Vincent Cohen-Addad

Moses Charikar, Vincent Cohen-Addad, Ruiquan Gao et al.

Clustering is a basic task in data analysis and machine learning, and the optimization of clustering objectives are well-studied optimization problems; amongst these, the $k$-Means objective is arguably the most well known. Given a collection of points in a metric space, the goal is to partition them into $k$ clusters, each with an associated center, so as to minimize the sum of squared distances of points to their cluster centers. In this paper, we present a polynomial-time $3+2\sqrt{2}+ε<5.83$-approximation algorithm for $k$-Means in general metrics. This substantially improves on the current-best $(9+ε)$-approximation in [Ahmadian, Norouzi-Fard, Svensson, Ward - FOCS'17, SICOMP'20], and even slightly improves on the $5.92$-approximation in [Cohen-Addad, Esfandiari, Mirrokni, Narayanan - STOC'22] for the Euclidean special case. A natural approach for $k$-Means is to leverage Lagrangian Multiplier Preserving (LMP) approximations for the facility location problem. The previous best results for $k$-Means build upon an adaptation of an LMP $3$-approximation for facility location with metric connection costs in [Jain, Vazirani - J.ACM'01] based on a primal-dual method, rather than on the improved LMP greedy $2$-approximation for the same problem in [Jain, Mahdian, Markakis, Saberi, Vazirani - J.ACM'03]. The barrier to using the improved LMP algorithm was that no adaptation of this algorithm and its analysis to the case of squared metric connection costs was known (since squared distances violate triangle inequality). Our main contribution is overcoming this barrier by providing such an adaptation. This new LMP approximation algorithm is then combined with the framework recently introduced in [Cohen-Addad, Grandoni, Lee, Schwiegelshohn, Svensson - STOC'25] for the related (metric) $k$-Median problem.

Vincent Cohen-Addad, Hossein Esfandiari, Vahab Mirrokni et al.

Motivated by data analysis and machine learning applications, we consider the popular high-dimensional Euclidean $k$-median and $k$-means problems. We propose a new primal-dual algorithm, inspired by the classic algorithm of Jain and Vazirani and the recent algorithm of Ahmadian, Norouzi-Fard, Svensson, and Ward. Our algorithm achieves an approximation ratio of $2.406$ and $5.912$ for Euclidean $k$-median and $k$-means, respectively, improving upon the 2.633 approximation ratio of Ahmadian et al. and the 6.1291 approximation ratio of Grandoni, Ostrovsky, Rabani, Schulman, and Venkat. Our techniques involve a much stronger exploitation of the Euclidean metric than previous work on Euclidean clustering. In addition, we introduce a new method of removing excess centers using a variant of independent sets over graphs that we dub a "nested quasi-independent set". In turn, this technique may be of interest for other optimization problems in Euclidean and $\ell_p$ metric spaces.

Yanfei Chen, Jinsung Yoon, Devendra Singh Sachan et al. · mila

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.

Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee et al.

In the classical NP-hard metric $k$-median problem, we are given a set of $n$ clients and centers with metric distances between them, along with an integer parameter $k\geq 1$. The objective is to select a subset of $k$ open centers that minimizes the total distance from each client to its closest open center. In their seminal work, Jain, Mahdian, Markakis, Saberi, and Vazirani presented the Greedy algorithm for facility location, which implies a $2$-approximation algorithm for $k$-median that opens $k$ centers in expectation. Since then, substantial research has aimed at narrowing the gap between their algorithm and the best achievable approximation by an algorithm guaranteed to open exactly $k$ centers. During the last decade, all improvements have been achieved by leveraging their algorithm or a small improvement thereof, followed by a second step called bi-point rounding, which inherently increases the approximation guarantee. Our main result closes this gap: for any $ε>0$, we present a $(2+ε)$-approximation algorithm for $k$-median, improving the previous best-known approximation factor of $2.613$. Our approach builds on a combination of two algorithms. First, we present a non-trivial modification of the Greedy algorithm that operates with $O(\log n/ε^2)$ adaptive phases. Through a novel walk-between-solutions approach, this enables us to construct a $(2+ε)$-approximation algorithm for $k$-median that consistently opens at most $k + O(\log n{/ε^2})$ centers. Second, we develop a novel $(2+ε)$-approximation algorithm tailored for stable instances, where removing any center from an optimal solution increases the cost by at least an $Ω(ε^3/\log n)$ fraction. Achieving this involves a sampling approach inspired by the $k$-means++ algorithm and a reduction to submodular optimization subject to a partition matroid.

Vincent Cohen-Addad, Alessandro Epasto, Silvio Lattanzi et al.

