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A cornerstone of social choice theory is Condorcet's paradox which says that in an election where $n$ voters rank $m$ candidates it is possible that, no matter which candidate is declared the winner, a majority of voters would have preferred an alternative candidate. Instead, can we always choose a small committee of winning candidates that is preferred to any alternative candidate by a majority of voters? Elkind, Lang, and Saffidine raised this question and called such a committee a Condorcet winning set. They showed that winning sets of size $2$ may not exist, but sets of size logarithmic in the number of candidates always do. In this work, we show that Condorcet winning sets of size $6$ always exist, regardless of the number of candidates or the number of voters. More generally, we show that if $\frac{\alpha}{1 - \ln \alpha} \geq \frac{2}{k + 1}$, then there always exists a committee of size $k$ such that less than an $\alpha$ fraction of the voters prefer an alternate candidate. These are the first nontrivial positive results that apply for all $k \geq 2$. Our proof uses the probabilistic method and the minimax theorem, inspired by recent work on approximately stable committee selection. We construct a distribution over committees that performs sufficiently well (when compared against any candidate on any small subset of the voters) so that this distribution must contain a committee with the desired property in its support., Comment: Added relevant reference

In many applications, SMT solvers are used to solve similar or identical tasks over time. When the performance of the solver varies significantly despite only small changes, this leads to frustration for users. This has been called the stability problem, and it represents an important usability challenge for SMT solvers. In this paper, we introduce an approach for mitigating the stability problem based on normalizing solver inputs. We show that a perfect normalizing algorithm exists but is computationally expensive. We then describe an approximate algorithm and evaluate it on a set of benchmarks from related work, as well as a large set of benchmarks sampled from SMT-LIB. Our evaluation shows that our approximate normalizer reduces runtime variability with minimal overhead and is able to normalize a large class of mutated benchmarks to a unique normal form.

Recent advances in large language models have shown capabilities that are extraordinary and near-superhuman. These models operate with such complexity that reliably evaluating and aligning them proves challenging for humans. This leads to the natural question: can guidance from weak models (like humans) adequately direct the capabilities of strong models? In a recent and somewhat surprising work, Burns et al. (2023) empirically demonstrated that when strong models (like GPT-4) are finetuned using labels generated by weak supervisors (like GPT-2), the strong models outperform their weaker counterparts -- a phenomenon they term weak-to-strong generalization. In this work, we present a theoretical framework for understanding weak-to-strong generalization. Specifically, we show that the improvement in performance achieved by strong models over their weaker counterparts is quantified by the misfit error incurred by the strong model on labels generated by the weaker model. Our theory reveals several curious algorithmic insights. For instance, we can predict the amount by which the strong model will improve over the weak model, and also choose among different weak models to train the strong model, based on its misfit error. We validate our theoretical findings through various empirical assessments., Comment: 19 pages; NeurIPS 2024 camera-ready version with additional experiments, references and discussion

Many data analytics systems store and process large datasets in partitions containing millions of rows. By mapping rows to partitions in an optimized way, it is possible to improve query performance by skipping over large numbers of irrelevant partitions during query processing. This mapping is referred to as a data layout. Recent works have shown that customizing the data layout to the anticipated query workload greatly improves query performance, but the performance benefits may disappear if the workload changes. Reorganizing data layouts to accommodate workload drift can resolve this issue, but reorganization costs could exceed query savings if not done carefully. In this paper, we present an algorithmic framework OReO that makes online reorganization decisions to balance the benefits of improved query performance with the costs of reorganization. Our framework extends results from Metrical Task Systems to provide a tight bound on the worst-case performance guarantee for online reorganization, without prior knowledge of the query workload. Through evaluation on real-world datasets and query workloads, our experiments demonstrate that online reorganization with OReO can lead to an up to 32% improvement in combined query and reorganization time compared to using a single, optimized data layout for the entire workload., Comment: ICDE 2024

We study the classic correlation clustering in the dynamic setting. Given $n$ objects and a complete labeling of the object-pairs as either similar or dissimilar, the goal is to partition the objects into arbitrarily many clusters while minimizing disagreements with the labels. In the dynamic setting, an update consists of a flip of a label of an edge. In a breakthrough result, [BDHSS, FOCS'19] showed how to maintain a 3-approximation with polylogarithmic update time by providing a dynamic implementation of the Pivot algorithm of [ACN, STOC'05]. Since then, it has been a major open problem to determine whether the 3-approximation barrier can be broken in the fully dynamic setting. In this paper, we resolve this problem. Our algorithm, Modified Pivot, locally improves the output of Pivot by moving some vertices to other existing clusters or new singleton clusters. We present an analysis showing that this modification does indeed improve the approximation to below 3. We also show that its output can be maintained in polylogarithmic time per update.

In the kernel density estimation (KDE) problem one is given a kernel $K(x, y)$ and a dataset $P$ of points in a Euclidean space, and must prepare a data structure that can quickly answer density queries: given a point $q$, output a $(1+\epsilon)$-approximation to $\mu:=\frac1{|P|}\sum_{p\in P} K(p, q)$. The classical approach to KDE is the celebrated fast multipole method of [Greengard and Rokhlin]. The fast multipole method combines a basic space partitioning approach with a multidimensional Taylor expansion, which yields a $\approx \log^d (n/\epsilon)$ query time (exponential in the dimension $d$). A recent line of work initiated by [Charikar and Siminelakis] achieved polynomial dependence on $d$ via a combination of random sampling and randomized space partitioning, with [Backurs et al.] giving an efficient data structure with query time $\approx \mathrm{poly}{\log(1/\mu)}/\epsilon^2$ for smooth kernels. Quadratic dependence on $\epsilon$, inherent to the sampling methods, is prohibitively expensive for small $\epsilon$. This issue is addressed by quasi-Monte Carlo methods in numerical analysis. The high level idea in quasi-Monte Carlo methods is to replace random sampling with a discrepancy based approach -- an idea recently applied to coresets for KDE by [Phillips and Tai]. The work of Phillips and Tai gives a space efficient data structure with query complexity $\approx 1/(\epsilon \mu)$. This is polynomially better in $1/\epsilon$, but exponentially worse in $1/\mu$. We achieve the best of both: a data structure with $\approx \mathrm{poly}{\log(1/\mu)}/\epsilon$ query time for smooth kernel KDE. Our main insight is a new way to combine discrepancy theory with randomized space partitioning inspired by, but significantly more efficient than, that of the fast multipole methods. We hope that our techniques will find further applications to linear algebra for kernel matrices.

