We show that every algorithm for testing n-variate Boolean functions for monotonicityhas query complexity Ω(n1/4). All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only Ω(logn). Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015)recently showed that non-adaptive algorithms require almost Ω(n1/2) queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity O(logn) when the input is a regular LTF.
我们证明了每个用于测试n变量布尔函数单调性的算法都具有查询复杂度Ω(n1/4)。该问题的所有先前的下界都是为非自适应算法设计的，因此，一般(可能自适应)单调性测试器的最佳先前下界仅为Ω(logn)。结合Khot, Minzer和Safra (FOCS 2015)的非自适应单调性测试仪的查询复杂度，我们的下界表明，自适应最多可以使测试单调性的查询复杂度降低二次。通过对比，我们发现用于测试正则线性阈值函数(ltf)单调性的自适应和非自适应算法的查询复杂度之间存在指数差距。Chen, De, Servedio和Tan (STOC 2015)最近表明，非自适应算法几乎需要Ω(n /2)次查询才能完成此任务。提出了一种新的自适应单调性测试算法，该算法在输入为正则LTF时查询复杂度为O(logn)。
{"title":"A polynomial lower bound for testing monotonicity","authors":"Aleksandrs Belovs, Eric Blais","doi":"10.1145/2897518.2897567","DOIUrl":"https://doi.org/10.1145/2897518.2897567","url":null,"abstract":"We show that every algorithm for testing n-variate Boolean functions for monotonicityhas query complexity Ω(n1/4). All previous lower bounds for this problem were designed for non-adaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only Ω(logn). Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015)recently showed that non-adaptive algorithms require almost Ω(n1/2) queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity O(logn) when the input is a regular LTF.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116268119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Daskalakis, Anindya De, Gautam Kamath, Christos Tzamos
An (n,k)-Poisson Multinomial Distribution (PMD) is the distribution of the sum of n independent random vectors supported on the set Bk={e1,…,ek} of standard basis vectors in ℝk. We show that any (n,k)-PMD is poly(k/σ)-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on n from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structural characterization of PMDs shown in recent work by Daskalakis, Kamath and Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS for approximate Nash equilibria in anonymous games, significantly improving the state of the art, and matching qualitatively the running time dependence on n and 1/є of the best known algorithm for two-strategy anonymous games. Our new CLT also enables the construction of covers for the set of (n,k)-PMDs, which are proper and whose size is shown to be essentially optimal. Our cover construction combines our CLT with the Shapley-Folkman theorem and recent sparsification results for Laplacian matrices by Batson, Spielman, and Srivastava. Our cover size lower bound is based on an algebraic geometric construction. Finally, leveraging the structural properties of the Fourier spectrum of PMDs we show that these distributions can be learned from Ok(1/є2) samples in polyk(1/є)-time, removing the quasi-polynomial dependence of the running time on 1/є from prior work.
