Approximation Algorithms for Cost-Balanced Clustering

Published in Preprint, 2019

Clustering points in the Euclidean space is a fundamental problem in the theory of algorithms and in unsupervised learning. Various clustering objectives to quantify the quality of clustering have been proposed and studied; the $k$-means and $k$-median clustering objective are the most popular ones. In some cases, the $k$-means or the $k$-median objective may result in a few clusters of very large cost and many clusters of extremely small cost. Even when the optimal clusters are balanced in size, some of them may have a huge variance. This is undesirable for quantization or when we have a budget constraint on the cost of each cluster. Motivated by this, we study the cost-balanced $k$-means and the cost-balanced $k$-median problem. For a $k$-clustering $O_1, \ldots, O_k$ of a given set of $n$ points $X \subset \mathbb R^d$, we define its cost-balanced $k$-means cost as:

$ \boxed{\mathcal K ({O_1, \ldots, O_k}) \stackrel{def}{=} \max_{l \in [k]} \sum_{x \in O_l} \left\lVert{x - \mu_l}\right\rVert^2, \qquad \textrm{where } \mu_l = \frac{1}{\left\lvert{O_l}\right\rvert} \sum_{x \in O_l} x ~} $

In other words, we want to minimize the cost of the heaviest cluster or balance the cost of each cluster. For any $\varepsilon > 0$, we give a randomized algorithm with running time $\mathcal O\big({2^{poly \big({k/\varepsilon}\big)} n d }\big)$ that gives a $(1+\varepsilon)$-approximation to the optimal cost-balanced $k$-means and the similarly defined optimal cost-balanced $k$-median clustering, using $k$ clusters, with a constant probability. We define a more general version of the $k$-median clustering and the cost-balanced $k$-median clustering, and we name them $\ell_p$ cost $k$-clustering and $\ell_p$ cost-balanced $k$-clustering, respectively. Given a black-box algorithm which gives a constant factor approximation to the $\ell_p$ cost $k$-clustering, we show a procedure that runs in time $poly(n,k,p)$ which gives a bi-criteria $\mathcal O\big({1/\varepsilon^{1/p}}\big)$-approximation to the optimal $\ell_p$ cost-balanced $k$-clustering, using $(1+\varepsilon)k$ clusters.

Recommended citation: Amit Deshpande, Anand Louis, Deval Patel, Apoorv Singh. Approximation Algorithm for Cost-Balanced Clustering. In Submission.

On Euclidean k-means with α-Center Proximity

Published in AISTATS, 2019

$k$-means clustering is NP-hard in the worst case but previous work has shown efficient algorithms assuming the optimal $k$-means clusters are stable under additive or multiplicative perturbation of data. This has two caveats. First, we do not know how to efficiently verify this property of optimal solutions that are NP-hard to compute in the first place. Second, the stability assumptions required for polynomial time $k$-means algorithms are often unreasonable when compared to the ground-truth clusters in real-world data. A consequence of multiplicative perturbation resilience is center proximity, that is, every point is closer to the center of its own cluster than the center of any other cluster, by some multiplicative factor $ \alpha > 1 $.
We study the problem of minimizing the Euclidean $k$-means objective only over clusterings that satisfy $ \alpha $-center proximity. We give a simple algorithm to find the optimal $ \alpha $-center-proximal $k$-means clustering in running time exponential in k and $ 1/(\alpha−1) $ but linear in the number of points and the dimension. We define an analogous $ \alpha $-center proximity condition for outliers, and give similar algorithmic guarantees for $k$-means with outliers and $ \alpha $-center proximity. On the hardness side we show that for any $ \alpha’ > 1 $, there exists an $ \alpha \leq \alpha’ $, $ (\alpha > 1) $, and an $ \varepsilon_0>0 $ such that minimizing the $k$-means objective over clusterings that satisfy $ \alpha $-center proximity is NP-hard to approximate within a multiplicative $(1+\varepsilon_0)$ factor. Find the full paper here.

Recommended citation: Amit Deshpande, Anand Louis, Apoorv Singh ; Proceedings of Machine Learning Research, PMLR 89:2087-2095, 2019