Given an arbitrary set of high dimensional points in $\ell_1$, there are known negative results that preclude the possibility of always mapping them to a low dimensional $\ell_1$ space while preserving distances with small multiplicative distortion. This is in stark contrast with dimension reduction in Euclidean space ($\ell_2$) where such mappings are always possible. While the first non-trivial lower bounds for $\ell_1$ dimension reduction were established almost 20 years ago, there has been limited progress in understanding what sets of points in $\ell_1$ are conducive to a low-dimensional mapping. In this work, we study a new characterization of $\ell_1$ metrics that are conducive to dimension reduction in $\ell_1$. Our characterization focuses on metrics that are defined by the disagreement of binary variables over a probability distribution -- any $\ell_1$ metric can be represented in this form. We show that, for configurations of $n$ points in $\ell_1$ obtained from tree Ising models, we can reduce dimension to $\mathrm{polylog}(n)$ with constant distortion. In doing so, we develop technical tools for embedding truncated metrics which have been studied because of their applications in computer vision, and are objects of independent interest in metric geometry. Among other tools, we show how any $\ell_1$ metric can be truncated with $O(1)$ distortion and $O(\log(n))$ blowup in dimension.

We study the Ultrametric Violation Distance problem introduced by Cohen-Addad, Fan, Lee, and Mesmay [FOCS, 2022]. Given pairwise distances $x\in \mathbb{R}_{>0}^{\binom{[n]}{2}}$ as input, the goal is to modify the minimum number of distances so as to make it a valid ultrametric. In other words, this is the problem of fitting an ultrametric to given data, where the quality of the fit is measured by the $\ell_0$ norm of the error; variants of the problem for the $\ell_\infty$ and $\ell_1$ norms are well-studied in the literature. Our main result is a 5-approximation algorithm for Ultrametric Violation Distance, improving the previous best large constant factor ($\geq 1000$) approximation algorithm. We give an $O(\min\{L,\log n\})$-approximation for weighted Ultrametric Violation Distance where the weights satisfy triangle inequality and $L$ is the number of distinct values in the input. We also give a $16$-approximation for the problem on $k$-partite graphs, where the input is specified on pairs of vertices that form a complete $k$-partite graph. All our results use a unified algorithmic framework with small modifications for the three cases., Comment: SODA 2024

Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. Popular clustering algorithms such as Lloyd's algorithm and $k$-means++ can take $\Omega(ndk)$ time when clustering $n$ points in a $d$-dimensional space (represented by an $n\times d$ matrix $X$) into $k$ clusters. In applications with moderate to large $k$, the multiplicative $k$ factor can become very expensive. We introduce a simple randomized clustering algorithm that provably runs in expected time $O(\mathrm{nnz}(X) + n\log n)$ for arbitrary $k$. Here $\mathrm{nnz}(X)$ is the total number of non-zero entries in the input dataset $X$, which is upper bounded by $nd$ and can be significantly smaller for sparse datasets. We prove that our algorithm achieves approximation ratio $\smash{\widetilde{O}(k^4)}$ on any input dataset for the $k$-means objective. We also believe that our theoretical analysis is of independent interest, as we show that the approximation ratio of a $k$-means algorithm is approximately preserved under a class of projections and that $k$-means++ seeding can be implemented in expected $O(n \log n)$ time in one dimension. Finally, we show experimentally that our clustering algorithm gives a new tradeoff between running time and cluster quality compared to previous state-of-the-art methods for these tasks., Comment: 41 pages, 6 figures, to appear in NeurIPS 2023

We introduce a new class of objectives for optimal transport computations of datasets in high-dimensional Euclidean spaces. The new objectives are parametrized by $\rho \geq 1$, and provide a metric space $\mathcal{R}_{\rho}(\cdot, \cdot)$ for discrete probability distributions in $\mathbb{R}^d$. As $\rho$ approaches $1$, the metric approaches the Earth Mover's distance, but for $\rho$ larger than (but close to) $1$, admits significantly faster algorithms. Namely, for distributions $\mu$ and $\nu$ supported on $n$ and $m$ vectors in $\mathbb{R}^d$ of norm at most $r$ and any $\epsilon > 0$, we give an algorithm which outputs an additive $\epsilon r$-approximation to $\mathcal{R}_{\rho}(\mu, \nu)$ in time $(n+m) \cdot \mathrm{poly}((nm)^{(\rho-1)/\rho} \cdot 2^{\rho / (\rho-1)} / \epsilon)$., Comment: in COLT 2023

We consider the following well-studied problem of metric distortion in social choice. Suppose we have an election with $n$ voters and $m$ candidates located in a shared metric space. We would like to design a voting rule that chooses a candidate whose average distance to the voters is small. However, instead of having direct access to the distances in the metric space, the voting rule obtains, from each voter, a ranked list of the candidates in order of distance. Can we design a rule that regardless of the election instance and underlying metric space, chooses a candidate whose cost differs from the true optimum by only a small factor (known as the distortion)? A long line of work culminated in finding optimal deterministic voting rules with metric distortion $3$. However, for randomized voting rules, there is still a gap in our understanding: Even though the best lower bound is $2.112$, the best upper bound is still $3$, attained even by simple rules such as Random Dictatorship. Finding a randomized rule that guarantees distortion $3 - \epsilon$ has been a major challenge in computational social choice, as prevalent approaches to designing voting rules are known to be insufficient. Such a voting rule must use information beyond aggregate comparisons between pairs of candidates, and cannot only assign positive probability to candidates that are voters' top choices. In this work, we give a rule that guarantees distortion less than $2.753$. To do so we study a handful of voting rules that are new to the problem. One is Maximal Lotteries, a rule based on the Nash equilibrium of a natural zero-sum game which dates back to the 60's. The others are novel rules that can be thought of as hybrids of Random Dictatorship and the Copeland rule. Though none of these rules can beat distortion $3$ alone, a randomization between Maximal Lotteries and any of the novel rules can., Comment: JACM, to appear

