Geometric Identification in Predict-Then-Optimize Learning
Abstract
Decision-focused surrogates can recover downstream decisions without identifying the quotient report. We characterize the equality set of the convex Smart Predict-then-Optimize surrogate (SPO+) population risk. Under central symmetry, the centered mean class is the unique Bayes minimizer exactly when every nonzero effective displacement makes the old optimizer leave the shifted optimal face with positive probability. This condition separates face crossing from selected-oracle disagreement and gives quantitative local coercivity. Without symmetry, strict crossing alone need not identify the mean; selection balance with reflected crossing restores quotient-report identification, and conditional versions extend the result to measurable predictors. These are population statements, without finite-sample report-recovery or generic transfer-regret guarantees. Closed-form mechanisms reproduce the analytic identities and rates. Portfolio, complete-matrix KuaiRec, and Energy/Storage studies measure predictive fidelity, shifted regret, and fitted-report geometry. A known data-generating process (DGP) companion retains their application geometries while isolating conditional-mean recovery and crossing, without testing the original observational assumptions.
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BibTeX
@misc{xu2026geometric,
title = {{Geometric Identification in Predict-Then-Optimize Learning}},
author = {Jiaxiao Xu and Changhong Mou and Keji Liu and Dinghua Xu and Yeyu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33472},
eprint = {2609.33472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33472},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric Identification in Predict-Then-Optimize Learning
AU - Jiaxiao Xu
AU - Changhong Mou
AU - Keji Liu
AU - Dinghua Xu
AU - Yeyu Zhang
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33472
UR - https://arxiv.org/abs/2609.33472
AB - Decision-focused surrogates can recover downstream decisions without identifying the quotient report. We characterize the equality set of the convex Smart Predict-then-Optimize surrogate (SPO+) population risk. Under central symmetry, the centered mean class is the unique Bayes minimizer exactly when every nonzero effective displacement makes the old optimizer leave the shifted optimal face with positive probability. This condition separates face crossing from selected-oracle disagreement and gives quantitative local coercivity. Without symmetry, strict crossing alone need not identify the mean; selection balance with reflected crossing restores quotient-report identification, and conditional versions extend the result to measurable predictors. These are population statements, without finite-sample report-recovery or generic transfer-regret guarantees. Closed-form mechanisms reproduce the analytic identities and rates. Portfolio, complete-matrix KuaiRec, and Energy/Storage studies measure predictive fidelity, shifted regret, and fitted-report geometry. A known data-generating process (DGP) companion retains their application geometries while isolating conditional-mean recovery and crossing, without testing the original observational assumptions.
ER -