# An Online-Learning Approach to Inverse Optimization

arXiv: https://arxiv.org/abs/1810.12997 · HTML: https://mlsanity.com/paper/1810.12997/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Oct 2018.

## Built on by (most cited first, 18 of 18)

- [Contextual Inverse Optimization: Offline and Online Learning](https://mlsanity.com/paper/2106.14015/) (2021)
- [Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming](https://mlsanity.com/paper/2301.11260/) (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)
- [Repeated Principal-Agent Games with Unobserved Agent Rewards and Perfect-Knowledge Agents](https://mlsanity.com/paper/2304.07407/) (2023)
- [Simple Projection-Free Algorithm for Contextual Recommendation with Logarithmic Regret and Robustness](https://mlsanity.com/paper/2603.20826/) (2026)
- [Estimating and Incentivizing Imperfect-Knowledge Agents with Hidden Rewards](https://mlsanity.com/paper/2308.06717/) (2023)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [Using Inverse Optimization to Learn Cost Functions in Generalized Nash Games](https://mlsanity.com/paper/2102.12415/) (2021)
- [A proof of convergence of inverse reinforcement learning for multi-objective optimization](https://mlsanity.com/paper/2305.06137/) (2023)
- [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)
- [Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes](https://mlsanity.com/paper/2609.09809/) (2026)
- [Tight Regret Bound for Online Inverse Linear Optimization via Multiscale Matrix Weights](https://mlsanity.com/paper/2609.26978/) (2026)
- [Optimal and Efficient Online Inverse Optimization](https://mlsanity.com/paper/2610.08735/) (2026)
- [Outperformance Inverse Optimization: Learning Objective Functions that Outperform Agent Decisions](https://mlsanity.com/paper/2610.09890/) (2026)

## Abstract (arXiv)

In this paper, we demonstrate how to learn the objective function of a decision-maker while only observing the problem input data and the decision-maker's corresponding decisions over multiple rounds. We present exact algorithms for this online version of inverse optimization which converge at a rate of  O(1/√T)  in the number of observations~T and compare their further properties. Especially, they all allow taking decisions which are essentially as good as those of the observed decision-maker already after relatively few iterations, but are suited best for different settings each. Our approach is based on online learning and works for linear objectives over arbitrary feasible sets for which we have a linear optimization oracle. As such, it generalizes previous approaches based on KKT-system decomposition and dualization. We also introduce several generalizations, such as the approximate learning of non-linear objective functions, dynamically changing as well as parameterized objectives and the case of suboptimal observed decisions. When applied to the stochastic offline case, our algorithms are able to give guarantees on the quality of the learned objectives in expectation. Finally, we show the effectiveness and possible applications of our methods in indicative computational experiments.