Suppose that we have $n$ agents and $n$ items which lie in a shared metric space. We would like to match the agents to items such that the total distance from agents to their matched items is as small as possible. However, instead of having direct access to distances in the metric, we only have each agent's ranking of the items in order of distance. Given this limited information, what is the minimum possible worst-case approximation ratio (known as the distortion) that a matching mechanism can guarantee? Previous work by Caragiannis et al. proved that the (deterministic) Serial Dictatorship mechanism has distortion at most $2^n - 1$. We improve this by providing a simple deterministic mechanism that has distortion $O(n^2)$. We also provide the first nontrivial lower bound on this problem, showing that any matching mechanism (deterministic or randomized) must have worst-case distortion $\Omega(\log n)$. In addition to these new bounds, we show that a large class of truthful mechanisms derived from Deferred Acceptance all have worst-case distortion at least $2^n - 1$, and we find an intriguing connection between thin matchings (analogous to the well-known thin trees conjecture) and the distortion gap between deterministic and randomized mechanisms., Comment: to appear at EC'23

A classical result in learning theory shows the equivalence of PAC learnability of binary hypothesis classes and the finiteness of VC dimension. Extending this to the multiclass setting was an open problem, which was settled in a recent breakthrough result characterizing multiclass PAC learnability via the DS dimension introduced earlier by Daniely and Shalev-Shwartz. In this work we consider list PAC learning where the goal is to output a list of $k$ predictions. List learning algorithms have been developed in several settings before and indeed, list learning played an important role in the recent characterization of multiclass learnability. In this work we ask: when is it possible to $k$-list learn a hypothesis class? We completely characterize $k$-list learnability in terms of a generalization of DS dimension that we call the $k$-DS dimension. Generalizing the recent characterization of multiclass learnability, we show that a hypothesis class is $k$-list learnable if and only if the $k$-DS dimension is finite.

We provide an efficient unified plug-in approach for estimating symmetric properties of distributions given $n$ independent samples. Our estimator is based on profile-maximum-likelihood (PML) and is sample optimal for estimating various symmetric properties when the estimation error $\epsilon \gg n^{-1/3}$. This result improves upon the previous best accuracy threshold of $\epsilon \gg n^{-1/4}$ achievable by polynomial time computable PML-based universal estimators [ACSS21, ACSS20]. Our estimator reaches a theoretical limit for universal symmetric property estimation as [Han21] shows that a broad class of universal estimators (containing many well known approaches including ours) cannot be sample optimal for every $1$-Lipschitz property when $\epsilon \ll n^{-1/3}$., Comment: Accepted at NeurIPS 2022

A crucial task in the political redistricting problem is to sample redistricting plans i.e. a partitioning of the graph of census blocks into districts. We show that Recombination [DeFord-Duchin-Solomon'21]-a popular Markov chain to sample redistricting plans-is exponentially slow mixing on simple subgraph of $\mathbb{Z}_2.$ We show an alternative way to sample balance, compact and contiguous redistricting plans using a "relaxed" version of ReCom and rejection sampling., Comment: Correcting the definition of Markov chain to sample from spanning tree distribution

We study parallel algorithms for correlation clustering. Each pair among $n$ objects is labeled as either "similar" or "dissimilar". The goal is to partition the objects into arbitrarily many clusters while minimizing the number of disagreements with the labels. Our main result is an algorithm that for any $\epsilon > 0$ obtains a $(3+\epsilon)$-approximation in $O(1/\epsilon)$ rounds (of models such as massively parallel computation, local, and semi-streaming). This is a culminating point for the rich literature on parallel correlation clustering. On the one hand, the approximation (almost) matches a natural barrier of 3 for combinatorial algorithms. On the other hand, the algorithm's round-complexity is essentially constant. To achieve this result, we introduce a simple $O(1/\epsilon)$-round parallel algorithm. Our main result is to provide an analysis of this algorithm, showing that it achieves a $(3+\epsilon)$-approximation. Our analysis draws on new connections to sublinear-time algorithms. Specifically, it builds on the work of Yoshida, Yamamoto, and Ito [STOC'09] on bounding the "query complexity" of greedy maximal independent set. To our knowledge, this is the first application of this method in analyzing the approximation ratio of any algorithm.

We study the effect of Johnson-Lindenstrauss transforms in various projective clustering problems, generalizing recent results which only applied to center-based clustering [MMR19]. We ask the general question: for a Euclidean optimization problem and an accuracy parameter $\epsilon \in (0, 1)$, what is the smallest target dimension $t \in \mathbb{N}$ such that a Johnson-Lindenstrauss transform $\Pi \colon \mathbb{R}^d \to \mathbb{R}^t$ preserves the cost of the optimal solution up to a $(1+\epsilon)$-factor. We give a new technique which uses coreset constructions to analyze the effect of the Johnson-Lindenstrauss transform. Our technique, in addition applying to center-based clustering, improves on (or is the first to address) other Euclidean optimization problems, including: $\bullet$ For $(k,z)$-subspace approximation: we show that $t = \tilde{O}(zk^2 / \epsilon^3)$ suffices, whereas the prior best bound, of $O(k/\epsilon^2)$, only applied to the case $z = 2$ [CEMMP15]. $\bullet$ For $(k,z)$-flat approximation: we show $t = \tilde{O}(zk^2/\epsilon^3)$ suffices, completely removing the dependence on $n$ from the prior bound $\tilde{O}(zk^2 \log n/\epsilon^3)$ of [KR15]. $\bullet$ For $(k,z)$-line approximation: we show $t = O((k \log \log n + z + \log(1/\epsilon)) / \epsilon^3)$ suffices, and ours is the first to give any dimension reduction result.