{"title":"A size-free CLT for poisson multinomials and its applications","authors":"C. Daskalakis, Anindya De, Gautam Kamath, Christos Tzamos","doi":"10.1145/2897518.2897519","DOIUrl":"https://doi.org/10.1145/2897518.2897519","url":null,"abstract":"An (n,k)-Poisson Multinomial Distribution (PMD) is the distribution of the sum of n independent random vectors supported on the set Bk={e1,…,ek} of standard basis vectors in ℝk. We show that any (n,k)-PMD is poly(k/σ)-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on n from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structural characterization of PMDs shown in recent work by Daskalakis, Kamath and Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS for approximate Nash equilibria in anonymous games, significantly improving the state of the art, and matching qualitatively the running time dependence on n and 1/є of the best known algorithm for two-strategy anonymous games. Our new CLT also enables the construction of covers for the set of (n,k)-PMDs, which are proper and whose size is shown to be essentially optimal. Our cover construction combines our CLT with the Shapley-Folkman theorem and recent sparsification results for Laplacian matrices by Batson, Spielman, and Srivastava. Our cover size lower bound is based on an algebraic geometric construction. Finally, leveraging the structural properties of the Fourier spectrum of PMDs we show that these distributions can be learned from Ok(1/є2) samples in polyk(1/є)-time, removing the quasi-polynomial dependence of the running time on 1/є from prior work.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"29 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123498620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An (n, k)-Poisson Multinomial Distribution (PMD) is a random variable of the form X = ∑i=1n Xi, where the Xi’s are independent random vectors supported on the set of standard basis vectors in k. In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is approximately sparse, i.e., its L1-norm is small outside a small set. By building on this result, we obtain the following applications: Learning Theory. We give the first computationally efficient learning algorithm for PMDs under the total variation distance. Our algorithm learns an (n, k)-PMD within variation distance ε using a near-optimal sample size of Ok(1/ε2), and runs in time Ok(1/ε2) · logn. Previously, no algorithm with a (1/ε) runtime was known, even for k=3. Game Theory. We give the first efficient polynomial-time approximation scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized anonymous games with n players and k strategies, our algorithm computes a well-supported ε-Nash equilibrium in time nO(k3) · (k/ε)O(k3log(k/ε)/loglog(k/ε))k−1. The best previous algorithm for this problem had running time n(f(k)/ε)k, where f(k) = Ω(kk2), for any k>2. Statistics. We prove a multivariate central limit theorem (CLT) that relates an arbitrary PMD to a discretized multivariate Gaussian with the same mean and covariance, in total variation distance. Our new CLT strengthens the CLT of Valiant and Valiant by removing the dependence on n in the error bound. Along the way we prove several new structural results of independent interest about PMDs. These include: (i) a robust moment-matching lemma, roughly stating that two PMDs that approximately agree on their low-degree parameter moments are close in variation distance; (ii) near-optimal size proper ε-covers for PMDs in total variation distance (constructive upper bound and nearly-matching lower bound). In addition to Fourier analysis, we employ a number of analytic tools, including the saddlepoint method from complex analysis, that may find other applications.
{"title":"The fourier transform of poisson multinomial distributions and its algorithmic applications","authors":"Ilias Diakonikolas, D. Kane, Alistair Stewart","doi":"10.1145/2897518.2897552","DOIUrl":"https://doi.org/10.1145/2897518.2897552","url":null,"abstract":"An (n, k)-Poisson Multinomial Distribution (PMD) is a random variable of the form X = ∑i=1n Xi, where the Xi’s are independent random vectors supported on the set of standard basis vectors in k. In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is approximately sparse, i.e., its L1-norm is small outside a small set. By building on this result, we obtain the following applications: Learning Theory. We give the first computationally efficient learning algorithm for PMDs under the total variation distance. Our algorithm learns an (n, k)-PMD within variation distance ε using a near-optimal sample size of Ok(1/ε2), and runs in time Ok(1/ε2) · logn. Previously, no algorithm with a (1/ε) runtime was known, even for k=3. Game Theory. We give the first efficient polynomial-time approximation scheme (EPTAS) for computing Nash equilibria in anonymous games. For normalized anonymous games with n players and k strategies, our algorithm computes a well-supported ε-Nash equilibrium in time nO(k3) · (k/ε)O(k3log(k/ε)/loglog(k/ε))k−1. The best previous algorithm for this problem had running time n(f(k)/ε)k, where f(k) = Ω(kk2), for any k>2. Statistics. We prove a multivariate central limit theorem (CLT) that relates an arbitrary PMD to a discretized multivariate Gaussian with the same mean and covariance, in total variation distance. Our new CLT strengthens the CLT of Valiant and Valiant by removing the dependence on n in the error bound. Along the way we prove several new structural results of independent interest about PMDs. These include: (i) a robust moment-matching lemma, roughly stating that two PMDs that approximately agree on their low-degree parameter moments are close in variation distance; (ii) near-optimal size proper ε-covers for PMDs in total variation distance (constructive upper bound and nearly-matching lower bound). In addition to Fourier analysis, we employ a number of analytic tools, including the saddlepoint method from complex analysis, that may find other applications.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133018193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Raef Bassily, Kobbi Nissim, Adam D. Smith, T. Steinke, Uri Stemmer, Jonathan Ullman
Adaptivity is an important feature of data analysis - the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated a general formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution P and a set of n independent samples x is drawn from P. We seek an algorithm that, given x as input, accurately answers a sequence of adaptively chosen ``queries'' about the unknown distribution P. How many samples n must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions towards resolving this question: We give upper bounds on the number of samples n that are needed to answer statistical queries. The bounds improve and simplify the work of Dwork et al. (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015; NIPS, 2015). We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries (alternatively, risk minimization queries). As in Dwork et al., our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that the stability notion guaranteed by differential privacy implies low generalization error. We also show that weaker stability guarantees such as bounded KL divergence and total variation distance lead to correspondingly weaker generalization guarantees.
