# The Curvature of Regret in Contextual Linear Optimization

arXiv: https://arxiv.org/abs/2610.01980 · HTML: https://mlsanity.com/paper/2610.01980/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 1 Oct 2026.

## Builds on (most cited first, 13 of 13)

- [From Softmax to Sparsemax: A Sparse Model of Attention and Multi-Label Classification](https://mlsanity.com/paper/1602.02068/) (2016)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Decision-Focused Learning: Foundations, State of the Art, Benchmark and Future Opportunities](https://mlsanity.com/paper/2307.13565/) (2023)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [SparseMAP: Differentiable Sparse Structured Inference](https://mlsanity.com/paper/1802.04223/) (2018)
- [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)
- [Gen-DFL: Decision-Focused Generative Learning for Robust Decision Making](https://mlsanity.com/paper/2502.05468/) (2025)
- [TaskMet: Task-Driven Metric Learning for Model Learning](https://mlsanity.com/paper/2312.05250/) (2023)
- [Diffusion-DFL: Decision-focused Diffusion Models for Stochastic Optimization](https://mlsanity.com/paper/2510.11590/) (2025)
- [Decision-Focused Learning via Tangent-Space Projection of Prediction Error](https://mlsanity.com/paper/2605.01361/) (2026)

## Abstract (arXiv)

Decision-focused learning for linear optimization is complicated by the discontinuity of the optimizer, where small cost errors may leave the decision unchanged or move it to a different vertex. We show that this non-smooth pointwise behavior becomes locally quadratic after averaging over the data distribution, and we derive the curvature in closed form, specifically, a matrix-valued measure supported on the walls of the normal fan. This measure depends only on the feasible set, with the data distribution entering only as a weight. We then offer a tractable approximation for this curvature, computable with just one projection to the feasible set. We prove that the approximation weakly converges to the true population curvature. We offer one application of our findings, a decision-aware scenario generation method for expected-cost linear optimization. Our experiments test the quadratic and weak convergence laws and show a 30.8% regret improvement over uniform allocation on battery arbitrage.