Learning-Augmented Algorithms for $k$-median via Online Learning

TL;DR

This paper introduces a new learning-augmented framework for the $k$-median problem, leveraging online learning to improve solutions across multiple instances, with theoretical guarantees and empirical validation.

Contribution

It proposes a novel online learning-inspired model for learning-augmented algorithms applied to $k$-median, providing an efficient algorithm with competitive average performance.

Findings

Algorithm closely matches the best fixed $k$-median solution in hindsight.

Empirical results show near-optimal performance and adaptation to changing data.

Framework generalizes learning-augmented approaches to multiple instances.

Abstract

The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm's solution and that of an optimal solution for that instance. We apply this framework to the classic $k$-median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed $k$-median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence.