Sample-and-Search: An Effective Algorithm for Learning-Augmented k-Median Clustering in High dimensions

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.

We study the problem of k-means clustering on large datasets. The state-of-the-art for the problem is given by coresets-based approaches, which build small weighted summaries of the input and derive approximate solutions with rigorous quality guarantees from them. One of the most popular and advanced approaches to derive coresets for k-means is sensitivity sampling. However, sensitivity sampling requires to compute the importance of each input point with respect to the whole dataset over all possible choices of centers. Since the exact computation of such quantities is unfeasible, current approaches work by approximating the sensitivity values. Nevertheless, the runtime of such approaches is still impractical for large datasets. In this work, we propose to reduce the runtime of sensitivity-based approaches for k-means by leveraging predictions to approximate the importance of input points. We first formally prove that current theoretical results on coresets construction via sensitivity sampling hold for coarser approximations of sensitivities compared to the one required by existing approaches. This implies that even fairly noisy predictors can be leveraged for sensitivity-sampling approaches. We then propose a natural predictor, which applies to the common scenario where clustering is performed (over time) on a sequence of datasets from the same problem. We prove that when the datasets in the sequence come from the same (unknown) distribution, centers resulting in a low error on one dataset can be used as predictions for sensitivity sampling in subsequent datasets, with guarantees on their quality. We perform an extensive experimental evaluation showing that our approach significantly improves, in terms of clustering cost vs runtime, over uniform sampling and state-of-the-art sensitivity sampling approaches when applied to sequences of datasets.

Clustering problems such as $k$-means and $k$-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects. In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation, in graph metrics or in Euclidean space, where our algorithm matches or improves the state-of-the-art.