We consider coresets for $k$-clustering problems, where the goal is to assign points to centers minimizing powers of distances. A popular example is the $k$-median objective $\sum_{p}\min_{c\in C}dist(p,C)$. Given a point set $P$, a coreset $\Omega$ is a small weighted subset that approximates the cost of $P$ for all candidate solutions $C$ up to a $(1\pm\varepsilon )$ multiplicative factor. In this paper, we give a sharp VC-dimension based analysis for coreset construction. As a consequence, we obtain improved $k$-median coreset bounds for the following metrics: Coresets of size $\tilde{O}\left(k\varepsilon^{-2}\right)$ for shortest path metrics in planar graphs, improving over the bounds $\tilde{O}\left(k\varepsilon^{-6}\right)$ by [Cohen-Addad, Saulpic, Schwiegelshohn, STOC'21] and $\tilde{O}\left(k^2\varepsilon^{-4}\right)$ by [Braverman, Jiang, Krauthgamer, Wu, SODA'21]. Coresets of size $\tilde{O}\left(kd\ell\varepsilon^{-2}\log m\right)$ for clustering $d$-dimensional polygonal curves of length at most $m$ with curves of length at most $\ell$ with respect to Frechet metrics, improving over the bounds $\tilde{O}\left(k^3d\ell\varepsilon^{-3}\log m\right)$ by [Braverman, Cohen-Addad, Jiang, Krauthgamer, Schwiegelshohn, Toftrup, and Wu, FOCS'22] and $\tilde{O}\left(k^2d\ell\varepsilon^{-2}\log m \log |P|\right)$ by [Conradi, Kolbe, Psarros, Rohde, SoCG'24].

The $\ell_2^2$ min-sum $k$-clustering problem is to partition an input set into clusters $C_1,\ldots,C_k$ to minimize $\sum_{i=1}^k\sum_{p,q\in C_i}\|p-q\|_2^2$. Although $\ell_2^2$ min-sum $k$-clustering is NP-hard, it is not known whether it is NP-hard to approximate $\ell_2^2$ min-sum $k$-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the $\ell_2^2$ min-sum $k$-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than $1.056$ and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving the first $(1+\varepsilon)$-coreset construction for $\ell_2^2$ min-sum $k$-clustering. Our coreset uses $\mathcal{O}\left(k^{\varepsilon^{-4}}\right)$ space and can be leveraged to achieve a polynomial-time approximation scheme with runtime $nd\cdot f(k,\varepsilon^{-1})$, where $d$ is the underlying dimension of the input dataset and $f$ is a fixed function. Finally, we consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label $i\in[k]$ for input point, thereby implicitly partitioning the input dataset into $k$ clusters that induce an approximately optimal solution, up to some amount of adversarial error $\alpha\in\left[0,\frac{1}{2}\right)$. We give a polynomial-time algorithm that outputs a $\frac{1+\gamma\alpha}{(1-\alpha)^2}$-approximation to $\ell_2^2$ min-sum $k$-clustering, for a fixed constant $\gamma>0$.

We explore the use of distributed differentially private computations across multiple servers, balancing the tradeoff between the error introduced by the differentially private mechanism and the computational efficiency of the resulting distributed algorithm. We introduce the linear-transformation model, where clients have access to a trusted platform capable of applying a public matrix to their inputs. Such computations can be securely distributed across multiple servers using simple and efficient secure multiparty computation techniques. The linear-transformation model serves as an intermediate model between the highly expressive central model and the minimal local model. In the central model, clients have access to a trusted platform capable of applying any function to their inputs. However, this expressiveness comes at a cost, as it is often expensive to distribute such computations, leading to the central model typically being implemented by a single trusted server. In contrast, the local model assumes no trusted platform, which forces clients to add significant noise to their data. The linear-transformation model avoids the single point of failure for privacy present in the central model, while also mitigating the high noise required in the local model. We demonstrate that linear transformations are very useful for differential privacy, allowing for the computation of linear sketches of input data. These sketches largely preserve utility for tasks such as private low-rank approximation and private ridge regression, while introducing only minimal error, critically independent of the number of clients. Previously, such accuracy had only been achieved in the more expressive central model.

