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.
Figures & tables
(∇t)i:=−∑x∈Rtwx(M(x)−min(M(x),d(x,vi)).
Algorithm 1 Algorithm for an instance of Learn-Bounded-Median
Figure 1: For i.i.d. instances (Uniform Square and Multiple Clusters ), the optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions (top figure) and approximation ratios - avg. and std. dev. over 10 random instances (bottom figure).
Figure 2: Experimental Results for Dynamic Instances
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 3: (Uniform Square): The optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios – avg. and std. dev. over 10 random instances (right) for k=2,3,6 .
Figure 4: (Uniform Rectangle): The optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios – avg. and std. dev. over 10 random instances (right) for k=2,3,6 .
Figure 5: (Multiple Clusters): the optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios - avg. and std. dev. over 10 random instances (right) for k=4,8,12,16 .
Figure 6: (Uniform Hypersphere): Fractional mass of algorithm over time at the center (avg. and std. dev. over 10 runs), d=2,8
Figure 7: (Oscillating Instances): Fractional mass of algorithm and OPT over time in one of the clusters (left) and corresponding dynamic approximation ratios (right), for k=1,2,3,4 .
Figure 8: (Scale Changes): Fractional mass of algorithm over time in one of the clusters (left) and corresponding dynamic approximation ratios (right), for k=1,2,3 .
Figure 9: (Scale Changes): Dynamic approximation ratio plots with T=50000 , for k=2,3.
Figure 10: (Small Drifts): The optimal (black plus), deterministic (blue cross), and randomized (red diamond) solutions for one of the random instances (left) and approximation ratios – avg. and std. dev. over 10 random instances (right) for k=1,2,3 .