Given $n$ points in $\ell_p^d$, we consider the problem of partitioning points into $k$ clusters with associated centers. The cost of a clustering is the sum of $p^{\text{th}}$ powers of distances of points to their cluster centers. For $p \in [1,2]$, we design sketches of size poly$(\log(nd),k,1/\epsilon)$ such that the cost of the optimal clustering can be estimated to within factor $1+\epsilon$, despite the fact that the compressed representation does not contain enough information to recover the cluster centers or the partition into clusters. This leads to a streaming algorithm for estimating the clustering cost with space poly$(\log(nd),k,1/\epsilon)$. We also obtain a distributed memory algorithm, where the $n$ points are arbitrarily partitioned amongst $m$ machines, each of which sends information to a central party who then computes an approximation of the clustering cost. Prior to this work, no such streaming or distributed-memory algorithm was known with sublinear dependence on $d$ for $p \in [1,2)$., Comment: ICALP 2022

Consider the following social choice problem. Suppose we have a set of $n$ voters and $m$ candidates that lie in a metric space. The goal is to design a mechanism to choose a candidate whose average distance to the voters is as small as possible. However, the mechanism does not get direct access to the metric space. Instead, it gets each voter's ordinal ranking of the candidates by distance. Given only this partial information, what is the smallest worst-case approximation ratio (known as the distortion) that a mechanism can guarantee? A simple example shows that no deterministic mechanism can guarantee distortion better than $3$, and no randomized mechanism can guarantee distortion better than $2$. It has been conjectured that both of these lower bounds are optimal, and recently, Gkatzelis, Halpern, and Shah proved this conjecture for deterministic mechanisms. We disprove the conjecture for randomized mechanisms for $m \geq 3$ by constructing elections for which no randomized mechanism can guarantee distortion better than $2.0261$ for $m = 3$, $2.0496$ for $m = 4$, up to $2.1126$ as $m \to \infty$. We obtain our lower bounds by identifying a class of simple metrics that appear to capture much of the hardness of the problem, and we show that any randomized mechanism must have high distortion on one of these metrics. We provide a nearly matching upper bound for this restricted class of metrics as well. Finally, we conjecture that these bounds give the optimal distortion for every $m$, and provide a proof for $m = 3$, thereby resolving that case., Comment: To appear in SODA2022

Many clustering algorithms are guided by certain cost functions such as the widely-used $k$-means cost. These algorithms divide data points into clusters with often complicated boundaries, creating difficulties in explaining the clustering decision. In a recent work, Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020) introduced explainable clustering, where the cluster boundaries are axis-parallel hyperplanes and the clustering is obtained by applying a decision tree to the data. The central question here is: how much does the explainability constraint increase the value of the cost function? Given $d$-dimensional data points, we show an efficient algorithm that finds an explainable clustering whose $k$-means cost is at most $k^{1 - 2/d}\,\mathrm{poly}(d\log k)$ times the minimum cost achievable by a clustering without the explainability constraint, assuming $k,d\ge 2$. Taking the minimum of this bound and the $k\,\mathrm{polylog} (k)$ bound in independent work by Makarychev-Shan (ICML 2021), Gamlath-Jia-Polak-Svensson (2021), or Esfandiari-Mirrokni-Narayanan (2021), we get an improved bound of $k^{1 - 2/d}\,\mathrm{polylog}(k)$, which we show is optimal for every choice of $k,d\ge 2$ up to a poly-logarithmic factor in $k$. For $d = 2$ in particular, we show an $O(\log k\log\log k)$ bound, improving near-exponentially over the previous best bound of $O(k\log k)$ by Laber and Murtinho (ICML 2021)., Comment: 34 pages, 2 figures, to appear in SODA 2022

We give a $0.5368$-competitive algorithm for edge-weighted online bipartite matching. Prior to our work, the best competitive ratio was $0.5086$ due to Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020). They achieved their breakthrough result by developing a subroutine called \emph{online correlated selection} (OCS) which takes as input a sequence of pairs and selects one item from each pair. Importantly, the selections the OCS makes are negatively correlated. We achieve our result by defining \emph{multiway} OCSes which receive arbitrarily many elements at each step, rather than just two. In addition to better competitive ratios, our formulation allows for a simpler reduction from edge-weighted online bipartite matching to OCSes. While Fahrbach et al. used a factor-revealing linear program to optimize the competitive ratio, our analysis directly connects the competitive ratio to the parameters of the multiway OCS. Finally, we show that the formulation of Farhbach et al. can achieve a competitive ratio of at most $0.5239$, confirming that multiway OCSes are strictly more powerful., Comment: Added motivation and a more thorough overview of our paper in the first few sections, fixed a few typos

In the non-negative matrix factorization (NMF) problem, the input is an $m\times n$ matrix $M$ with non-negative entries and the goal is to factorize it as $M\approx AW$. The $m\times k$ matrix $A$ and the $k\times n$ matrix $W$ are both constrained to have non-negative entries. This is in contrast to singular value decomposition, where the matrices $A$ and $W$ can have negative entries but must satisfy the orthogonality constraint: the columns of $A$ are orthogonal and the rows of $W$ are also orthogonal. The orthogonal non-negative matrix factorization (ONMF) problem imposes both the non-negativity and the orthogonality constraints, and previous work showed that it leads to better performances than NMF on many clustering tasks. We give the first constant-factor approximation algorithm for ONMF when one or both of $A$ and $W$ are subject to the orthogonality constraint. We also show an interesting connection to the correlation clustering problem on bipartite graphs. Our experiments on synthetic and real-world data show that our algorithm achieves similar or smaller errors compared to previous ONMF algorithms while ensuring perfect orthogonality (many previous algorithms do not satisfy the hard orthogonality constraint)., Comment: 26 pages, 5 figures. To be published in AISTATS 2021. Font size increased