We study the theoretical and practical runtime limits of k-means and k-median clustering on large datasets. Since effectively all clustering methods are slower than the time it takes to read the dataset, the fastest approach is to quickly compress the data and perform the clustering on the compressed representation. Unfortunately, there is no universal best choice for compressing the number of points - while random sampling runs in sublinear time and coresets provide theoretical guarantees, the former does not enforce accuracy while the latter is too slow as the numbers of points and clusters grow. Indeed, it has been conjectured that any sensitivity-based coreset construction requires super-linear time in the dataset size. We examine this relationship by first showing that there does exist an algorithm that obtains coresets via sensitivity sampling in effectively linear time - within log-factors of the time it takes to read the data. Any approach that significantly improves on this must then resort to practical heuristics, leading us to consider the spectrum of sampling strategies across both real and artificial datasets in the static and streaming settings. Through this, we show the conditions in which coresets are necessary for preserving cluster validity as well as the settings in which faster, cruder sampling strategies are sufficient. As a result, we provide a comprehensive theoretical and practical blueprint for effective clustering regardless of data size. Our code is publicly available and has scripts to recreate the experiments.

Given a set of points, clustering consists of finding a partition of a point set into $k$ clusters such that the center to which a point is assigned is as close as possible. Most commonly, centers are points themselves, which leads to the famous $k$-median and $k$-means objectives. One may also choose centers to be $j$ dimensional subspaces, which gives rise to subspace clustering. In this paper, we consider learning bounds for these problems. That is, given a set of $n$ samples $P$ drawn independently from some unknown, but fixed distribution $\mathcal{D}$, how quickly does a solution computed on $P$ converge to the optimal clustering of $\mathcal{D}$? We give several near optimal results. In particular, For center-based objectives, we show a convergence rate of $\tilde{O}\left(\sqrt{{k}/{n}}\right)$. This matches the known optimal bounds of [Fefferman, Mitter, and Narayanan, Journal of the Mathematical Society 2016] and [Bartlett, Linder, and Lugosi, IEEE Trans. Inf. Theory 1998] for $k$-means and extends it to other important objectives such as $k$-median. For subspace clustering with $j$-dimensional subspaces, we show a convergence rate of $\tilde{O}\left(\sqrt{\frac{kj^2}{n}}\right)$. These are the first provable bounds for most of these problems. For the specific case of projective clustering, which generalizes $k$-means, we show a convergence rate of $\Omega\left(\sqrt{\frac{kj}{n}}\right)$ is necessary, thereby proving that the bounds from [Fefferman, Mitter, and Narayanan, Journal of the Mathematical Society 2016] are essentially optimal.

We study oblivious sketching for $k$-sparse linear regression under various loss functions such as an $\ell_p$ norm, or from a broad class of hinge-like loss functions, which includes the logistic and ReLU losses. We show that for sparse $\ell_2$ norm regression, there is a distribution over oblivious sketches with $\Theta(k\log(d/k)/\varepsilon^2)$ rows, which is tight up to a constant factor. This extends to $\ell_p$ loss with an additional additive $O(k\log(k/\varepsilon)/\varepsilon^2)$ term in the upper bound. This establishes a surprising separation from the related sparse recovery problem, which is an important special case of sparse regression. For this problem, under the $\ell_2$ norm, we observe an upper bound of $O(k \log (d)/\varepsilon + k\log(k/\varepsilon)/\varepsilon^2)$ rows, showing that sparse recovery is strictly easier to sketch than sparse regression. For sparse regression under hinge-like loss functions including sparse logistic and sparse ReLU regression, we give the first known sketching bounds that achieve $o(d)$ rows showing that $O(\mu^2 k\log(\mu n d/\varepsilon)/\varepsilon^2)$ rows suffice, where $\mu$ is a natural complexity parameter needed to obtain relative error bounds for these loss functions. We again show that this dimension is tight, up to lower order terms and the dependence on $\mu$. Finally, we show that similar sketching bounds can be achieved for LASSO regression, a popular convex relaxation of sparse regression, where one aims to minimize $\|Ax-b\|_2^2+\lambda\|x\|_1$ over $x\in\mathbb{R}^d$. We show that sketching dimension $O(\log(d)/(\lambda \varepsilon)^2)$ suffices and that the dependence on $d$ and $\lambda$ is tight.