{"title":"Algorithmic stability for adaptive data analysis","authors":"Raef Bassily, Kobbi Nissim, Adam D. Smith, T. Steinke, Uri Stemmer, Jonathan Ullman","doi":"10.1145/2897518.2897566","DOIUrl":"https://doi.org/10.1145/2897518.2897566","url":null,"abstract":"Adaptivity is an important feature of data analysis - the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by Dwork et al. (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated a general formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution P and a set of n independent samples x is drawn from P. We seek an algorithm that, given x as input, accurately answers a sequence of adaptively chosen ``queries'' about the unknown distribution P. How many samples n must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions towards resolving this question: We give upper bounds on the number of samples n that are needed to answer statistical queries. The bounds improve and simplify the work of Dwork et al. (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015; NIPS, 2015). We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries (alternatively, risk minimization queries). As in Dwork et al., our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that the stability notion guaranteed by differential privacy implies low generalization error. We also show that weaker stability guarantees such as bounded KL divergence and total variation distance lead to correspondingly weaker generalization guarantees.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133959710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter is a reasonable approach when the bidders can be classified into a few categories, but there are many instances where the classification of bidders is a continuum. For example, the classification of the bidders may be based on their annual income, their propensity to buy an item based on past behavior, or in the case of ad auctions, the click through rate of their ads. We introduce an alternate model that captures this aspect, where bidders are a priori identical, but can be distinguished based (only) on some side information the auctioneer obtains at the time of the auction. We extend the sample complexity approach of Dhangwatnotai et al. and Cole and Roughgarden to this model and obtain almost matching upper and lower bounds. As an aside, we obtain a revenue monotonicity lemma which may be of independent interest. We also show how to use Empirical Risk Minimization techniques to improve the sample complexity bound of Cole and Roughgarden for the non-identical but independent value distribution case.
