Moses Charikar

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.

Moses Charikar, Chirag Pabbaraju, Ambuj Tewari

Curvature adaptivity is a classical theme in online optimization: for convex Lipschitz losses, adaptive methods interpolate between the optimal $O(\sqrt{T})$ regret for general convex losses and $O(\log T)$ regret under strong convexity. Recent work has shown that Follow-the-Perturbed-Leader (FTPL) achieves optimal $O(\sqrt{T})$ regret even for online non-convex Lipschitz losses, assuming access to an approximate offline-optimization oracle, but these guarantees do not exploit curvature. We show that FTPL can be made curvature-adaptive in the non-convex setting, without knowing in advance how curvature will accumulate over time. Our algorithm replaces the fixed perturbation scale of standard FTPL with a time-varying scale chosen using only past information. We give a simple follow-the-leader tuning rule for this scale and show that it competes, up to constants, with the best choice in hindsight. The resulting method achieves $O(\sqrt{T})$ regret for arbitrary non-convex Lipschitz losses and improves as cumulative curvature grows; with sufficiently accurate oracle calls, it achieves $O(\log T)$ regret when cumulative curvature grows linearly, which includes the classical strongly convex regime. We complement these upper bounds with matching lower bounds for prescribed cumulative-curvature sequences, already for one-dimensional convex losses, showing that the tradeoff between worst-case non-convex regret and curvature-driven fast rates is intrinsic.

Moses Charikar, Prasanna Ramakrishnan, Kangning Wang et al.

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 $3 - ε$ 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.

Moses Charikar, Chirag Pabbaraju

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.

Moses Charikar, Monika Henzinger, Lunjia Hu et al.

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 $Ω(ndk)$ 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.

Moses Charikar, Zhihao Jiang, Kirankumar Shiragur et al.

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 $ε\gg n^{-1/3}$. This result improves upon the previous best accuracy threshold of $ε\gg n^{-1/4}$ 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 $ε\ll n^{-1/3}$.

Moses Charikar, Chirag Pabbaraju, Ambuj Tewari

We study the problem of language identification in the limit, where given a sequence of examples from a target language, the goal of the learner is to output a sequence of guesses for the target language such that all the guesses beyond some finite time are correct. Classical results of Gold showed that language identification in the limit is impossible for essentially any interesting collection of languages. Later, Angluin gave a precise characterization of language collections for which this task is possible. Motivated by recent positive results for the related problem of language generation, we revisit the classic language identification problem in the setting where the learner is given the additional power of producing a list of $k$ guesses at each time step. The goal is to ensure that beyond some finite time, one of the guesses is correct at each time step. We give an exact characterization of collections of languages that can be $k$-list identified in the limit, based on a recursive version of Angluin's characterization (for language identification with a list of size $1$). This further leads to a conceptually appealing characterization: A language collection can be $k$-list identified in the limit if and only if the collection can be decomposed into $k$ collections of languages, each of which can be identified in the limit (with a list of size $1$). We also use our characterization to establish rates for list identification in the statistical setting where the input is drawn as an i.i.d. stream from a distribution supported on some language in the collection. Our results show that if a collection is $k$-list identifiable in the limit, then the collection can be $k$-list identified at an exponential rate, and this is best possible. On the other hand, if a collection is not $k$-list identifiable in the limit, then it cannot be $k$-list identified at any rate that goes to zero.

Moses Charikar, Chirag Pabbaraju

The recent work of Kleinberg & Mullainathan [KM24] provides a concrete model for language generation in the limit: given a sequence of examples from an unknown target language, the goal is to generate new examples from the target language such that no incorrect examples are generated beyond some point. In sharp contrast to strong negative results for the closely related problem of language identification, they establish positive results for language generation in the limit for all countable collections of languages. Follow-up work by Raman & Tewari [RT24] studies bounds on the number of distinct inputs required by an algorithm before correct language generation is achieved -- namely, whether this is a constant for all languages in the collection (uniform generation) or a language-dependent constant (non-uniform generation). We show that every countable language collection has a generator which has the stronger property of non-uniform generation in the limit. However, while the generation algorithm of [KM24] can be implemented using membership queries, we show that any algorithm cannot non-uniformly generate even for collections of just two languages, using only membership queries. We also formalize the tension between validity and breadth in the generation algorithm of [KM24] by introducing a definition of exhaustive generation, and show a strong negative result for exhaustive generation. Our result shows that a tradeoff between validity and breadth is inherent for generation in the limit. We also provide a precise characterization of the language collections for which exhaustive generation is possible. Finally, inspired by algorithms that can choose to obtain feedback, we consider a model of uniform generation with feedback, completely characterizing language collections for which such uniform generation with feedback is possible in terms of a complexity measure of the collection.

