# Geometric Identification in Predict-Then-Optimize Learning

arXiv: https://arxiv.org/abs/2609.33472 · HTML: https://mlsanity.com/paper/2609.33472/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 22 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Decision-Focused Learning: Through the Lens of Learning to Rank](https://mlsanity.com/paper/2112.03609/) (2021)
- [Generalization Bounds in the Predict-then-Optimize Framework](https://mlsanity.com/paper/1905.11488/) (2019)
- [Fast Rates for Contextual Linear Optimization](https://mlsanity.com/paper/2011.03030/) (2020)
- [Risk Bounds and Calibration for a Smart Predict-then-Optimize Method](https://mlsanity.com/paper/2108.08887/) (2021)
- [On Structured Prediction Theory with Calibrated Convex Surrogate Losses](https://mlsanity.com/paper/1703.02403/) (2017)
- [Risk Guarantees for End-to-End Prediction and Optimization Processes](https://mlsanity.com/paper/2012.15046/) (2020)
- [Surrogate Regret Bounds for Polyhedral Losses](https://mlsanity.com/paper/2110.14031/) (2021)
- [A General Theory for Structured Prediction with Smooth Convex Surrogates](https://mlsanity.com/paper/1902.01958/) (2019)
- [An Embedding Framework for the Design and Analysis of Consistent Polyhedral Surrogates](https://mlsanity.com/paper/2206.14707/) (2022)
- [Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming](https://mlsanity.com/paper/2301.11260/) (2023)
- [What Data Enables Optimal Decisions? An Exact Characterization for Linear Optimization](https://mlsanity.com/paper/2505.21692/) (2025)
- [A Solver-Free Training Method for Predict-then-Optimize](https://mlsanity.com/paper/2606.19587/) (2026)
- [Sufficient Decision Proxies for Decision-Focused Learning](https://mlsanity.com/paper/2505.03953/) (2025)
- [Smart Surrogate Losses for Contextual Stochastic Linear Optimization with Robust Constraints](https://mlsanity.com/paper/2505.22881/) (2025)
- [From Non-Identifiability to Goal-Integrated Decision-Making in Parametric Inverse Optimization](https://mlsanity.com/paper/2603.17033/) (2026)
- [Learning Decision-Sufficient Representations for Linear Optimization](https://mlsanity.com/paper/2603.18551/) (2026)

## Abstract (arXiv)

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.