotic size control and power consistency under a localized signal condition.","short_abstract":"We study post-learning inference for structural properties of data-dependent combinatorial optimizers. The target is whether an oracle optimizer, rather than a latent parameter or smooth functional, belongs to a prescribed class, such as a category-mix, inventory, or resource-feasibility class. We focus on a high-dimen...","url_abs":"https://arxiv.org/abs/2607.05024","url_pdf":"https://arxiv.org/pdf/2607.05024v1","authors":"[\"Pengyu Li\",\"Shuting Shen\"]","published":"2026-07-06T13:02:14Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}