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1. Label consistency in overfitted generalized k-means

   Zhang, Linfan ; Amini, Arash A. ( January 2021 , Advances in neural information processing systems)

   Ranzato, M. ; Beygelzimer, A. ; Dauphin, Y. ; Liang, P.S. ; Vaughan, J. W. (Ed.)

   We provide theoretical guarantees for label consistency in generalized k-means problems, with an emphasis on the overfitted case where the number of clusters used by the algorithm is more than the ground truth. We provide conditions under which the estimated labels are close to a refinement of the true cluster labels. We consider both exact and approximate recovery of the labels. Our results hold for any constant-factor approximation to the k-means problem. The results are also model-free and only based on bounds on the maximum or average distance of the data points to the true cluster centers. These centers themselves are loosely defined and can be taken to be any set of points for which the aforementioned distances can be controlled. We show the usefulness of the results with applications to some manifold clustering problems. 

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2. Locally Private k-Means Clustering with Constant Multiplicative Approximation and Near-Optimal Additive Error

   https://doi.org/10.1609/aaai.v36i6.20565  

   Chaturvedi, Anamay ; Jones, Matthew ; Nguyen, Huy L. ( January 2022 , Proceedings of the AAAI Conference on Artificial Intelligence)

   Given a data set of size n in d'-dimensional Euclidean space, the k-means problem asks for a set of k points (called centers) such that the sum of the l_2^2-distances between the data points and the set of centers is minimized. Previous work on this problem in the local differential privacy setting shows how to achieve multiplicative approximation factors arbitrarily close to optimal, but suffers high additive error. The additive error has also been seen to be an issue in implementations of differentially private k-means clustering algorithms in both the central and local settings. In this work, we introduce a new locally private k-means clustering algorithm that achieves near-optimal additive error whilst retaining constant multiplicative approximation factors and round complexity. Concretely, given any c>sqrt(2), our algorithm achieves O(k^(1 + O(1/(2c^2-1))) * sqrt(d' n) * log d' * poly log n) additive error with an O(c^2) multiplicative approximation factor. 

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3. Convergence Rate of Stochastic k-means

   Tang, Cheng ; Monteleoni, Claire ( January 2017 , Proceedings of the 20th International Conference on Artificial Intelligence and Statistics)

   We analyze online (Bottou & Bengio, 1994) and mini-batch (Sculley, 2010) k-means variants. Both scale up the widely used Lloyd’s algorithm via stochastic approximation, and have become popular for large-scale clustering and unsupervised feature learning. We show, for the first time, that they have global convergence towards “local optima” at rate O(1/t) under general conditions. In addition, we show that if the dataset is clusterable, stochastic k-means with suitable initialization converges to an optimal k-means solution at rate O(1/t) with high probability. The k-means objective is non-convex and non-differentiable; we exploit ideas from non-convex gradient-based optimization by providing a novel characterization of the trajectory of the k-means algorithm on its solution space, and circumvent its non-differentiability via geometric insights about the k-means update. 

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4. Near-optimal Algorithms for Explainable k-Medians and k-Means

   Makarychev, Konstantin ; Shan, Liren ( July 2021 , International Conference on Machine Learning)

   null (Ed.)

   We consider the problem of explainable k-medians and k-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the k-medians or k-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is O(log k) competitive with k-medians with ℓ1 norm and O(k) competitive with k-means. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^{3}{2}k) competitive for k-medians with ℓ2 norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for k-medians; in this work, we prove a lower bound of Ω(k) for k-means. We also provide a lower bound of Ω(log k) for k-medians with ℓ2 norm. 

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5. Convergence of online k-means

   So, G. ; Mahajan, G. ; Dasgupta, S. ( January 2022 , Proceedings of The 25th International Conference on Artificial Intelligence and Statistics)

   We prove asymptotic convergence for a general class of k-means algorithms performed over streaming data from a distribution— the centers asymptotically converge to the set of stationary points of the k-means objective function. To do so, we show that online k-means over a distribution can be interpreted as stochastic gradient descent with a stochastic learning rate schedule. Then, we prove convergence by extending techniques used in optimization literature to handle settings where center-specific learning rates may depend on the past trajectory of the centers. 

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