Colonies of the arboreal turtle ant create networks of trails that link nests and food sources on the graph formed by branches and vines in the canopy of the tropical forest. Ants put down a volatile pheromone on edges as they traverse them. At each vertex, the next edge to traverse is chosen using a decision rule based on the current pheromone level. There is a bidirectional flow of ants around the network. In a field study, Chandrasekhar et al. (2021) observed that the trail networks approximately minimize the number of vertices, thus solving a variant of the popular shortest path problem without any central control and with minimal computational resources. We propose a biologically plausible model, based on a variant of the reinforced random walk on a graph, which explains this observation and suggests surprising algorithms for the shortest path problem and its variants. Through simulations and analysis, we show that when the rate of flow of ants does not change, the dynamics converges to the path with the minimum number of vertices, as observed in the field. The dynamics converges to the shortest path when the rate of flow increases with time, so the colony can solve the shortest path problem merely by increasing the flow rate. We also show that to guarantee convergence to the shortest path, bidirectional flow and a decision rule dividing the flow in proportion to the pheromone level are necessary, but convergence to approximately short paths is possible with other decision rules., Comment: Changes in presentation. Published in PNAS 2023

In this paper we revisit the kernel density estimation problem: given a kernel $K(x, y)$ and a dataset of $n$ points in high dimensional Euclidean space, prepare a data structure that can quickly output, given a query $q$, a $(1+\epsilon)$-approximation to $\mu:=\frac1{|P|}\sum_{p\in P} K(p, q)$. First, we give a single data structure based on classical near neighbor search techniques that improves upon or essentially matches the query time and space complexity for all radial kernels considered in the literature so far. We then show how to improve both the query complexity and runtime by using recent advances in data-dependent near neighbor search. We achieve our results by giving a new implementation of the natural importance sampling scheme. Unlike previous approaches, our algorithm first samples the dataset uniformly (considering a geometric sequence of sampling rates), and then uses existing approximate near neighbor search techniques on the resulting smaller dataset to retrieve the sampled points that lie at an appropriate distance from the query. We show that the resulting sampled dataset has strong geometric structure, making approximate near neighbor search return the required samples much more efficiently than for worst case datasets of the same size. As an example application, we show that this approach yields a data structure that achieves query time $\mu^{-(1+o(1))/4}$ and space complexity $\mu^{-(1+o(1))}$ for the Gaussian kernel. Our data dependent approach achieves query time $\mu^{-0.173-o(1)}$ and space $\mu^{-(1+o(1))}$ for the Gaussian kernel. The data dependent analysis relies on new techniques for tracking the geometric structure of the input datasets in a recursive hashing process that we hope will be of interest in other applications in near neighbor search.

In this paper we provide a new efficient algorithm for approximately computing the profile maximum likelihood (PML) distribution, a prominent quantity in symmetric property estimation. We provide an algorithm which matches the previous best known efficient algorithms for computing approximate PML distributions and improves when the number of distinct observed frequencies in the given instance is small. We achieve this result by exploiting new sparsity structure in approximate PML distributions and providing a new matrix rounding algorithm, of independent interest. Leveraging this result, we obtain the first provable computationally efficient implementation of PseudoPML, a general framework for estimating a broad class of symmetric properties. Additionally, we obtain efficient PML-based estimators for distributions with small profile entropy, a natural instance-based complexity measure. Further, we provide a simpler and more practical PseudoPML implementation that matches the best-known theoretical guarantees of such an estimator and evaluate this method empirically., Comment: Accepted at Thirty-fourth Conference on Neural Information Processing Systems (NeurIPS 2020)

In the W-streaming model, an algorithm is given $O(n \mathrm{polylog} n)$ space and must process a large graph of up to $O(n^2)$ edges. In this short note we give two algorithms for edge colouring under the W-streaming model. For edge colouring in W-streaming, a colour for every edge must be determined by the time all the edges are streamed. Our first algorithm uses $\Delta + o(\Delta)$ colours in $O(n \log^2 n)$ space when the edges arrive according to a uniformly random permutation. The second algorithm uses $(1 + o(1))\Delta^2 / s$ colours in $\tilde{O}(n s)$ space when edges arrival adversarially., Comment: To appear in SOSA21. Updated to contain references to relevant work from SODA21

Embedding into hyperbolic space is emerging as an effective representation technique for datasets that exhibit hierarchical structure. This development motivates the need for algorithms that are able to effectively extract knowledge and insights from datapoints embedded in negatively curved spaces. We focus on the problem of nearest neighbor search, a fundamental problem in data analysis. We present efficient algorithmic solutions that build upon established methods for nearest neighbor search in Euclidean space, allowing for easy adoption and integration with existing systems. We prove theoretical guarantees for our techniques and our experiments demonstrate the effectiveness of our approach on real datasets over competing algorithms.

