# Contextual Inverse Optimization: Offline and Online Learning

arXiv: https://arxiv.org/abs/2106.14015 · HTML: https://mlsanity.com/paper/2106.14015/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 26 Jun 2021.

## Builds on (most cited first, 18 of 18)

- [An Algorithmic Perspective on Imitation Learning](https://mlsanity.com/paper/1811.06711/) (2018)
- [A Survey of Inverse Reinforcement Learning: Challenges, Methods and Progress](https://mlsanity.com/paper/1806.06877/) (2018)
- [Inverse Optimization with Noisy Data](https://mlsanity.com/paper/1507.03266/) (2015)
- [Data-driven Inverse Optimization with Imperfect Information](https://mlsanity.com/paper/1512.05489/) (2015)
- [Data-Driven Estimation in Equilibrium Using Inverse Optimization](https://mlsanity.com/paper/1308.3397/) (2013)
- [Repeated Inverse Reinforcement Learning](https://mlsanity.com/paper/1705.05427/) (2017)
- [Multidimensional Binary Search for Contextual Decision-Making](https://mlsanity.com/paper/1611.00829/) (2016)
- [Contextual Search via Intrinsic Volumes](https://mlsanity.com/paper/1804.03195/) (2018)
- [On Structured Prediction Theory with Calibrated Convex Surrogate Losses](https://mlsanity.com/paper/1703.02403/) (2017)
- [An Online-Learning Approach to Inverse Optimization](https://mlsanity.com/paper/1810.12997/) (2018)
- [Watch and Learn: Optimizing from Revealed Preferences Feedback](https://mlsanity.com/paper/1504.01033/) (2015)
- [Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit](https://mlsanity.com/paper/1511.04650/) (2015)
- [Generalized Inverse Optimization through Online Learning](https://mlsanity.com/paper/1810.01920/) (2018)
- [Learning Economic Parameters from Revealed Preferences](https://mlsanity.com/paper/1407.7937/) (2014)
- [Online Learning and Profit Maximization from Revealed Preferences](https://mlsanity.com/paper/1407.7294/) (2014)
- [Model-Free Assortment Pricing with Transaction Data](https://mlsanity.com/paper/2101.02251/) (2021)
- [Learning from Rational Behavior: Predicting Solutions to Unknown Linear Programs](https://mlsanity.com/paper/1506.02162/) (2015)
- [Learning to Emulate an Expert Projective Cone Scheduler](https://mlsanity.com/paper/1801.09821/) (2018)

## Built on by (most cited first, 20 of 20)

- [Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming](https://mlsanity.com/paper/2301.11260/) (2023)
- [Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms](https://mlsanity.com/paper/2305.07730/) (2023)
- [Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound](https://mlsanity.com/paper/2501.14349/) (2025)
- [Revisiting Online Learning Approach to Inverse Linear Optimization: A Fenchel-Young Loss Perspective and Gap-Dependent Regret Analysis](https://mlsanity.com/paper/2501.13648/) (2025)
- [Finite and Corruption-Robust Regret Bounds in Online Inverse Linear Optimization under M-Convex Action Sets](https://mlsanity.com/paper/2602.01682/) (2026)
- [Simple Projection-Free Algorithm for Contextual Recommendation with Logarithmic Regret and Robustness](https://mlsanity.com/paper/2603.20826/) (2026)
- [Tight Generalization Bounds for Noiseless Inverse Optimization](https://mlsanity.com/paper/2605.08866/) (2026)
- [Multiscale Reward Hedging from Correct Demonstrations](https://mlsanity.com/paper/2608.06825/) (2026)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [From Inverse Optimization to Feasibility to ERM](https://mlsanity.com/paper/2402.17890/) (2024)
- [Conformalized Decision Risk Assessment](https://mlsanity.com/paper/2505.13243/) (2025)
- [Efficient Online Inverse Optimization with O(d) Regret](https://mlsanity.com/paper/2609.13440/) (2026)
- [Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions](https://mlsanity.com/paper/2510.04455/) (2025)
- [Explicit Iteration Complexity of Exact Data-Driven Inverse Optimization for Integer Linear Programs](https://mlsanity.com/paper/2607.22263/) (2026)
- [Tight Regret Bound for Online Inverse Linear Optimization via Multiscale Matrix Weights](https://mlsanity.com/paper/2609.26978/) (2026)
- [Misalignment, Learning, and Ranking: Harnessing Users Limited Attention](https://mlsanity.com/paper/2402.14013/) (2024)
- [Deviation-Based Learning: Training Recommender Systems Using Informed User Choice](https://mlsanity.com/paper/2109.09816/) (2021)
- [Contextual Preference Distribution Learning](https://mlsanity.com/paper/2603.17139/) (2026)
- [OTSS: Output-Targeted Soft Segmentation for Contextual Decision-Weight Learning](https://mlsanity.com/paper/2605.00193/) (2026)
- [TACIT: Optimization Models that Learn from Their Mistakes](https://mlsanity.com/paper/2609.38434/) (2026)

## Abstract (arXiv)

We study the problems of offline and online contextual optimization with feedback information, where instead of observing the loss, we observe, after-the-fact, the optimal action an oracle with full knowledge of the objective function would have taken. We aim to minimize regret, which is defined as the difference between our losses and the ones incurred by an all-knowing oracle. In the offline setting, the decision-maker has information available from past periods and needs to make one decision, while in the online setting, the decision-maker optimizes decisions dynamically over time based a new set of feasible actions and contextual functions in each period. For the offline setting, we characterize the optimal minimax policy, establishing the performance that can be achieved as a function of the underlying geometry of the information induced by the data. In the online setting, we leverage this geometric characterization to optimize the cumulative regret. We develop an algorithm that yields the first regret bound for this problem that is logarithmic in the time horizon. Finally, we show via simulation that our proposed algorithms outperform previous methods from the literature.