We study the private $k$-median and $k$-means clustering problem in $d$ dimensional Euclidean space. By leveraging tree embeddings, we give an efficient and easy to implement algorithm, that is empirically competitive with state of the art non private methods. We prove that our method computes a solution with cost at most $O(d^{3/2}\log n)\cdot OPT + O(k d^2 \log^2 n / ε^2)$, where $ε$ is the privacy guarantee. (The dimension term, $d$, can be replaced with $O(\log k)$ using standard dimension reduction techniques.) Although the worst-case guarantee is worse than that of state of the art private clustering methods, the algorithm we propose is practical, runs in near-linear, $\tilde{O}(nkd)$, time and scales to tens of millions of points. We also show that our method is amenable to parallelization in large-scale distributed computing environments. In particular we show that our private algorithms can be implemented in logarithmic number of MPC rounds in the sublinear memory regime. Finally, we complement our theoretical analysis with an empirical evaluation demonstrating the algorithm's efficiency and accuracy in comparison to other privacy clustering baselines.

David P. Woodruff, Vincent Cohen-Addad, Lalit Jain et al.

Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.

Jacob Imola, Alessandro Epasto, Mohammad Mahdian et al.

Hierarchical Clustering is a popular unsupervised machine learning method with decades of history and numerous applications. We initiate the study of differentially private approximation algorithms for hierarchical clustering under the rigorous framework introduced by (Dasgupta, 2016). We show strong lower bounds for the problem: that any $ε$-DP algorithm must exhibit $O(|V|^2/ ε)$-additive error for an input dataset $V$. Then, we exhibit a polynomial-time approximation algorithm with $O(|V|^{2.5}/ ε)$-additive error, and an exponential-time algorithm that meets the lower bound. To overcome the lower bound, we focus on the stochastic block model, a popular model of graphs, and, with a separation assumption on the blocks, propose a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly. Finally, we perform an empirical study of our algorithms and validate their performance.

Hongjie Chen, Vincent Cohen-Addad, Tommaso d'Orsi et al.

We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient $(ε, δ)$-differentially private algorithm for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. For the latter, we design an $(ε, δ)$-differentially private algorithm that recovers the centers of the $k$-mixture when the minimum separation is at least $ O(k^{1/t}\sqrt{t})$. For all choices of $t$, this algorithm requires sample complexity $n\geq k^{O(1)}d^{O(t)}$ and time complexity $(nd)^{O(t)}$. Prior work required minimum separation at least $O(\sqrt{k})$ as well as an explicit upper bound on the Euclidean norm of the centers.

Nairen Cao, Vincent Cohen-Addad, Euiwoong Lee et al.

Correlation clustering is a well-studied problem, first proposed by Bansal, Blum, and Chawla [Mach. Learn. '04]. The input is an unweighted, undirected graph. The problem is to cluster the vertices so as to minimize the number of edges between vertices in different clusters and missing edges between vertices inside the same cluster. This problem has a wide application in data mining and machine learning. We introduce a general framework that transforms existing static correlation clustering algorithms into fully-dynamic ones that work against an adaptive adversary. We show how to apply our framework to known efficient correlation clustering algorithms, starting from the classic 3-approximate Pivot algorithm from Ailon, Charikar and Newman [JACM'08]. Applied to the most recent sublinear $1.485$-approximation algorithm from Cao, Cohen-Addad, Lee, Li, Lolck, Newman, Thorup, Vogl, Yan and Zhang [STOC'25], we get a $1.485$-approximation fully-dynamic algorithm that works with worst-case constant update time. The original static algorithm gets its approximation factor with constant probability, and we get the same against an adaptive adversary in the sense that for any given update step, not known to our algorithm, our solution is a $1.485$-approximation with constant probability when we reach this update. Most of previous dynamic algorithms, including the celebrated result from Behnezhad, Charikar, Ma and Tan [FOCS'19], had approximation factors around $3$ in expectation, and they could only handle an oblivious adversary. A recent algorithm by Braverman, Dharangutte, Pai, Shah, and Wang [AISTATS'25] could handle an adaptive adversary, but it has a large unspecified constant approximation ratio. This contrasts with our general transformation, which works with all the best approximation factors known for the static case.

Vincent Cohen-Addad, Chenglin Fan, Silvio Lattanzi et al.

Correlation clustering is a central problem in unsupervised learning, with applications spanning community detection, duplicate detection, automated labelling and many more. In the correlation clustering problem one receives as input a set of nodes and for each node a list of co-clustering preferences, and the goal is to output a clustering that minimizes the disagreement with the specified nodes' preferences. In this paper, we introduce a simple and computationally efficient algorithm for the correlation clustering problem with provable privacy guarantees. Our approximation guarantees are stronger than those shown in prior work and are optimal up to logarithmic factors.

Miltiadis Stouras, Vincent Cohen-Addad, Silvio Lattanzi et al.

Retrieval-augmented generation (RAG) systems typically rely on a single retriever and a single set of hyperparameters, despite facing highly heterogeneous queries that range from simple factoid questions to complex multi-hop reasoning. We propose a method that automatically selects a small, diverse subset of retrievers (a portfolio) from a large pool of candidates, to cover different regions of the target query distribution. We formalize this setting via an expected best-of-$k$ objective over the query distribution and show that it admits an efficient portfolio construction algorithm with near-optimal guarantees. Across multiple QA benchmarks, our learned portfolios and router pipeline consistently outperform single-retriever and naive multi-retriever baselines on both retrieval metrics and answer quality. In addition, compared to inference-time hyperparameter tuning approaches, fixed portfolios enable parallel retrieval and LLM calls, achieving comparable (and sometimes better) accuracy with substantially lower latency and token cost.

