Geometric Identification in Predict-Then-Optimize Learning

2026-09-27

AI Summary

This study addresses the inability of decision-focused agents to identify reported quotients in predict-then-optimize learning. It characterizes the SPO+ risk equivalence sets, establishing mean uniqueness under symmetry and identification conditions for asymmetric cases. Methodologically, this work proposes facet-intersection and selection-balancing mechanisms, achieving a theoretical breakthrough in population-level geometric identification. Technically, it integrates convex analysis with closed-form derivations, applying them to empirical studies in portfolio investment and energy storage. The project validates the geometric properties of prediction fidelity, shifted regret, and fitted reports, thereby providing solid theoretical foundations for decision-oriented 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.

Problem

• Predict-Then-Optimize
• Geometric Identification
• Decision-focused surrogate
• SPO+
• Quotient report identification

Innovation

• Predict-Then-Optimize
• SPO+ Surrogate
• Geometric Identification
• Face Crossing
• Decision-Focused Learning