传统上，贝叶斯最优拍卖设计问题要么考虑竞标者的价值是独立的，要么考虑每个竞标者的价值分布是可识别的。当投标人可以分为几个类别时，后者是一种合理的方法，但在许多情况下，投标人的分类是一个连续体。例如，对竞标者的分类可能是基于他们的年收入，他们基于过去的行为购买物品的倾向，或者在广告拍卖的情况下，他们的广告的点击率。我们引入了一个捕捉这方面的替代模型，其中竞标者是先验相同的，但可以根据拍卖师在拍卖时获得的一些附带信息来区分。我们将Dhangwatnotai et al.和Cole and Roughgarden的样本复杂度方法推广到该模型，得到了几乎匹配的上界和下界。作为题外话，我们得到了一个有独立意义的收入单调引理。我们还展示了如何使用经验风险最小化技术来改进Cole和Roughgarden的样本复杂性界，以解决非相同但独立的值分布情况。
{"title":"The sample complexity of auctions with side information","authors":"Nikhil R. Devanur, Zhiyi Huang, Alexandros Psomas","doi":"10.1145/2897518.2897553","DOIUrl":"https://doi.org/10.1145/2897518.2897553","url":null,"abstract":"Traditionally, the Bayesian optimal auction design problem has been considered either when the bidder values are i.i.d, or when each bidder is individually identifiable via her value distribution. The latter is a reasonable approach when the bidders can be classified into a few categories, but there are many instances where the classification of bidders is a continuum. For example, the classification of the bidders may be based on their annual income, their propensity to buy an item based on past behavior, or in the case of ad auctions, the click through rate of their ads. We introduce an alternate model that captures this aspect, where bidders are a priori identical, but can be distinguished based (only) on some side information the auctioneer obtains at the time of the auction. We extend the sample complexity approach of Dhangwatnotai et al. and Cole and Roughgarden to this model and obtain almost matching upper and lower bounds. As an aside, we obtain a revenue monotonicity lemma which may be of independent interest. We also show how to use Empirical Risk Minimization techniques to improve the sample complexity bound of Cole and Roughgarden for the non-identical but independent value distribution case.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116639683","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.
{"title":"Separations in query complexity using cheat sheets","authors":"S. Aaronson, S. Ben-David, Robin Kothari","doi":"10.1145/2897518.2897644","DOIUrl":"https://doi.org/10.1145/2897518.2897644","url":null,"abstract":"We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127537742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r
{"title":"Base collapse of holographic algorithms","authors":"Mingji Xia","doi":"10.1145/2897518.2897560","DOIUrl":"https://doi.org/10.1145/2897518.2897560","url":null,"abstract":"A holographic algorithm solves a problem in a domain of size n, by reducing it to counting perfect matchings in planar graphs. It may simulate a n-value variable by a bunch of t matchgate bits, which has 2t values. The transformation in the simulation can be expressed as a n × 2t matrix M, called the base of the holographic algorithm. We wonder whether more matchgate bits bring us more powerful holographic algorithms. In another word, whether we can solve the same original problem, with a collapsed base of size n × 2r, where r<t. Base collapse is discovered for small domain n=2,3,4. For n=3, 4, the base collapse is proved under the condition that there is a full rank generator. We prove for any n, the base collapse to a r≤ ⌊ logn ⌋, under some similar conditions. One of the conditions is that the original problem is defined by one symmetric function. In the proof, we utilize elementary matchgate transformations instead of matchgate identities.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126233523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, Decelle et al. conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulie proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make `helpful' changes that strengthen ties within each community and break ties between them. We show a surprising result that these `helpful' changes can shift the information-theoretic threshold, making the community detection problem strictly harder. We complement this by showing that an algorithm based on semidefinite programming (which was known to get close to the threshold) continues to work in the semirandom model (even for partial recovery). This suggests that algorithms based on semidefinite programming are robust in ways that any algorithm meeting the information-theoretic threshold cannot be. These results point to an interesting new direction: Can we find robust, semirandom analogues to some of the classical, average-case thresholds in statistics? We also explore this question in the broadcast tree model, and we show that the viewpoint of semirandom models can help explain why some algorithms are preferred to others in practice, in spite of the gaps in their statistical performance on random models.