Kexin Rong, Paul Liu, Sarah Ashok Sonje et al.

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.

Moses Charikar, Chirag Pabbaraju

Kleinberg and Mullainathan (2024) recently proposed an interesting model for language generation in the limit: Given a countable collection of languages, and an adversary enumerating the strings of some language $L$ from the collection, the objective is to generate new strings from the target language, such that all strings generated beyond some finite time are valid. Li, Raman and Tewari (2024) and Charikar and Pabbaraju (2024) showed strong non-uniform generation guarantees in this model, giving algorithms that generate new valid strings from $L$ after seeing a number of distinct input strings $t(L)$ that depends only on $L$ (and the collection), but not the enumeration order. However, for both these works, the language-wise generation times $t(L)$ of the algorithm can be strictly sub-optimal. In this work, we study Pareto-optimality of non-uniform language generation in the limit. We propose an algorithm, whose generation times $t^\star(L)$ are (almost) Pareto-optimal: any other algorithm whose generation time for some language $L$ is strictly smaller than $t^\star(L)$, must satisfy that its generation time for some other language $L'$ is strictly worse than $t^\star(L')$. Pareto-optimality is essentially the best that one can achieve for non-uniform generation. Our algorithmic framework conveniently adapts to further give Pareto-optimal non-uniform generation algorithms in the practically motivated settings of noisy as well as representative generation.

Moses Charikar, Lunjia Hu

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).

Nima Anari, Moses Charikar, Kirankumar Shiragur et al.

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.

Nima Anari, Moses Charikar, Kirankumar Shiragur et al.

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.

Moses Charikar, Kirankumar Shiragur, Aaron Sidford

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.

Moses Charikar, Kirankumar Shiragur, Aaron Sidford

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 $ε= Ω(n^{-0.166})$. 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.

Hongyang R. Zhang, Vatsal Sharan, Moses Charikar et al.

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.

Moses Charikar, Vaggos Chatziafratis, Rad Niazadeh

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.

Vaggos Chatziafratis, Rad Niazadeh, Moses Charikar

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.

Jacob Steinhardt, Moses Charikar, Gregory Valiant

We introduce a criterion, resilience, which allows properties of a dataset (such as its mean or best low rank approximation) to be robustly computed, even in the presence of a large fraction of arbitrary additional data. Resilience is a weaker condition than most other properties considered so far in the literature, and yet enables robust estimation in a broader variety of settings. We provide new information-theoretic results on robust distribution learning, robust estimation of stochastic block models, and robust mean estimation under bounded $k$th moments. We also provide new algorithmic results on robust distribution learning, as well as robust mean estimation in $\ell_p$-norms. Among our proof techniques is a method for pruning a high-dimensional distribution with bounded $1$st moments to a stable "core" with bounded $2$nd moments, which may be of independent interest.

Yuchen Zhang, Percy Liang, Moses Charikar

We study the Stochastic Gradient Langevin Dynamics (SGLD) algorithm for non-convex optimization. The algorithm performs stochastic gradient descent, where in each step it injects appropriately scaled Gaussian noise to the update. We analyze the algorithm's hitting time to an arbitrary subset of the parameter space. Two results follow from our general theory: First, we prove that for empirical risk minimization, if the empirical risk is point-wise close to the (smooth) population risk, then the algorithm achieves an approximate local minimum of the population risk in polynomial time, escaping suboptimal local minima that only exist in the empirical risk. Second, we show that SGLD improves on one of the best known learnability results for learning linear classifiers under the zero-one loss.

Moses Charikar, Jacob Steinhardt, Gregory Valiant

The vast majority of theoretical results in machine learning and statistics assume that the available training data is a reasonably reliable reflection of the phenomena to be learned or estimated. Similarly, the majority of machine learning and statistical techniques used in practice are brittle to the presence of large amounts of biased or malicious data. In this work we consider two frameworks in which to study estimation, learning, and optimization in the presence of significant fractions of arbitrary data. The first framework, list-decodable learning, asks whether it is possible to return a list of answers, with the guarantee that at least one of them is accurate. For example, given a dataset of $n$ points for which an unknown subset of $αn$ points are drawn from a distribution of interest, and no assumptions are made about the remaining $(1-α)n$ points, is it possible to return a list of $\operatorname{poly}(1/α)$ answers, one of which is correct? The second framework, which we term the semi-verified learning model, considers the extent to which a small dataset of trusted data (drawn from the distribution in question) can be leveraged to enable the accurate extraction of information from a much larger but untrusted dataset (of which only an $α$-fraction is drawn from the distribution). We show strong positive results in both settings, and provide an algorithm for robust learning in a very general stochastic optimization setting. This general result has immediate implications for robust estimation in a number of settings, including for robustly estimating the mean of distributions with bounded second moments, robustly learning mixtures of such distributions, and robustly finding planted partitions in random graphs in which significant portions of the graph have been perturbed by an adversary.