We study the problem of estimating the edit distance between two $n$-character strings. While exact computation in the worst case is believed to require near-quadratic time, previous work showed that in certain regimes it is possible to solve the following {\em gap edit distance} problem in sub-linear time: distinguish between inputs of distance $\le k$ and $>k^2$. Our main result is a very simple algorithm for this benchmark that runs in time $\tilde O(n/\sqrt{k})$, and in particular settles the open problem of obtaining a truly sublinear time for the entire range of relevant $k$. Building on the same framework, we also obtain a $k$-vs-$k^2$ algorithm for the one-sided preprocessing model with $\tilde O(n)$ preprocessing time and $\tilde O(n/k)$ query time (improving over a recent $\tilde O(n/k+k^2)$-query time algorithm for the same problem [GRS'20]., Comment: 26 pages

In this paper we consider the problem of computing the likelihood of the profile of a discrete distribution, i.e., the probability of observing the multiset of element frequencies, and computing a profile maximum likelihood (PML) distribution, i.e., a distribution with the maximum profile likelihood. For each problem we provide polynomial time algorithms that given $n$ i.i.d.\ samples from a discrete distribution, achieve an approximation factor of $\exp\left(-O(\sqrt{n} \log n) \right)$, improving upon the previous best-known bound achievable in polynomial time of $\exp(-O(n^{2/3} \log n))$ (Charikar, Shiragur and Sidford, 2019). Through the work of Acharya, Das, Orlitsky and Suresh (2016), this implies a polynomial time universal estimator for symmetric properties of discrete distributions in a broader range of error parameter. We achieve these results by providing new bounds on the quality of approximation of the Bethe and Sinkhorn permanents (Vontobel, 2012 and 2014). We show that each of these are $\exp(O(k \log(N/k)))$ approximations to the permanent of $N \times N$ matrices with non-negative rank at most $k$, improving upon the previous known bounds of $\exp(O(N))$. To obtain our results on PML, we exploit the fact that the PML objective is proportional to the permanent of a certain Vandermonde matrix with $\sqrt{n}$ distinct columns, i.e. with non-negative rank at most $\sqrt{n}$. As a by-product of our work we establish a surprising connection between the convex relaxation in prior work (CSS19) and the well-studied Bethe and Sinkhorn approximations., Comment: 59 pages

In this paper we provide a general framework for estimating symmetric properties of distributions from i.i.d. samples. For a broad class of symmetric properties we identify the easy region where empirical estimation works and the difficult region where more complex estimators are required. We show that by approximately computing the profile maximum likelihood (PML) distribution \cite{ADOS16} in this difficult region we obtain a symmetric property estimation framework that is sample complexity optimal for many properties in a broader parameter regime than previous universal estimation approaches based on PML. The resulting algorithms based on these pseudo PML distributions are also more practical., Comment: Published in Neural Information Processing Systems 2019

An emerging class of data systems partition their data and precompute approximate summaries (i.e., sketches and samples) for each segment to reduce query costs. They can then aggregate and combine the segment summaries to estimate results without scanning the raw data. However, given limited storage space each summary introduces approximation errors that affect query accuracy. For instance, systems that use existing mergeable summaries cannot reduce query error below the error of an individual precomputed summary. We introduce Storyboard, a query system that optimizes item frequency and quantile summaries for accuracy when aggregating over multiple segments. Compared to conventional mergeable summaries, Storyboard leverages additional memory available for summary construction and aggregation to derive a more precise combined result. This reduces error by up to 25x over interval aggregations and 4.4x over data cube aggregations on industrial datasets compared to standard summarization methods, with provable worst-case error guarantees.

Roughgarden, Vassilvitskii, and Wang (JACM 18) recently introduced a novel framework for proving lower bounds for Massively Parallel Computation using techniques from boolean function complexity. We extend their framework in two different ways, to capture two common features of Massively Parallel Computation: $\circ$ Adaptivity, where machines can write to and adaptively read from shared memory throughout the execution of the computation. Recent work of Behnezhad et al. (SPAA 19) showed that adaptivity enables significantly improved round complexities for a number of central graph problems. $\circ$ Promise problems, where the algorithm only has to succeed on certain inputs. These inputs may have special structure that is of particular interest, or they may be representative of hard instances of the overall problem. Using this extended framework, we give the first unconditional lower bounds on the complexity of distinguishing whether an input graph is a cycle of length $n$ or two cycles of length $n/2$. This promise problem, 1v2-Cycle, has emerged as a central problem in the study of Massively Parallel Computation. We prove that any adaptive algorithm for the 1v2-Cycle problem with I/O capacity $O(n^{\varepsilon})$ per machine requires $\Omega(1/\varepsilon)$ rounds, matching a recent upper bound of Behnezhad et al. In addition to strengthening the connections between Massively Parallel Computation and boolean function complexity, we also develop new machinery to reason about the latter. At the heart of our proofs are optimal lower bounds on the query complexity and approximate certificate complexity of the 1v2-Cycle problem.

Pattern counting in graphs is a fundamental primitive for many network analysis tasks, and a number of methods have been developed for scaling subgraph counting to large graphs. Many real-world networks carry a natural notion of strength of connection between nodes, which are often modeled by a weighted graph, but existing scalable graph algorithms for pattern mining are designed for unweighted graphs. Here, we develop a suite of deterministic and random sampling algorithms that enable the fast discovery of the 3-cliques (triangles) with the largest weight in a graph, where weight is measured by a generalized mean of a triangle's edges. For example, one of our proposed algorithms can find the top-1000 weighted triangles of a weighted graph with billions of edges in thirty seconds on a commodity server, which is orders of magnitude faster than existing "fast" enumeration schemes. Our methods thus open the door towards scalable pattern mining in weighted graphs.