The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise distances to within $1 \pm \varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m \times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s \|x\|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JACM'14) shows that $s = O(\varepsilon ^{-1} \lg n)$ suffices. This is almost matched by a lower bound of $s = \Omega(\varepsilon ^{-1} \lg n/\lg(1/\varepsilon))$ by Nelson and Nguyen (STOC'13). Previous work thus suggests that we have near-optimal embeddings. In this work, we revisit sparse embeddings and identify a loophole in the lower bound. Concretely, it requires $d \geq n$, which in many applications is unrealistic. We exploit this loophole to give a sparser embedding when $d = o(n)$, achieving $s = O(\varepsilon^{-1}(\lg n/\lg(1/\varepsilon)+\lg^{2/3}n \lg^{1/3} d))$. We also complement our analysis by strengthening the lower bound of Nelson and Nguyen to hold also when $d \ll n$, thereby matching the first term in our new sparsity upper bound. Finally, we also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.

Given a set of $n$ points in $d$ dimensions, the Euclidean $k$-means problem (resp. the Euclidean $k$-median problem) consists of finding $k$ centers such that the sum of squared distances (resp. sum of distances) from every point to its closest center is minimized. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a $(1\pm \varepsilon)$ factor. The current state of the art coreset size is $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for Euclidean $k$-means and $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for Euclidean $k$-median. The best known lower bound for both problems is $\Omega(k \varepsilon^{-2})$. In this paper, we improve the upper bounds $\tilde O(\min(k^{3/2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for $k$-means and $\tilde O(\min(k^{4/3} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for $k$-median. In particular, ours is the first provable bound that breaks through the $k^2$ barrier while retaining an optimal dependency on $\varepsilon$.

As freelancing work keeps on growing almost everywhere due to a sharp decrease in communication costs and to the widespread of Internet-based labour marketplaces (e.g., guru.com, feelancer.com, mturk.com, upwork.com), many researchers and practitioners have started exploring the benefits of outsourcing and crowdsourcing. Since employers often use these platforms to find a group of workers to complete a specific task, researchers have focused their efforts on the study of team formation and matching algorithms and on the design of effective incentive schemes. Nevertheless, just recently, several concerns have been raised on possibly unfair biases introduced through the algorithms used to carry out these selection and matching procedures. For this reason, researchers have started studying the fairness of algorithms related to these online marketplaces, looking for intelligent ways to overcome the algorithmic bias that frequently arises. Broadly speaking, the aim is to guarantee that, for example, the process of hiring workers through the use of machine learning and algorithmic data analysis tools does not discriminate, even unintentionally, on grounds of nationality or gender. In this short paper, we define the Fair Team Formation problem in the following way: given an online labour marketplace where each worker possesses one or more skills, and where all workers are divided into two or more not overlapping classes (for examples, men and women), we want to design an algorithm that is able to find a team with all the skills needed to complete a given task, and that has the same number of people from all classes. We provide inapproximability results for the Fair Team Formation problem together with four algorithms for the problem itself. We also tested the effectiveness of our algorithmic solutions by performing experiments using real data from an online labor marketplace.

Coresets are one of the central methods to facilitate the analysis of large data. We continue a recent line of research applying the theory of coresets to logistic regression. First, we show the negative result that no strongly sublinear sized coresets exist for logistic regression. To deal with intractable worst-case instances we introduce a complexity measure $\mu(X)$, which quantifies the hardness of compressing a data set for logistic regression. $\mu(X)$ has an intuitive statistical interpretation that may be of independent interest. For data sets with bounded $\mu(X)$-complexity, we show that a novel sensitivity sampling scheme produces the first provably sublinear $(1\pm\eps)$-coreset. We illustrate the performance of our method by comparing to uniform sampling as well as to state of the art methods in the area. The experiments are conducted on real world benchmark data for logistic regression.