Limor Gultchin, Vincent Cohen-Addad, Sophie Giffard-Roisin et al.

Among the various aspects of algorithmic fairness studied in recent years, the tension between satisfying both \textit{sufficiency} and \textit{separation} -- e.g. the ratios of positive or negative predictive values, and false positive or false negative rates across groups -- has received much attention. Following a debate sparked by COMPAS, a criminal justice predictive system, the academic community has responded by laying out important theoretical understanding, showing that one cannot achieve both with an imperfect predictor when there is no equal distribution of labels across the groups. In this paper, we shed more light on what might be still possible beyond the impossibility -- the existence of a trade-off means we should aim to find a good balance within it. After refining the existing theoretical result, we propose an objective that aims to balance \textit{sufficiency} and \textit{separation} measures, while maintaining similar accuracy levels. We show the use of such an objective in two empirical case studies, one involving a multi-objective framework, and the other fine-tuning of a model pre-trained for accuracy. We show promising results, where better trade-offs are achieved compared to existing alternatives.

Natalia Ponomareva, Zheng Xu, H. Brendan McMahan et al.

High quality data is needed to unlock the full potential of AI for end users. However finding new sources of such data is getting harder: most publicly-available human generated data will soon have been used. Additionally, publicly available data often is not representative of users of a particular system -- for example, a research speech dataset of contractors interacting with an AI assistant will likely be more homogeneous, well articulated and self-censored than real world commands that end users will issue. Therefore unlocking high-quality data grounded in real user interactions is of vital interest. However, the direct use of user data comes with significant privacy risks. Differential Privacy (DP) is a well established framework for reasoning about and limiting information leakage, and is a gold standard for protecting user privacy. The focus of this work, \emph{Differentially Private Synthetic data}, refers to synthetic data that preserves the overall trends of source data,, while providing strong privacy guarantees to individuals that contributed to the source dataset. DP synthetic data can unlock the value of datasets that have previously been inaccessible due to privacy concerns and can replace the use of sensitive datasets that previously have only had rudimentary protections like ad-hoc rule-based anonymization. In this paper we explore the full suite of techniques surrounding DP synthetic data, the types of privacy protections they offer and the state-of-the-art for various modalities (image, tabular, text and decentralized). We outline all the components needed in a system that generates DP synthetic data, from sensitive data handling and preparation, to tracking the use and empirical privacy testing. We hope that work will result in increased adoption of DP synthetic data, spur additional research and increase trust in DP synthetic data approaches.

Lorenzo Beretta, Vincent Cohen-Addad, Silvio Lattanzi et al.

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.

Ainesh Bakshi, Vincent Cohen-Addad, Samuel B. Hopkins et al.

Metric embeddings are a widely used method in algorithm design, where generally a ``complex'' metric is embedded into a simpler, lower-dimensional one. Historically, the theoretical computer science community has focused on bi-Lipschitz embeddings, which guarantee that every pairwise distance is approximately preserved. In contrast, alternative embedding objectives that are commonly used in practice avoid bi-Lipschitz distortion; yet these approaches have received comparatively less study in theory. In this paper, we focus on Multi-dimensional Scaling (MDS), where we are given a set of non-negative dissimilarities $\{d_{i,j}\}_{i,j\in [n]}$ over $n$ points, and the goal is to find an embedding $\{x_1,\dots,x_n\} \subset R^k$ that minimizes $$\textrm{OPT}=\min_{x}\mathbb{E}_{i,j\in [n]}\left(1-\frac{\|x_i - x_j\|}{d_{i,j}}\right)^2.$$ Despite its popularity, our theoretical understanding of MDS is extremely limited. Recently, Demaine et. al. (arXiv:2109.11505) gave the first approximation algorithm with provable guarantees for this objective, which achieves an embedding in constant dimensional Euclidean space with cost $\textrm{OPT} +ε$ in $n^2\cdot 2^{\textrm{poly}(Δ/ε)}$ time, where $Δ$ is the aspect ratio of the input dissimilarities. For metrics that admit low-cost embeddings, $Δ$ scales polynomially in $n$. In this work, we give the first approximation algorithm for MDS with quasi-polynomial dependency on $Δ$: for constant dimensional Euclidean space, we achieve a solution with cost $O(\log Δ)\cdot \textrm{OPT}^{Ω(1)}+ε$ in time $n^{O(1)} \cdot 2^{\text{poly}((\log(Δ)/ε))}$. Our algorithms are based on a novel geometry-aware analysis of a conditional rounding of the Sherali-Adams LP Hierarchy, allowing us to avoid exponential dependency on the aspect ratio, which would typically result from this rounding.

Vincent Cohen-Addad, Karthik C. S., David Saulpic et al.