Estimating symmetric properties of a distribution, e.g. support size, coverage, entropy, distance to uniformity, are among the most fundamental problems in algorithmic statistics. While each of these properties have been studied extensively and separate optimal estimators are known for each, in striking recent work, Acharya et al. 2016 showed that there is a single estimator that is competitive for all symmetric properties. This work proved that computing the distribution that approximately maximizes \emph{profile likelihood (PML)}, i.e. the probability of observed frequency of frequencies, and returning the value of the property on this distribution is sample competitive with respect to a broad class of estimators of symmetric properties. Further, they showed that even computing an approximation of the PML suffices to achieve such a universal plug-in estimator. Unfortunately, prior to this work there was no known polynomial time algorithm to compute an approximate PML and it was open to obtain a polynomial time universal plug-in estimator through the use of approximate PML. In this paper we provide a algorithm (in number of samples) that, given $n$ samples from a distribution, computes an approximate PML distribution up to a multiplicative error of $\exp(n^{2/3} \mathrm{poly} \log(n))$ in time nearly linear in $n$. Generalizing work of Acharya et al. 2016 on the utility of approximate PML we show that our algorithm provides a nearly linear time universal plug-in estimator for all symmetric functions up to accuracy $\epsilon = \Omega(n^{-0.166})$. Further, we show how to extend our work to provide efficient polynomial-time algorithms for computing a $d$-dimensional generalization of PML (for constant $d$) that allows for universal plug-in estimation of symmetric relationships between distributions., Comment: 68 pages

We resolve the randomized one-way communication complexity of Dynamic Time Warping (DTW) distance. We show that there is an efficient one-way communication protocol using $\widetilde{O}(n/\alpha)$ bits for the problem of computing an $\alpha$-approximation for DTW between strings $x$ and $y$ of length $n$, and we prove a lower bound of $\Omega(n / \alpha)$ bits for the same problem. Our communication protocol works for strings over an arbitrary metric of polynomial size and aspect ratio, and we optimize the logarithmic factors depending on properties of the underlying metric, such as when the points are low-dimensional integer vectors equipped with various metrics or have bounded doubling dimension. We also consider linear sketches of DTW, showing that such sketches must have size $\Omega(n)$.

Recent works on Hierarchical Clustering (HC), a well-studied problem in exploratory data analysis, have focused on optimizing various objective functions for this problem under arbitrary similarity measures. In this paper we take the first step and give novel scalable algorithms for this problem tailored to Euclidean data in R^d and under vector-based similarity measures, a prevalent model in several typical machine learning applications. We focus primarily on the popular Gaussian kernel and other related measures, presenting our results through the lens of the objective introduced recently by Moseley and Wang [2017]. We show that the approximation factor in Moseley and Wang [2017] can be improved for Euclidean data. We further demonstrate both theoretically and experimentally that our algorithms scale to very high dimension d, while outperforming average-linkage and showing competitive results against other less scalable approaches.

We consider the tensor completion problem of predicting the missing entries of a tensor. The commonly used CP model has a triple product form, but an alternate family of quadratic models, which are the sum of pairwise products instead of a triple product, have emerged from applications such as recommendation systems. Non-convex methods are the method of choice for learning quadratic models, and this work examines their sample complexity and error guarantee. Our main result is that with the number of samples being only linear in the dimension, all local minima of the mean squared error objective are global minima and recover the original tensor. We substantiate our theoretical results with experiments on synthetic and real-world data, showing that quadratic models have better performance than CP models where there are a limited amount of observations available., Comment: 16 pages. Appeared in AISTATS 2019

Pattern counting in graphs is fundamental to network science tasks, and there are many scalable methods for approximating counts of small patterns, often called motifs, in large graphs. However, modern graph datasets now contain richer structure, and incorporating temporal information in particular has become a critical part of network analysis. Temporal motifs, which are generalizations of small subgraph patterns that incorporate temporal ordering on edges, are an emerging part of the network analysis toolbox. However, there are no algorithms for fast estimation of temporal motifs counts; moreover, we show that even counting simple temporal star motifs is NP-complete. Thus, there is a need for fast and approximate algorithms. Here, we present the first frequency estimation algorithms for counting temporal motifs. More specifically, we develop a sampling framework that sits as a layer on top of existing exact counting algorithms and enables fast and accurate memory-efficient estimates of temporal motif counts. Our results show that we can achieve one to two orders of magnitude speedups with minimal and controllable loss in accuracy on a number of datasets., Comment: 9 pages, 4 figures

An important question that arises in the study of high dimensional vector representations learned from data is: given a set $\mathcal{D}$ of vectors and a query $q$, estimate the number of points within a specified distance threshold of $q$. We develop two estimators, LSH Count and Multi-Probe Count that use locality sensitive hashing to preprocess the data to accurately and efficiently estimate the answers to such questions via importance sampling. A key innovation is the ability to maintain a small number of hash tables via preprocessing data structures and algorithms that sample from multiple buckets in each hash table. We give bounds on the space requirements and sample complexity of our schemes, and demonstrate their effectiveness in experiments on a standard word embedding dataset., Comment: Preliminary version appeared in ICML 2018

Given a set of points $P\subset \mathbb{R}^{d}$ and a kernel $k$, the Kernel Density Estimate at a point $x\in\mathbb{R}^{d}$ is defined as $\mathrm{KDE}_{P}(x)=\frac{1}{|P|}\sum_{y\in P} k(x,y)$. We study the problem of designing a data structure that given a data set $P$ and a kernel function, returns *approximations to the kernel density* of a query point in *sublinear time*. We introduce a class of unbiased estimators for kernel density implemented through locality-sensitive hashing, and give general theorems bounding the variance of such estimators. These estimators give rise to efficient data structures for estimating the kernel density in high dimensions for a variety of commonly used kernels. Our work is the first to provide data-structures with theoretical guarantees that improve upon simple random sampling in high dimensions., Comment: A preliminary version of this paper appeared in FOCS 2017

Hierarchical Clustering (HC) is a widely studied problem in exploratory data analysis, usually tackled by simple agglomerative procedures like average-linkage, single-linkage or complete-linkage. In this paper we focus on two objectives, introduced recently to give insight into the performance of average-linkage clustering: a similarity based HC objective proposed by [Moseley and Wang, 2017] and a dissimilarity based HC objective proposed by [Cohen-Addad et al., 2018]. In both cases, we present tight counterexamples showing that average-linkage cannot obtain better than 1/3 and 2/3 approximations respectively (in the worst-case), settling an open question raised in [Moseley and Wang, 2017]. This matches the approximation ratio of a random solution, raising a natural question: can we beat average-linkage for these objectives? We answer this in the affirmative, giving two new algorithms based on semidefinite programming with provably better guarantees.

