[Paper Review] Data-driven Inverse Optimization with Incomplete Information

Peyman Mohajerin Esfahani, Soroosh Shafieezadeh-Abadeh+2 more|arXiv (Cornell University)|Dec 17, 2015

Advanced Bandit Algorithms Research4 citations

TL;DR

This paper proposes a data-driven inverse optimization framework that learns an agent's objective function from incomplete or noisy signal-response data by minimizing worst-case risk between estimated and actual decisions. It reformulates the problem as a tractable convex program when prediction error is measured in objective value space, offering strong out-of-sample performance guarantees under bounded rationality, measurement noise, or model misspecification.

ABSTRACT

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the {\em estimated} decision ({\em i.e.}, the decision implied by a particular candidate objective) differs from the agent's {\em actual} response to a random signal. We show that our framework offers attractive out-of-sample performance guarantees for different prediction errors and that the emerging inverse optimization problems can be reformulated as (or approximated by) tractable convex programs when the prediction error is measured in the space of objective values. A main strength of the proposed approach is that it naturally generalizes to situations where the observer has imperfect information, {\em e.g.}, when the agent's true objective function is not contained in the space of candidate objectives, when the agent suffers from bounded rationality or implementation errors, or when the observed signal-response pairs are corrupted by measurement noise.

Motivation & Objective

• To address the challenge of learning an agent's true objective function when historical data is incomplete, noisy, or corrupted.
• To formalize inverse optimization as a distributionally robust program that minimizes worst-case risk between estimated and actual decisions.
• To provide theoretical guarantees on out-of-sample performance under various forms of uncertainty, including bounded rationality and measurement error.
• To develop a tractable convex optimization reformulation when prediction error is measured in objective value space.
• To generalize existing inverse optimization approaches to settings where the true objective is not in the candidate set of objectives.

Proposed method

• Formalize the inverse optimization problem as a distributionally robust program that minimizes the worst-case risk of decision mismatch between estimated and actual responses.
• Model uncertainty in the agent's true objective by considering a set of plausible objective functions consistent with observed data.
• Use a risk measure based on the worst-case expected deviation between estimated and actual decisions across possible signals.
• Reformulate the inverse problem as a tractable convex program when prediction error is measured in the space of objective values.
• Incorporate robustness to bounded rationality and implementation errors by allowing for deviations between optimal and observed actions.
• Handle measurement noise in signal-response pairs by embedding uncertainty sets that reflect plausible data corruption.

Experimental results

Research questions

• RQ1 How can inverse optimization be made robust to incomplete or corrupted historical data in learning agent preferences?
• RQ2 What is the impact of bounded rationality and implementation errors on the reliability of learned objectives in inverse optimization?
• RQ3 Can a distributionally robust framework ensure strong out-of-sample performance guarantees in inverse optimization under model uncertainty?
• RQ4 Under what conditions can the inverse optimization problem be reformulated as a tractable convex program?
• RQ5 How does the proposed method generalize to cases where the true objective function lies outside the candidate set of objectives?

Key findings

• The proposed framework provides strong out-of-sample performance guarantees by minimizing worst-case risk between estimated and actual decisions.
• The inverse optimization problem can be reformulated as a tractable convex program when prediction error is measured in the space of objective values.
• The method naturally accommodates bounded rationality, implementation errors, and measurement noise without requiring exact knowledge of the true objective.
• Robustness to model misspecification is achieved by considering a set of plausible objectives consistent with observed data.
• The approach generalizes existing inverse optimization methods by allowing for imperfect information and uncertainty in both signals and responses.
• The framework maintains theoretical tractability while offering flexibility in modeling various sources of uncertainty in real-world applications.