A Tight VC-Dimension Analysis of Clustering Coresets with Applications

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].

Re-Invoke: Tool Invocation Rewriting for Zero-Shot Tool Retrieval

Recent advances in large language models (LLMs) have enabled autonomous agents with complex reasoning and task-fulfillment capabilities using a wide range of tools. However, effectively identifying the most relevant tools for a given task becomes a key bottleneck as the toolset size grows, hindering reliable tool utilization. To address this, we introduce Re-Invoke, an unsupervised tool retrieval method designed to scale effectively to large toolsets without training. Specifically, we first generate a diverse set of synthetic queries that comprehensively cover different aspects of the query space associated with each tool document during the tool indexing phase. Second, we leverage LLM's query understanding capabilities to extract key tool-related context and underlying intents from user queries during the inference phase. Finally, we employ a novel multi-view similarity ranking strategy based on intents to pinpoint the most relevant tools for each query. Our evaluation demonstrates that Re-Invoke significantly outperforms state-of-the-art alternatives in both single-tool and multi-tool scenarios, all within a fully unsupervised setting. Notably, on the ToolE datasets, we achieve a 20% relative improvement in nDCG@5 for single-tool retrieval and a 39% improvement for multi-tool retrieval.

Dynamic Correlation Clustering in Sublinear Update Time

We study the classic problem of correlation clustering in dynamic node streams. In this setting, nodes are either added or randomly deleted over time, and each node pair is connected by a positive or negative edge. The objective is to continuously find a partition which minimizes the sum of positive edges crossing clusters and negative edges within clusters. We present an algorithm that maintains an $O(1)$-approximation with $O$(polylog $n$) amortized update time. Prior to our work, Behnezhad, Charikar, Ma, and L. Tan achieved a $5$-approximation with $O(1)$ expected update time in edge streams which translates in node streams to an $O(D)$-update time where $D$ is the maximum possible degree. Finally we complement our theoretical analysis with experiments on real world data.

Perturb-and-Project: Differentially Private Similarities and Marginals

We revisit the input perturbations framework for differential privacy where noise is added to the input $A\in \mathcal{S}$ and the result is then projected back to the space of admissible datasets $\mathcal{S}$. Through this framework, we first design novel efficient algorithms to privately release pair-wise cosine similarities. Second, we derive a novel algorithm to compute $k$-way marginal queries over $n$ features. Prior work could achieve comparable guarantees only for $k$ even. Furthermore, we extend our results to $t$-sparse datasets, where our efficient algorithms yields novel, stronger guarantees whenever $t\le n^{5/6}/\log n\,.$ Finally, we provide a theoretical perspective on why \textit{fast} input perturbation algorithms works well in practice. The key technical ingredients behind our results are tight sum-of-squares certificates upper bounding the Gaussian complexity of sets of solutions.

A Near-Linear Time Approximation Algorithm for Beyond-Worst-Case Graph Clustering

We consider the semi-random graph model of [Makarychev, Makarychev and Vijayaraghavan, STOC'12], where, given a random bipartite graph with $\alpha$ edges and an unknown bipartition $(A, B)$ of the vertex set, an adversary can add arbitrary edges inside each community and remove arbitrary edges from the cut $(A, B)$ (i.e. all adversarial changes are \textit{monotone} with respect to the bipartition). For this model, a polynomial time algorithm is known to approximate the Balanced Cut problem up to value $O(\alpha)$ [MMV'12] as long as the cut $(A, B)$ has size $\Omega(\alpha)$. However, it consists of slow subroutines requiring optimal solutions for logarithmically many semidefinite programs. We study the fine-grained complexity of the problem and present the first near-linear time algorithm that achieves similar performances to that of [MMV'12]. Our algorithm runs in time $O(|V(G)|^{1+o(1)} + |E(G)|^{1+o(1)})$ and finds a balanced cut of value $O(\alpha)$. Our approach appears easily extendible to related problem, such as Sparsest Cut, and also yields an near-linear time $O(1)$-approximation to Dagupta's objective function for hierarchical clustering [Dasgupta, STOC'16] for the semi-random hierarchical stochastic block model inputs of [Cohen-Addad, Kanade, Mallmann-Trenn, Mathieu, JACM'19].

Multi-View Stochastic Block Models

Graph clustering is a central topic in unsupervised learning with a multitude of practical applications. In recent years, multi-view graph clustering has gained a lot of attention for its applicability to real-world instances where one has access to multiple data sources. In this paper we formalize a new family of models, called \textit{multi-view stochastic block models} that captures this setting. For this model, we first study efficient algorithms that naively work on the union of multiple graphs. Then, we introduce a new efficient algorithm that provably outperforms previous approaches by analyzing the structure of each graph separately. Furthermore, we complement our results with an information-theoretic lower bound studying the limits of what can be done in this model. Finally, we corroborate our results with experimental evaluations.