A basic computational primitive in the analysis of massive datasets is summing simple functions over a large number of objects. Modern applications pose an additional challenge in that such functions often depend on a parameter vector $y$ (query) that is unknown a priori. Given a set of points $X\subset \mathbb{R}^{d}$ and a pairwise function $w:\mathbb{R}^{d}\times \mathbb{R}^{d}\to [0,1]$, we study the problem of designing a data-structure that enables sublinear-time approximation of the summation $Z_{w}(y)=\frac{1}{|X|}\sum_{x\in X}w(x,y)$ for any query $y\in \mathbb{R}^{d}$. By combining ideas from Harmonic Analysis (partitions of unity and approximation theory) with Hashing-Based-Estimators [Charikar, Siminelakis FOCS'17], we provide a general framework for designing such data structures through hashing that reaches far beyond what previous techniques allowed. A key design principle is a collection of $T\geq 1$ hashing schemes with collision probabilities $p_{1},\ldots, p_{T}$ such that $\sup_{t\in [T]}\{p_{t}(x,y)\} = \Theta(\sqrt{w(x,y)})$. This leads to a data-structure that approximates $Z_{w}(y)$ using a sub-linear number of samples from each hash family. Using this new framework along with Distance Sensitive Hashing [Aumuller, Christiani, Pagh, Silvestri PODS'18], we show that such a collection can be constructed and evaluated efficiently for any log-convex function $w(x,y)=e^{\phi(\langle x,y\rangle)}$ of the inner product on the unit sphere $x,y\in \mathcal{S}^{d-1}$. Our method leads to data structures with sub-linear query time that significantly improve upon random sampling and can be used for Kernel Density or Partition Function Estimation. We provide extensions of our result from the sphere to $\mathbb{R}^{d}$ and from scalar functions to vector functions., Comment: 39 pages, 3 figures

Hierarchical clustering is a popular unsupervised data analysis method. For many real-world applications, we would like to exploit prior information about the data that imposes constraints on the clustering hierarchy, and is not captured by the set of features available to the algorithm. This gives rise to the problem of "hierarchical clustering with structural constraints". Structural constraints pose major challenges for bottom-up approaches like average/single linkage and even though they can be naturally incorporated into top-down divisive algorithms, no formal guarantees exist on the quality of their output. In this paper, we provide provable approximation guarantees for two simple top-down algorithms, using a recently introduced optimization viewpoint of hierarchical clustering with pairwise similarity information [Dasgupta, 2016]. We show how to find good solutions even in the presence of conflicting prior information, by formulating a constraint-based regularization of the objective. We further explore a variation of this objective for dissimilarity information [Cohen-Addad et al., 2018] and improve upon current techniques. Finally, we demonstrate our approach on a real dataset for the taxonomy application., Comment: In Proc. 35th International Conference on Machine Learning (ICML 2018)

Edit distance is a fundamental measure of distance between strings and has been widely studied in computer science. While the problem of estimating edit distance has been studied extensively, the equally important question of actually producing an alignment (i.e., the sequence of edits) has received far less attention. Somewhat surprisingly, we show that any algorithm to estimate edit distance can be used in a black-box fashion to produce an approximate alignment of strings, with modest loss in approximation factor and small loss in run time. Plugging in the result of Andoni, Krauthgamer, and Onak, we obtain an alignment that is a $(\log n)^{O(1/\varepsilon^2)}$ approximation in time $\tilde{O}(n^{1 + \varepsilon})$. Closely related to the study of approximation algorithms is the study of metric embeddings for edit distance. We show that min-hash techniques can be useful in designing edit distance embeddings through three results: (1) An embedding from Ulam distance (edit distance over permutations) to Hamming space that matches the best known distortion of $O(\log n)$ and also implicitly encodes a sequence of edits between the strings; (2) In the case where the edit distance between the input strings is known to have an upper bound $K$, we show that embeddings of edit distance into Hamming space with distortion $f(n)$ can be modified in a black-box fashion to give distortion $O(f(\operatorname{poly}(K)))$ for a class of periodic-free strings; (3) A randomized dimension-reduction map with contraction $c$ and asymptotically optimal expected distortion $O(c)$, improving on the previous $\tilde{O}(c^{1 + 2 / \log \log \log n})$ distortion result of Batu, Ergun, and Sahinalp.

Modern networks run "middleboxes" that offer services ranging from network address translation and server load balancing to firewalls, encryption, and compression. In an industry trend known as Network Functions Virtualization (NFV), these middleboxes run as virtual machines on any commodity server, and the switches steer traffic through the relevant chain of services. Network administrators must decide how many middleboxes to run, where to place them, and how to direct traffic through them, based on the traffic load and the server and network capacity. Rather than placing specific kinds of middleboxes on each processing node, we argue that server virtualization allows each server node to host all middlebox functions, and simply vary the fraction of resources devoted to each one. This extra flexibility fundamentally changes the optimization problem the network administrators must solve to a new kind of multi-commodity flow problem, where the traffic flows consume bandwidth on the links as well as processing resources on the nodes. We show that allocating resources to maximize the processed flow can be optimized exactly via a linear programming formulation, and to arbitrary accuracy via an efficient combinatorial algorithm. Our experiments with real traffic and topologies show that a joint optimization of node and link resources leads to an efficient use of bandwidth and processing capacity. We also study a class of design problems that decide where to provide node capacity to best process and route a given set of demands, and demonstrate both approximation algorithms and hardness results for these problems.

We study the recently introduced problem of finding dense common subgraphs: Given a sequence of graphs that share the same vertex set, the goal is to find a subset of vertices $S$ that maximizes some aggregate measure of the density of the subgraphs induced by $S$ in each of the given graphs. Different choices for the aggregation function give rise to variants of the problem that were studied recently. We settle many of the questions left open by previous works, showing NP-hardness, hardness of approximation, non-trivial approximation algorithms, and an integrality gap for a natural relaxation.