This article is an interpretation of the ICLR 2022 selected paper, Online Facility Location with Predictions. The paper was completed by Jiang Shaofeng's research group of Peking University Frontier Computing Research Center in collaboration with Shanghai University of Finance and Economics and Shanghai Jiao Tong University, and the authors are sorted alphabetically according to the practice in the field of theoretical computing. The paper proposes an algorithm that uses prediction information to solve the problem of online facility site selection, and in the experiment, the algorithm has better performance than the traditional algorithm when the prediction is more accurate.
Graphic | Zhang Yubo
Jiang Shaofeng Research Group of Peking University
Thesis link: https://arxiv.org/abs/2110.08840
1 Research background
As machine learning flourishes, people are also thinking about whether machine learning techniques can provide new solutions to other problems in computer science. In recent years, some researchers have set their sights on online issues. Due to the limitations of online problems, algorithms need to make global decisions with only a subset of input. If we can explore the laws in the data through techniques such as machine learning and provide predictions for future inputs for online algorithms, then we can go beyond traditional worst-case analysis and achieve better approximations.
The online issue examined in this article is The Online Facility Location (OFL). Facility Location is a problem similar to k-clustering, such as k-means/k-median. In this problem, we also need to divide the input set of points into clusters and pay for each point the distance to the corresponding cluster center. Unlike k-clusters, Facility Location allows for any number of clustering centers, each at the cost of w. The OFL problem, based on its offline version, instead gives the set of input points in some order, giving only one point at a time, and asks the algorithm to immediately output its corresponding cluster center.
We use a competitive ratio to measure the performance of online algorithms. The competitive ratio is a widely used method of measuring the accuracy of an online algorithm, and for OFL, the competitive ratio of an online algorithm is defined as the ratio of the cost of the clustering scheme produced by the algorithm to the cost of the optimal solution under the offline/all-information version. This reflects the cost of incomplete information in the online setting of the algorithm.
Adam Meyerson first studied the OFL problem in 2001 and proposed an online algorithm for competitive ratios. Dimitris Fotakis then proposed an algorithm for the competition ratio and proved that any algorithm would have at least a worst-case competition ratio, optimizing the OFL problem. We hope to break through the traditional worst-case analysis by introducing new predictive techniques to provide new solutions to OFL problems.
2 Theoretical results
This article is the first to examine how predictions can be used to solve of THEL problems in the case of Non-Uniform Cost (Non Uniform Cost, i.e., the cost of setting cluster centers at different points is different, expressed as a cost function w).
The prediction model used in this article is very simple, and our algorithm will accept an additional prediction as input on top of the original input. For each point entered, the predictor predicts that the approximate ratio of the cluster center algorithm corresponding to this point in the optimal solution is related to the predictor's prediction accuracy, and the more accurate the predictor is accepted, the smaller the approximation ratio of the algorithm.
We use the distance of the actual output of the prediction and the optimal solution as a measure of prediction accuracy (the smaller the distance, the more accurate it is).
Note: Based on the approximate ratio of theoretically lower bounds
As can be seen from the above figure, the algorithm proposed in this paper can achieve constant approximation when the prediction is completely accurate, breaking through the traditional worst-case analysis, and the performance in the worst-case prediction situation is almost the same as the previous optimal algorithm. In addition, the theoretical lower bound of the approximation ratio under the prediction model in this paper is further analyzed, and the analysis shows that the approximation ratio of the proposed algorithm is close to optimal.
3 Algorithm design
The algorithm proposed in this paper is based on the Meyerson algorithm, which meyerson originally proposed when he studied the OFL problem. Although the approximation ratio of the algorithm is not optimal under Non-Uniform Cost, it is concise, easy to analyze, and has good properties.
The analysis of Meyerson's algorithm is based on a hierarchical approach: considering a cluster center f* in the optimal solution and the set of points assigned to f* in the data, if we have now set up a cluster center no more than d distance from f*, then the algorithm can guarantee an approximate ratio of future inputs.
Using this property, we designed an auxiliary algorithm that runs both the auxiliary algorithm and the Meyerson algorithm. The auxiliary algorithm is designed to ensure that the cost must not exceed the cost of the Meyerson algorithm, so it will not affect the final approximation ratio analysis. The execution of the algorithm is divided into two phases:
In the first stage, with the help of predictions, auxiliary algorithms can build approximate solutions that are only possible from the optimal solution. This phase will soon end, and the auxiliary algorithm guarantees that the total cost of the Meyerson algorithm at this stage will not affect the final approximation ratio.
In the second stage, due to the preparation work of the previous stage, the approximate ratio of meyerson's algorithm at this stage is to achieve a better approximation ratio.
4 Experimental results
We implement the above algorithm ideas and test them on four datasets commonly used by clustering algorithms. There are three datasets in Euclidean space and one dataset for the shortest-circuit metric on the graph, three of which are Uniform Cost and one of which is Non-Uniform Cost. The parameters of the dataset are summarized in the following figure.
We provide the algorithm with two reference objects, the first is the original Meyerson algorithm that does not rely on predictions at all, and the second is the algorithm that directly outputs the prediction results. It should be noted that the Meyerson algorithm has performed very well under Uniform Cost, and can achieve constant approximation under random sequential input. The following graph shows the approximate ratio of the three algorithms at different prediction qualities.
5 Summary and Outlook
In this paper, an algorithm for solving OFL problems using predictions is proposed, which can not only surpass the theoretical nether that is not based on predictions when predictions are more accurate, but also achieve approximate optimal prediction when predicting the worst. In the experiment we designed, the algorithm exhibited a level that matched expectations.
How to design algorithms that effectively use predictions is still a new research direction, and there are many geometric problems similar to OFL that need to be further studied. In the future, we expect the techniques in this paper to be used not only to solve problems directly related to OFL, but also to have a wider range of applications on other geometric problems.