The $k$-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find $k$ representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of $k$-means in Euclidean spaces parameterized by the number of clusters, $k$. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a $(1+\varepsilon)$-approximation for high-dimensional Euclidean $k$-means in time $2^{(k/\varepsilon)^{O(1)}} \cdot dn^{O(1)}$. In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean $k$-means by showing that there is no $2^{(k/\varepsilon)^{1-o(1)}} \cdot n^{O(1)}$ time algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is $2^{\tilde{O}(k/\varepsilon)} + O(ndk)$. Furthermore, assuming XXH, we show that the seminal $O(n^{kd+1})$ runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for $k$-means is optimal for small values of $k$.

Vincent Cohen-Addad, Karthik C. S., David Saulpic et al.

The $k$-median and $k$-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the $k$-median (resp. $k$-means) problem is to find $k$ representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a $(1+\varepsilon)$-factor approximation in low-dimensional Euclidean metric for both the $k$-median and $k$-means problems in near-linear time $2^{(1/\varepsilon)^{O(d^2)}} n \cdot \text{polylog}(n)$ (where $d$ is the dimension and $n$ is the number of input points). We improve this running time to $2^{\tilde{O}(1/\varepsilon)^{d-1}} \cdot n \cdot \text{polylog}(n)$, and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no $2^{{o}(1/\varepsilon^{d-1})} n^{O(1)}$ algorithm achieving a $(1+\varepsilon)$-approximation for $k$-means.

Gheorghe Comanici, Eric Bieber, Mike Schaekermann et al. · amazon-science, baidu

In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving.

MohammadHossein Bateni, Vincent Cohen-Addad, Yuzhou Gu et al.

Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan for generating and combining a set of solutions can be thought of as an algorithm for reasoning using a probabilistic oracle. We introduce a theoretical framework for analyzing such reasoning algorithms. This framework formalizes the principles underlying popular techniques for iterative improvement and answer aggregation, providing a foundation for designing a new generation of more powerful reasoning methods. Unlike approaches for understanding models that rely on architectural specifics, our model is grounded in experimental evidence. As a result, it offers a general perspective that may extend to a wide range of current and future reasoning oracles.

Tom Wollschläger, Jannes Elstner, Simon Geisler et al.

The safety alignment of large language models (LLMs) can be circumvented through adversarially crafted inputs, yet the mechanisms by which these attacks bypass safety barriers remain poorly understood. Prior work suggests that a single refusal direction in the model's activation space determines whether an LLM refuses a request. In this study, we propose a novel gradient-based approach to representation engineering and use it to identify refusal directions. Contrary to prior work, we uncover multiple independent directions and even multi-dimensional concept cones that mediate refusal. Moreover, we show that orthogonality alone does not imply independence under intervention, motivating the notion of representational independence that accounts for both linear and non-linear effects. Using this framework, we identify mechanistically independent refusal directions. We show that refusal mechanisms in LLMs are governed by complex spatial structures and identify functionally independent directions, confirming that multiple distinct mechanisms drive refusal behavior. Our gradient-based approach uncovers these mechanisms and can further serve as a foundation for future work on understanding LLMs.

Kyriakos Axiotis, Vincent Cohen-Addad, Monika Henzinger et al.

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 Hölder 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 $\varepsilon λΦ_k$, where $Φ_k$ represents the $k$-means cost for the input embeddings and $λ$ is the Hölder 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.

MohammadHossein Bateni, Vincent Cohen-Addad, Alessandro Epasto et al.

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 $δ(x)$ 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 $δ(x)$ 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.

Simon Geisler, Tom Wollschläger, M. H. I. Abdalla et al.

To circumvent the alignment of large language models (LLMs), current optimization-based adversarial attacks usually craft adversarial prompts by maximizing the likelihood of a so-called affirmative response. An affirmative response is a manually designed start of a harmful answer to an inappropriate request. While it is often easy to craft prompts that yield a substantial likelihood for the affirmative response, the attacked model frequently does not complete the response in a harmful manner. Moreover, the affirmative objective is usually not adapted to model-specific preferences and essentially ignores the fact that LLMs output a distribution over responses. If low attack success under such an objective is taken as a measure of robustness, the true robustness might be grossly overestimated. To alleviate these flaws, we propose an adaptive and semantic optimization problem over the population of responses. We derive a generally applicable objective via the REINFORCE policy-gradient formalism and demonstrate its efficacy with the state-of-the-art jailbreak algorithms Greedy Coordinate Gradient (GCG) and Projected Gradient Descent (PGD). For example, our objective doubles the attack success rate (ASR) on Llama3 and increases the ASR from 2% to 50% with circuit breaker defense.

Vincent Cohen-Addad, Andrew Draganov, Matteo Russo et al.

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 $Ω$ 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].

