Contextual Inverse Optimization: Offline and Online Learning

TL;DR

This paper introduces a new framework for offline and online contextual optimization using feedback on the optimal actions, providing algorithms with logarithmic regret bounds and demonstrating superior performance through simulations.

Contribution

It characterizes the optimal minimax policy in offline settings and develops the first algorithm with logarithmic regret bounds for online contextual optimization.

Findings

Proposed algorithms outperform previous methods in simulations.

Established the performance limits based on data geometry.

Achieved logarithmic regret bounds in online optimization.

Abstract

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.