Active Learning For Contextual Linear Optimization: A Margin-Based Approach

TL;DR

This paper introduces a novel margin-based active learning algorithm for contextual linear optimization that minimizes label acquisition while maintaining low decision loss, with theoretical guarantees and practical applications.

Contribution

It is the first to develop an active learning method directly informed by the SPO loss, with theoretical bounds on label complexity and practical algorithms for decision-making tasks.

Findings

The algorithm achieves significantly lower label complexity than naive supervised learning.

Theoretical bounds demonstrate efficiency under margin conditions and surrogate losses.

Numerical experiments validate the practical effectiveness in pricing and routing problems.

Abstract

We develop the first active learning method for contextual linear optimization. Specifically, we introduce a label acquisition algorithm that sequentially decides whether to request the ``labels'' of feature samples from an unlabeled data stream, where the labels correspond to the coefficients of the objective in the linear optimization. Our method is the first to be directly informed by the decision loss induced by the predicted coefficients, referred to as the Smart Predict-then-Optimize (SPO) loss. Motivated by the structure of the SPO loss, our algorithm adopts a margin-based criterion utilizing the concept of distance to degeneracy. In particular, we design an efficient active learning algorithm with theoretical excess risk (i.e., generalization) guarantees. We derive upper bounds on the label complexity, defined as the number of samples whose labels are acquired to achieve a desired small level of SPO risk. These bounds show that our algorithm has a much smaller label complexity than the naive supervised learning approach that labels all samples, particularly when the SPO loss is minimized directly on the collected data. To address the discontinuity and nonconvexity of the SPO loss, we derive label complexity bounds under tractable surrogate loss functions. Under natural margin conditions, these bounds also outperform naive supervised learning. Using the SPO+ loss, a specialized surrogate of the SPO loss, we establish even tighter bounds under separability conditions. Finally, we present numerical evidence showing the practical value of our algorithms in settings such as personalized pricing and the shortest path problem.