Michael P. Brenner, Vincent Cohen-Addad, David Woodruff

This paper demonstrates that artificial intelligence can accelerate mathematical discovery by autonomously solving an open problem in theoretical physics. We present a neuro-symbolic system, combining the Gemini Deep Think large language model with a systematic Tree Search (TS) framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings. Specifically, the agent evaluated the core integral $I(N,α)$ for arbitrary loop geometries, directly improving upon recent AI-assisted attempts \cite{BCE+25} that only yielded partial asymptotic solutions. To substantiate our methodological claims regarding AI-accelerated discovery and to ensure transparency, we detail system prompts, search constraints, and intermittent feedback loops that guided the model. The agent identified a suite of 6 different analytical methods, the most elegant of which expands the kernel in Gegenbauer polynomials $C_l^{(3/2)}$ to naturally absorb the integrand's singularities. The methods lead to an asymptotic result for $I(N,α)$ at large $N$ that both agrees with numerical results and also connects to the continuous Feynman parameterization of Quantum Field Theory. We detail both the algorithmic methodology that enabled this discovery and the resulting mathematical derivations.

Mahdi Nikdan, Vincent Cohen-Addad, Dan Alistarh et al.

Effective data selection is critical for efficient training of modern Large Language Models (LLMs). This paper introduces Influence Distillation, a novel, mathematically-justified framework for data selection that employs second-order information to optimally weight training samples. By distilling each sample's influence on a target distribution, our method assigns model-specific weights that are used to select training data for LLM fine-tuning, guiding it toward strong performance on the target domain. We derive these optimal weights for both Gradient Descent and Adam optimizers. To ensure scalability and reduce computational cost, we propose a $\textit{landmark-based approximation}$: influence is precisely computed for a small subset of "landmark" samples and then efficiently propagated to all other samples to determine their weights. We validate Influence Distillation by applying it to instruction tuning on the Tulu V2 dataset, targeting a range of tasks including GSM8k, SQuAD, and MMLU, across several models from the Llama and Qwen families. Experiments show that Influence Distillation matches or outperforms state-of-the-art performance while achieving up to $3.5\times$ faster selection.

Vincent Cohen-Addad, Marina Drygala, Nathan Klein et al.

The Weighted Tree Augmentation Problem (WTAP) is a fundamental network design problem where the goal is to find a minimum-cost set of additional edges (links) to make an input tree 2-edge-connected. While a 2-approximation is standard and the integrality gap of the classic Cut LP relaxation is known to be at least 1.5, achieving approximation factors significantly below 2 has proven challenging. Recent advances of Traub and Zenklusen using local search culminated in a ratio of $1.5+ε$, establishing the state-of-the-art. In this work, we present a randomized approximation algorithm for WTAP with an approximation ratio below 1.49. Our approach is based on designing and rounding a strong linear programming relaxation for WTAP which incorporates variables that represent subsets of edges and the links used to cover them, inspired by lift-and-project methods like Sherali-Adams.

Sasan Tavakkol, Lin Chen, Max Springer et al.

Scarcity of labeled data, especially for rare events, hinders training effective machine learning models. This paper proposes SYNAPSE-G (Synthetic Augmentation for Positive Sampling via Expansion on Graphs), a novel pipeline leveraging Large Language Models (LLMs) to generate synthetic training data for rare event classification, addressing the cold-start problem. This synthetic data serve as seeds for semi-supervised label propagation on a similarity graph constructed between the seeds and a large unlabeled dataset. This identifies candidate positive examples, subsequently labeled by an oracle (human or LLM). The expanded dataset then trains/fine-tunes a classifier. We theoretically analyze how the quality (validity and diversity) of the synthetic data impacts the precision and recall of our method. Experiments on the imbalanced SST2 and MHS datasets demonstrate SYNAPSE-G's effectiveness in finding positive labels, outperforming baselines including nearest neighbor search.

Lorenzo Beretta, Vincent Cohen-Addad, Rajesh Jayaram et al.

We give a reduction from $(1+\varepsilon)$-approximate Earth Mover's Distance (EMD) to $(1+\varepsilon)$-approximate Closest Pair (CP). As a consequence, we improve the fastest known approximation algorithm for high-dimensional EMD. Here, given $p\in [1, 2]$ and two sets of $n$ points $X,Y \subseteq (\mathbb R^d,\ell_p)$, their EMD is the minimum cost of a perfect matching between $X$ and $Y$, where the cost of matching two vectors is their $\ell_p$ distance. Further, CP is the basic problem of finding a pair of points realizing $\min_{x \in X, y\in Y} ||x-y||_p$. Our contribution is twofold: we show that if a $(1+\varepsilon)$-approximate CP can be computed in time $n^{2-φ}$, then a $1+O(\varepsilon)$ approximation to EMD can be computed in time $n^{2-Ω(φ)}$; plugging in the fastest known algorithm for CP [Alman, Chan, Williams FOCS'16], we obtain a $(1+\varepsilon)$-approximation algorithm for EMD running in time $n^{2-\tildeΩ(\varepsilon^{1/3})}$ for high-dimensional point sets, which improves over the prior fastest running time of $n^{2-Ω(\varepsilon^2)}$ [Andoni, Zhang FOCS'23]. Our main technical contribution is a sublinear implementation of the Multiplicative Weights Update framework for EMD. Specifically, we demonstrate that the updates can be executed without ever explicitly computing or storing the weights; instead, we exploit the underlying geometric structure to perform the updates implicitly.

Felix Zhou, Samson Zhou, Vahab Mirrokni et al.