Data-Efficient Learning via Clustering-Based Sensitivity Sampling: Foundation Models and Beyond

We study the data selection problem, whose aim is to select a small representative subset of data that can be used to efficiently train a machine learning model. We present a new data selection approach based on $k$-means clustering and sensitivity sampling. Assuming access to an embedding representation of the data with respect to which the model loss is H\"older continuous, our approach provably allows selecting a set of ``typical'' $k + 1/\varepsilon^2$ elements whose average loss corresponds to the average loss of the whole dataset, up to a multiplicative $(1\pm\varepsilon)$ factor and an additive $\varepsilon \lambda \Phi_k$, where $\Phi_k$ represents the $k$-means cost for the input embeddings and $\lambda$ is the H\"older constant. We furthermore demonstrate the performance and scalability of our approach on fine-tuning foundation models and show that it outperforms state-of-the-art methods. We also show how it can be applied on linear regression, leading to a new sampling strategy that surprisingly matches the performances of leverage score sampling, while being conceptually simpler and more scalable.

A Scalable Algorithm for Individually Fair K-means Clustering

We present a scalable algorithm for the individually fair ($p$, $k$)-clustering problem introduced by Jung et al. and Mahabadi et al. Given $n$ points $P$ in a metric space, let $\delta(x)$ for $x\in P$ be the radius of the smallest ball around $x$ containing at least $n / k$ points. A clustering is then called individually fair if it has centers within distance $\delta(x)$ of $x$ for each $x\in P$. While good approximation algorithms are known for this problem no efficient practical algorithms with good theoretical guarantees have been presented. We design the first fast local-search algorithm that runs in ~$O(nk^2)$ time and obtains a bicriteria $(O(1), 6)$ approximation. Then we show empirically that not only is our algorithm much faster than prior work, but it also produces lower-cost solutions.

A quasi-polynomial time algorithm for Multi-Dimensional Scaling via LP hierarchies

Multi-dimensional Scaling (MDS) is a family of methods for embedding pair-wise dissimilarities between $n$ objects into low-dimensional space. MDS is widely used as a data visualization tool in the social and biological sciences, statistics, and machine learning. We study the Kamada-Kawai formulation of MDS: given a set of non-negative dissimilarities $\{d_{i,j}\}_{i , j \in [n]}$ over $n$ points, the goal is to find an embedding $\{x_1,\dots,x_n\} \subset \mathbb{R}^k$ that minimizes \[ \text{OPT} = \min_{x} \mathbb{E}_{i,j \in [n]} \left[ \left(1-\frac{\|x_i - x_j\|}{d_{i,j}}\right)^2 \right] \] Despite its popularity, our theoretical understanding of MDS is extremely limited. Recently, Demaine, Hesterberg, Koehler, Lynch, and Urschel (arXiv:2109.11505) gave the first approximation algorithm with provable guarantees for Kamada-Kawai, which achieves an embedding with cost $\text{OPT} +\epsilon$ in $n^2 \cdot 2^{\tilde{\mathcal{O}}(k \Delta^4 / \epsilon^2)}$ time, where $\Delta$ is the aspect ratio of the input dissimilarities. In this work, we give the first approximation algorithm for MDS with quasi-polynomial dependency on $\Delta$: for target dimension $k$, we achieve a solution with cost $\mathcal{O}(\text{OPT}^{ \hspace{0.04in}1/k } \cdot \log(\Delta/\epsilon) )+ \epsilon$ in time $n^{ \mathcal{O}(1)} \cdot 2^{\tilde{\mathcal{O}}( k^2 (\log(\Delta)/\epsilon)^{k/2 + 1} ) }$. Our approach is based on a novel analysis of a conditioning-based rounding scheme for the Sherali-Adams LP Hierarchy. Crucially, our analysis exploits the geometry of low-dimensional Euclidean space, allowing us to avoid an exponential dependence on the aspect ratio $\Delta$. We believe our geometry-aware treatment of the Sherali-Adams Hierarchy is an important step towards developing general-purpose techniques for efficient metric optimization algorithms.

Multi-Swap $k$-Means++

The $k$-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is often the practitioners' choice algorithm for optimizing the popular $k$-means clustering objective and is known to give an $O(\log k)$-approximation in expectation. To obtain higher quality solutions, Lattanzi and Sohler (ICML 2019) proposed augmenting $k$-means++ with $O(k \log \log k)$ local search steps obtained through the $k$-means++ sampling distribution to yield a $c$-approximation to the $k$-means clustering problem, where $c$ is a large absolute constant. Here we generalize and extend their local search algorithm by considering larger and more sophisticated local search neighborhoods hence allowing to swap multiple centers at the same time. Our algorithm achieves a $9 + \varepsilon$ approximation ratio, which is the best possible for local search. Importantly we show that our approach yields substantial practical improvements, we show significant quality improvements over the approach of Lattanzi and Sohler (ICML 2019) on several datasets.