{"title":"How robust are reconstruction thresholds for community detection?","authors":"Ankur Moitra, William Perry, Alexander S. Wein","doi":"10.1145/2897518.2897573","DOIUrl":"https://doi.org/10.1145/2897518.2897573","url":null,"abstract":"The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, Decelle et al. conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulie proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make `helpful' changes that strengthen ties within each community and break ties between them. We show a surprising result that these `helpful' changes can shift the information-theoretic threshold, making the community detection problem strictly harder. We complement this by showing that an algorithm based on semidefinite programming (which was known to get close to the threshold) continues to work in the semirandom model (even for partial recovery). This suggests that algorithms based on semidefinite programming are robust in ways that any algorithm meeting the information-theoretic threshold cannot be. These results point to an interesting new direction: Can we find robust, semirandom analogues to some of the classical, average-case thresholds in statistics? We also explore this question in the broadcast tree model, and we show that the viewpoint of semirandom models can help explain why some algorithms are preferred to others in practice, in spite of the gaps in their statistical performance on random models.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115367273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear change of basis. Two fundamental parameters in holographic algorithms are the domain size and the basis size. Roughly, the domain size is the range of colors involved in the counting problem at hand (e.g. counting graph k-colorings is a problem over domain size k), while the basis size captures the dimensionality of the representation of those colors. A major open problem has been: for a given k, what is the smallest ℓ for which any holographic algorithm for a problem over domain size k "collapses to" (can be simulated by) a holographic algorithm with basis size ℓ? Cai and Lu showed in 2008 that over domain size 2, basis size 1 suffices, opening the door to an extensive line of work on the structural theory of holographic algorithms over the Boolean domain. Cai and Fu later showed for signatures of full rank that over domain sizes 3 and 4, basis sizes 1 and 2, respectively, suffice, and they conjectured that over domain size k there is a collapse to basis size ⌊log2 k⌋. In this work, we resolve this conjecture in the affirmative for signatures of full rank for all k.
{"title":"Basis collapse for holographic algorithms over all domain sizes","authors":"Sitan Chen","doi":"10.1145/2897518.2897546","DOIUrl":"https://doi.org/10.1145/2897518.2897546","url":null,"abstract":"The theory of holographic algorithms introduced by Valiant represents a novel approach to achieving polynomial-time algorithms for seemingly intractable counting problems via a reduction to counting planar perfect matchings and a linear change of basis. Two fundamental parameters in holographic algorithms are the domain size and the basis size. Roughly, the domain size is the range of colors involved in the counting problem at hand (e.g. counting graph k-colorings is a problem over domain size k), while the basis size captures the dimensionality of the representation of those colors. A major open problem has been: for a given k, what is the smallest ℓ for which any holographic algorithm for a problem over domain size k \"collapses to\" (can be simulated by) a holographic algorithm with basis size ℓ? Cai and Lu showed in 2008 that over domain size 2, basis size 1 suffices, opening the door to an extensive line of work on the structural theory of holographic algorithms over the Boolean domain. Cai and Fu later showed for signatures of full rank that over domain sizes 3 and 4, basis sizes 1 and 2, respectively, suffice, and they conjectured that over domain size k there is a collapse to basis size ⌊log2 k⌋. In this work, we resolve this conjecture in the affirmative for signatures of full rank for all k.","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115430418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Brandt, O. Fischer, J. Hirvonen, Barbara Keller, Tuomo Lempiäinen, J. Rybicki, J. Suomela, Jara Uitto
We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Omega(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Omega(log* n) rounds [Chung et al. 2014].
我们证明了任何用于Lovász局部引理的随机蒙特卡罗分布式算法都需要Omega(log log n)轮通信，假设它以高概率找到正确的分配。我们的结果即使在d = O(1)的特殊情况下也成立，其中d是依赖图的最大程度。根据之前的工作，Lovász局部引理的分布式算法在有界度图中运行时间为O(log n)轮，在我们的工作之前，最好的下界是Omega(log* n)轮[Chung et al. 2014]。
{"title":"A lower bound for the distributed Lovász local lemma","authors":"S. Brandt, O. Fischer, J. Hirvonen, Barbara Keller, Tuomo Lempiäinen, J. Rybicki, J. Suomela, Jara Uitto","doi":"10.1145/2897518.2897570","DOIUrl":"https://doi.org/10.1145/2897518.2897570","url":null,"abstract":"We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Omega(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Omega(log* n) rounds [Chung et al. 2014].","PeriodicalId":442965,"journal":{"name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","volume":"386 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117090206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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