Deep neural networks often use large, high-quality datasets to achieve high performance on many machine learning tasks. When training involves potentially sensitive data, this process can raise privacy concerns, as large models have been shown to unintentionally memorize and reveal sensitive information, including reconstructing entire training samples. Differential privacy (DP) provides a robust framework for protecting individual data and in particular, a new approach to privately training deep neural networks is to approximate the input dataset with a privately generated synthetic dataset, before any subsequent training algorithm. We introduce a novel principled method for DP synthetic image embedding generation, based on fitting a Gaussian Mixture Model (GMM) in an appropriate embedding space using DP clustering. Our method provably learns a GMM under separation conditions. Empirically, a simple two-layer neural network trained on synthetically generated embeddings achieves state-of-the-art (SOTA) classification accuracy on standard benchmark datasets. Additionally, we demonstrate that our method can generate realistic synthetic images that achieve downstream classification accuracy comparable to SOTA methods. Our method is quite general, as the encoder and decoder modules can be freely substituted to suit different tasks. It is also highly scalable, consisting only of subroutines that scale linearly with the number of samples and/or can be implemented efficiently in distributed systems.

Sasan Tavakkol, Max Springer, Mohammadhossein Bateni et al.

Synthetic training data generation with Large Language Models (LLMs) like Google's Gemma and OpenAI's GPT offer a promising solution to the challenge of obtaining large, labeled datasets for training classifiers. When rapid model deployment is critical, such as in classifying emerging social media trends or combating new forms of online abuse tied to current events, the ability to generate training data is invaluable. While prior research has examined the comparability of synthetic data to human-labeled data, this study introduces a novel sampling algorithm, based on the maximum coverage problem, to select a representative subset from a synthetically generated dataset. Our results demonstrate that training a classifier on this contextually sampled subset achieves superior performance compared to training on the entire dataset. This "less is more" approach not only improves model accuracy but also reduces the volume of data required, leading to potentially more efficient model fine-tuning.

Vincent Cohen-Addad, Silvio Lattanzi, Andreas Maggiori et al.

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.

Vincent Cohen-Addad, Tommaso d'Orsi, Alessandro Epasto et al.

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.

Vincent Cohen-Addad, Tommaso d'Orsi, Silvio Lattanzi et al.

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.

Vincent Cohen-Addad, Tommaso d'Orsi, Aida Mousavifar

We consider the semi-random graph model of [Makarychev, Makarychev and Vijayaraghavan, STOC'12], where, given a random bipartite graph with $α$ 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 $O(α)$ [MMV'12] as long as the cut $(A, B)$ has size $Ω(α)$. 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 $O(α)$. 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].

Vincent Cohen-Addad, Kasper Green Larsen, David Saulpic et al.

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 $Ω(k \varepsilon^{-2})$. 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 $\varepsilon$.

Vincent Cohen-Addad, Kasper Green Larsen, David Saulpic et al.

Given a set of points in a metric space, the $(k,z)$-clustering problem consists of finding a set of $k$ points called centers, such that the sum of distances raised to the power of $z$ of every data point to its closest center is minimized. Special cases include the famous k-median problem ($z = 1$) and k-means problem ($z = 2$). The $k$-median and $k$-means problems are at the heart of modern data analysis and massive data applications have given raise to the notion of coreset: a small (weighted) subset of the input point set preserving the cost of any solution to the problem up to a multiplicative $(1 \pm \varepsilon)$ factor, hence reducing from large to small scale the input to the problem. In this paper, we present improved lower bounds for coresets in various metric spaces. In finite metrics consisting of $n$ points and doubling metrics with doubling constant $D$, we show that any coreset for $(k,z)$ clustering must consist of at least $Ω(k \varepsilon^{-2} \log n)$ and $Ω(k \varepsilon^{-2} D)$ points, respectively. Both bounds match previous upper bounds up to polylog factors. In Euclidean spaces, we show that any coreset for $(k,z)$ clustering must consists of at least $Ω(k\varepsilon^{-2})$ points. We complement these lower bounds with a coreset construction consisting of at most $\tilde{O}(k\varepsilon^{-2}\cdot \min(\varepsilon^{-z},k))$ points.

Amir Abboud, Mohammad Hossein Bateni, Vincent Cohen-Addad et al.

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=ω(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=Ω(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+ε)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

Vincent Cohen-Addad, Karthik C. S., Euiwoong Lee

K-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives in $\ell_p$-metrics. We introduce a new hypothesis called the Johnson Coverage Hypothesis (JCH), which roughly asserts that the well-studied max k-coverage problem on set systems is hard to approximate to a factor greater than 1-1/e, even when the membership graph of the set system is a subgraph of the Johnson graph. We then show that together with generalizations of the embedding techniques introduced by Cohen-Addad and Karthik (FOCS '19), JCH implies hardness of approximation results for k-median and k-means in $\ell_p$-metrics for factors which are close to the ones obtained for general metrics. In particular, assuming JCH we show that it is hard to approximate the k-means objective: $\bullet$ Discrete case: To a factor of 3.94 in the $\ell_1$-metric and to a factor of 1.73 in the $\ell_2$-metric; this improves upon the previous factor of 1.56 and 1.17 respectively, obtained under UGC. $\bullet$ Continuous case: To a factor of 2.10 in the $\ell_1$-metric and to a factor of 1.36 in the $\ell_2$-metric; this improves upon the previous factor of 1.07 in the $\ell_2$-metric obtained under UGC. We also obtain similar improvements under JCH for the k-median objective. Additionally, we prove a weak version of JCH using the work of Dinur et al. (SICOMP '05) on Hypergraph Vertex Cover, and recover all the results stated above of Cohen-Addad and Karthik (FOCS '19) to (nearly) the same inapproximability factors but now under the standard NP$\neq$P assumption (instead of UGC).