Jacob Steinhardt, Gregory Valiant, Moses Charikar

We consider a crowdsourcing model in which $n$ workers are asked to rate the quality of $n$ items previously generated by other workers. An unknown set of $αn$ workers generate reliable ratings, while the remaining workers may behave arbitrarily and possibly adversarially. The manager of the experiment can also manually evaluate the quality of a small number of items, and wishes to curate together almost all of the high-quality items with at most an $ε$ fraction of low-quality items. Perhaps surprisingly, we show that this is possible with an amount of work required of the manager, and each worker, that does not scale with $n$: the dataset can be curated with $\tilde{O}\Big(\frac{1}{βα^3ε^4}\Big)$ ratings per worker, and $\tilde{O}\Big(\frac{1}{βε^2}\Big)$ ratings by the manager, where $β$ is the fraction of high-quality items. Our results extend to the more general setting of peer prediction, including peer grading in online classrooms.

Pranjal Awasthi, Moses Charikar, Kevin A. Lai et al.

We resolve an open question from (Christiano, 2014b) posed in COLT'14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework the algorithm is shown a pair of items at each step, chosen from a set of $n$ items. The learner then predicts a label for each item, from a label set of size $L$ and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as collaborative filtering, online gambling, and online max cut among others. (Christiano, 2014a) designed an efficient online learning algorithm for this problem achieving a regret of $O(\sqrt{nL^3T})$, where $T$ is the number of rounds. Information theoretically, one can achieve a regret of $O(\sqrt{n \log L T})$. One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of (Christiano, 2014a), in fact achieves a regret of $O(\sqrt{nLT})$. Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.

Pranjal Awasthi, Afonso S. Bandeira, Moses Charikar et al.

We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: $k$-means and $k$-median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are $k$ clusters in $\mathbb{R}^m$ and data from each cluster consists of $n$ points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these $k$ clusters as the optimal integral solution? For the $k$-median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation $ε> 0$ between the balls. In other words, the pairwise center separation is $Δ> 2+ε$. Under the same distributional model, the $k$-means LP relaxation fails to recover such clusters at separation as large as $Δ= 4$. Yet, if we enforce PSD constraints on the $k$-means LP, we get exact cluster recovery at center separation $Δ> 2\sqrt2(1+\sqrt{1/m})$. In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the $k$-means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest.

Aditya Bhaskara, Moses Charikar, Ankur Moitra et al.

Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and tensors analogs of much of the matrix algebra toolkit are unlikely to exist because of hardness results. Efficient decomposition in the overcomplete case (where rank exceeds dimension) is particularly challenging. We introduce a smoothed analysis model for studying these questions and develop an efficient algorithm for tensor decomposition in the highly overcomplete case (rank polynomial in the dimension). In this setting, we show that our algorithm is robust to inverse polynomial error -- a crucial property for applications in learning since we are only allowed a polynomial number of samples. While algorithms are known for exact tensor decomposition in some overcomplete settings, our main contribution is in analyzing their stability in the framework of smoothed analysis. Our main technical contribution is to show that tensor products of perturbed vectors are linearly independent in a robust sense (i.e. the associated matrix has singular values that are at least an inverse polynomial). This key result paves the way for applying tensor methods to learning problems in the smoothed setting. In particular, we use it to obtain results for learning multi-view models and mixtures of axis-aligned Gaussians where there are many more "components" than dimensions. The assumption here is that the model is not adversarially chosen, formalized by a perturbation of model parameters. We believe this an appealing way to analyze realistic instances of learning problems, since this framework allows us to overcome many of the usual limitations of using tensor methods.

Aditya Bhaskara, Moses Charikar, Aravindan Vijayaraghavan

We give a robust version of the celebrated result of Kruskal on the uniqueness of tensor decompositions: we prove that given a tensor whose decomposition satisfies a robust form of Kruskal's rank condition, it is possible to approximately recover the decomposition if the tensor is known up to a sufficiently small (inverse polynomial) error. Kruskal's theorem has found many applications in proving the identifiability of parameters for various latent variable models and mixture models such as Hidden Markov models, topic models etc. Our robust version immediately implies identifiability using only polynomially many samples in many of these settings. This polynomial identifiability is an essential first step towards efficient learning algorithms for these models. Recently, algorithms based on tensor decompositions have been used to estimate the parameters of various hidden variable models efficiently in special cases as long as they satisfy certain "non-degeneracy" properties. Our methods give a way to go beyond this non-degeneracy barrier, and establish polynomial identifiability of the parameters under much milder conditions. Given the importance of Kruskal's theorem in the tensor literature, we expect that this robust version will have several applications beyond the settings we explore in this work.