Vincent Cohen-Addad, Silvio Lattanzi, Slobodan Mitrović et al.

Correlation clustering is a central topic in unsupervised learning, with many applications in ML and data mining. In correlation clustering, one receives as input a signed graph and the goal is to partition it to minimize the number of disagreements. In this work we propose a massively parallel computation (MPC) algorithm for this problem that is considerably faster than prior work. In particular, our algorithm uses machines with memory sublinear in the number of nodes in the graph and returns a constant approximation while running only for a constant number of rounds. To the best of our knowledge, our algorithm is the first that can provably approximate a clustering problem on graphs using only a constant number of MPC rounds in the sublinear memory regime. We complement our analysis with an experimental analysis of our techniques.

Vincent Cohen-Addad, Silvio Lattanzi, Ashkan Norouzi-Fard et al.

$k$-means++ \cite{arthur2007k} is a widely used clustering algorithm that is easy to implement, has nice theoretical guarantees and strong empirical performance. Despite its wide adoption, $k$-means++ sometimes suffers from being slow on large data-sets so a natural question has been to obtain more efficient algorithms with similar guarantees. In this paper, we present a near linear time algorithm for $k$-means++ seeding. Interestingly our algorithm obtains the same theoretical guarantees as $k$-means++ and significantly improves earlier results on fast $k$-means++ seeding. Moreover, we show empirically that our algorithm is significantly faster than $k$-means++ and obtains solutions of equivalent quality.

Vincent Cohen-Addad, Karthik C. S., Euiwoong Lee

The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this paper, we significantly improve upon the hardness of approximation factors known in the literature for these objectives. We show that if the input lies in a general metric space, it is NP-hard to approximate: $\bullet$ Continuous k-median to a factor of $2-o(1)$; this improves upon the previous inapproximability factor of 1.36 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ Continuous k-means to a factor of $4- o(1)$; this improves upon the previous inapproximability factor of 2.10 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ k-minsum to a factor of $1.415$; this improves upon the APX-hardness shown by Guruswami and Indyk (SODA '03). Our results shed new and perhaps counter-intuitive light on the differences between clustering problems in the continuous setting versus the discrete setting (where the candidate centers are given as part of the input).

Vincent Cohen-Addad, Karthik C. S., Guillaume Lagarde

A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic linkage algorithms (single, average, or complete). However, these methods on a data set of $n$ points in $Ω(\log n)$ dimensions exhibit a quite prohibitive running time of $Θ(n^2)$. In this paper, we provide a new algorithm which takes as input a set of points $P$ in $\mathbb{R}^d$, and for every $c\ge 1$, runs in time $n^{1+\fracρ{c^2}}$ (for some universal constant $ρ>1$) to output an ultrametric $Δ$ such that for any two points $u,v$ in $P$, we have $Δ(u,v)$ is within a multiplicative factor of $5c$ to the distance between $u$ and $v$ in the "best" ultrametric representation of $P$. Here, the best ultrametric is the ultrametric $\tildeΔ$ that minimizes the maximum distance distortion with respect to the $\ell_2$ distance, namely that minimizes $\underset{u,v \in P}{\max}\ \frac{\tildeΔ(u,v)}{\|u-v\|_2}$. We complement the above result by showing that under popular complexity theoretic assumptions, for every constant $\varepsilon>0$, no algorithm with running time $n^{2-\varepsilon}$ can distinguish between inputs in $\ell_\infty$-metric that admit isometric embedding and those that incur a distortion of $\frac{3}{2}$. Finally, we present empirical evaluation on classic machine learning datasets and show that the output of our algorithm is comparable to the output of the linkage algorithms while achieving a much faster running time.

Vincent Cohen-Addad, Benjamin Guedj, Varun Kanade et al.

We study the problem of online clustering where a clustering algorithm has to assign a new point that arrives to one of $k$ clusters. The specific formulation we use is the $k$-means objective: At each time step the algorithm has to maintain a set of k candidate centers and the loss incurred is the squared distance between the new point and the closest center. The goal is to minimize regret with respect to the best solution to the $k$-means objective ($\mathcal{C}$) in hindsight. We show that provided the data lies in a bounded region, an implementation of the Multiplicative Weights Update Algorithm (MWUA) using a discretized grid achieves a regret bound of $\tilde{O}(\sqrt{T})$ in expectation. We also present an online-to-offline reduction that shows that an efficient no-regret online algorithm (despite being allowed to choose a different set of candidate centres at each round) implies an offline efficient algorithm for the $k$-means problem. In light of this hardness, we consider the slightly weaker requirement of comparing regret with respect to $(1 + ε) \mathcal{C}$ and present a no-regret algorithm with runtime $O\left(T(\mathrm{poly}(log(T),k,d,1/ε)^{k(d+O(1))}\right)$. Our algorithm is based on maintaining an incremental coreset and an adaptive variant of the MWUA. We show that naïve online algorithms, such as \emph{Follow The Leader}, fail to produce sublinear regret in the worst case. We also report preliminary experiments with synthetic and real-world data.

Vincent Cohen-Addad, Frederik Mallmann-Trenn, Claire Mathieu

Motivated by crowdsourced computation, peer-grading, and recommendation systems, Braverman, Mao and Weinberg [STOC'16] studied the \emph{query} and \emph{round} complexity of fundamental problems such as finding the maximum (\textsc{max}), finding all elements above a certain value (\textsc{threshold-$v$}) or computing the top$-k$ elements (\textsc{Top}-$k$) in a noisy environment. For example, consider the task of selecting papers for a conference. This task is challenging due the crowdsourcing nature of peer reviews: the results of reviews are noisy and it is necessary to parallelize the review process as much as possible. We study the noisy value model and the noisy comparison model: In the \emph{noisy value model}, a reviewer is asked to evaluate a single element: "What is the value of paper $i$?" (\eg accept). In the \emph{noisy comparison model} (introduced in the seminal work of Feige, Peleg, Raghavan and Upfal [SICOMP'94]) a reviewer is asked to do a pairwise comparison: "Is paper $i$ better than paper $j$?" In this paper, we show optimal worst-case query complexity for the \textsc{max},\textsc{threshold-$v$} and \textsc{Top}-$k$ problems. For \textsc{max} and \textsc{Top}-$k$, we obtain optimal worst-case upper and lower bounds on the round vs query complexity in both models. For \textsc{threshold}-$v$, we obtain optimal query complexity and nearly-optimal round complexity, where $k$ is the size of the output) for both models. We then go beyond the worst-case and address the question of the importance of knowledge of the instance by providing, for a large range of parameters, instance-optimal algorithms with respect to the query complexity. Furthermore, we show that the value model is strictly easier than the comparison model.

Vincent Cohen-Addad, Varun Kanade, Frederik Mallmann-Trenn et al.

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a `good' hierarchical clustering is one that minimizes some cost function. He showed that this cost function has certain desirable properties. We take an axiomatic approach to defining `good' objective functions for both similarity and dissimilarity-based hierarchical clustering. We characterize a set of "admissible" objective functions (that includes Dasgupta's one) that have the property that when the input admits a `natural' hierarchical clustering, it has an optimal value. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better algorithms. For similarity-based hierarchical clustering, Dasgupta showed that the divisive sparsest-cut approach achieves an $O(\log^{3/2} n)$-approximation. We give a refined analysis of the algorithm and show that it in fact achieves an $O(\sqrt{\log n})$-approx. (Charikar and Chatziafratis independently proved that it is a $O(\sqrt{\log n})$-approx.). This improves upon the LP-based $O(\log n)$-approx. of Roy and Pokutta. For dissimilarity-based hierarchical clustering, we show that the classic average-linkage algorithm gives a factor 2 approx., and provide a simple and better algorithm that gives a factor 3/2 approx.. Finally, we consider `beyond-worst-case' scenario through a generalisation of the stochastic block model for hierarchical clustering. We show that Dasgupta's cost function has desirable properties for these inputs and we provide a simple 1 + o(1)-approximation in this setting.

Vincent Cohen-Addad, Chris Schwiegelshohn

We study the classic $k$-median and $k$-means clustering objectives in the beyond-worst-case scenario. We consider three well-studied notions of structured data that aim at characterizing real-world inputs: Distribution Stability (introduced by Awasthi, Blum, and Sheffet, FOCS 2010), Spectral Separability (introduced by Kumar and Kannan, FOCS 2010), Perturbation Resilience (introduced by Bilu and Linial, ICS 2010). We prove structural results showing that inputs satisfying at least one of the conditions are inherently "local". Namely, for any such input, any local optimum is close both in term of structure and in term of objective value to the global optima. As a corollary we obtain that the widely-used Local Search algorithm has strong performance guarantees for both the tasks of recovering the underlying optimal clustering and obtaining a clustering of small cost. This is a significant step toward understanding the success of local search heuristics in clustering applications.

Vincent Cohen-Addad, Varun Kanade

We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipschitz-continuous functions or functions with bounded gradients. 1 This is with good reason---any learning algorithm suffers linear regret even against piecewise constant functions that are chosen adversarially, arguably the simplest of non-Lipschitz continuous functions. The smoothed setting we consider is inspired by the seminal work of Spielman and Teng (2004) and the recent work of Gupta and Roughgarden---in this setting, the sequence of functions may be chosen by an adversary, however, with some uncertainty in the location of discontinuities. We give algorithms that achieve sublinear regret in the full information and bandit settings.