We give a sub exponential time approximation algorithm for the textsc{Unique Games} problem. The algorithms run in time that is exponential in an arbitrarily small polynomial of the input size, $n^{epsilon}$. The approximation guarantee depends on~$epsilon$, but not on the alphabet size or the number of variables. We also obtain a sub exponential algorithms with improved approximations for textsc{Small-Set Expansion} and textsc{Multicut}. For textsc{Max Cut}, textsc{Sparsest Cut}, and textsc{Vertex Cover}, we give sub exponential algorithms with improved approximations on some interesting subclasses of instances. Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for the textsc{Unique Games}. While our results stop short of refuting the UGC, they do suggest that textsc{Unique Games} is significantly easier than NP-hard problems such as textsc{Max 3Sat}, textsc{Max 3Lin}, textsc{Label Cover} and more, that are believed not to have a sub exponential algorithm achieving a non-trivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every $epsilon>0$ and every regular $n$-vertex graph~$G$, by changing at most $epsilon$ fraction of $G$'s edges, one can break~$G$ into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most $ n^{epsilon}$ eigenvalues larger than $1-eta$, where $eta$ depends polynomially on $epsilon$.
我们给出了textsc{独特的游戏}问题的次指数时间逼近算法。算法运行的时间是一个任意小的输入大小的多项式的指数，$n^{epsilon}$。近似保证取决于$epsilon$，但不取决于字母大小或变量数量。我们还得到了一种改进近似的次指数算法textsc{小集扩展}和textsc{多切口}。对于textsc{Max Cut}、textsc{最稀疏的切口}和textsc{顶点覆盖}，我们给出了对一些有趣的实例子类进行改进近似的次指数算法。Khot的Unique Games Conjecture (UGC)指出，实现近似保证(如textsc{独特的游戏})是np困难的。虽然我们的结果没有反驳UGC，但它们确实表明textsc{独特的游戏}比NP-hard问题(如textsc{最大3Sat}, textsc{最大3Lin}, textsc{标签封面}等)容易得多，这些问题被认为没有实现非平凡近似比的次指数算法。我们算法的主要组成部分是图分解的新结果，可能有其他应用。即对于每一个$epsilon>0$和每一个规则的$n$顶点图$G$，通过改变$G$的边的至多$epsilon$分数，可以将$G$分割成不相交的部分，使得每个部分上的诱导图的随机邻接矩阵的特征值至多$ n^{epsilon}$大于$1-eta$，其中$eta$多项式地依赖于$epsilon$。
{"title":"Subexponential Algorithms for Unique Games and Related Problems","authors":"Sanjeev Arora, B. Barak, David Steurer","doi":"10.1145/2775105","DOIUrl":"https://doi.org/10.1145/2775105","url":null,"abstract":"We give a sub exponential time approximation algorithm for the textsc{Unique Games} problem. The algorithms run in time that is exponential in an arbitrarily small polynomial of the input size, $n^{epsilon}$. The approximation guarantee depends on~$epsilon$, but not on the alphabet size or the number of variables. We also obtain a sub exponential algorithms with improved approximations for textsc{Small-Set Expansion} and textsc{Multicut}. For textsc{Max Cut}, textsc{Sparsest Cut}, and textsc{Vertex Cover}, we give sub exponential algorithms with improved approximations on some interesting subclasses of instances. Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for the textsc{Unique Games}. While our results stop short of refuting the UGC, they do suggest that textsc{Unique Games} is significantly easier than NP-hard problems such as textsc{Max 3Sat}, textsc{Max 3Lin}, textsc{Label Cover} and more, that are believed not to have a sub exponential algorithm achieving a non-trivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every $epsilon>0$ and every regular $n$-vertex graph~$G$, by changing at most $epsilon$ fraction of $G$'s edges, one can break~$G$ into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most $ n^{epsilon}$ eigenvalues larger than $1-eta$, where $eta$ depends polynomially on $epsilon$.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128529056","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 locally decodable code encodes a message by a codeword, such that even if the codeword is corrupted by noise, each message bit can be recovered with high probability by a randomized decoding procedure that reads only few bits of the codeword. Recently a new class of locally decodable codes, based on families of vectors with restricted dot products has been discovered. We refer to those codes as Matching Vector (MV) codes. In this work we develop a new view of MV codes and uncover certain similarities between them and classical Reed Muller codes. Our view allows us to obtain a deeper insight into the power and limitations of MV codes. We use it to construct codes that can tolerate more errors or are shorter than previously known codes for certain parameter settings. We also show super-linear lower bounds on the codeword length of any MV code.
{"title":"Matching Vector Codes","authors":"Zeev Dvir, Parikshit Gopalan, S. Yekhanin","doi":"10.1137/100804322","DOIUrl":"https://doi.org/10.1137/100804322","url":null,"abstract":"A locally decodable code encodes a message by a codeword, such that even if the codeword is corrupted by noise, each message bit can be recovered with high probability by a randomized decoding procedure that reads only few bits of the codeword. Recently a new class of locally decodable codes, based on families of vectors with restricted dot products has been discovered. We refer to those codes as Matching Vector (MV) codes. In this work we develop a new view of MV codes and uncover certain similarities between them and classical Reed Muller codes. Our view allows us to obtain a deeper insight into the power and limitations of MV codes. We use it to construct codes that can tolerate more errors or are shorter than previously known codes for certain parameter settings. We also show super-linear lower bounds on the codeword length of any MV code.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121699518","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 give new pseudorandom generators for emph{regular} read-once branching programs of small width. A branching program is regular if the in-degree of every vertex in it is either $0$ or $2$. For every width $d$ and length $n$, our pseudorandom generator uses a seed of length $O((log d + loglog n + log(1/epsilon))log n)$ to produce $n$ bits that cannot be distinguished from a uniformly random string by any regular width $d$ length $n$ read-once branching program, except with probability $epsilon$. We also give a result for general read-once branching programs, in the case that there are no vertices that are reached with small probability. We show that if a (possibly non-regular) branching program of length $n$ and width $d$ has the property that every vertex in the program is traversed with probability at least $gamma$ on a uniformly random input, then the error of the generator above is at most $2 epsilon/gamma^2$.
对于小宽度的emph{常规}读一次分支程序，我们给出了新的伪随机生成器。如果分支程序中每个顶点的in度为$0$或$2$，则分支程序是正则的。对于每个宽度$d$和长度$n$，我们的伪随机生成器使用长度$O((log d + loglog n + log(1/epsilon))log n)$的种子来生成$n$位，这些位不能被任何常规宽度$d$长度$n$读取一次的分支程序与均匀随机字符串区分开来，除非概率为$epsilon$。对于一般的只读一次分支程序，我们也给出了一个结果，在没有小概率到达的顶点的情况下。我们证明，如果一个长度为$n$，宽度为$d$的分支程序(可能是非规则的)具有这样的属性，即在均匀随机输入上，程序中的每个顶点都以至少$gamma$的概率遍历，那么上述生成器的误差最多为$2 epsilon/gamma^2$。
{"title":"Pseudorandom Generators for Regular Branching Programs","authors":"M. Braverman, Anup Rao, R. Raz, A. Yehudayoff","doi":"10.1137/120875673","DOIUrl":"https://doi.org/10.1137/120875673","url":null,"abstract":"We give new pseudorandom generators for emph{regular} read-once branching programs of small width. A branching program is regular if the in-degree of every vertex in it is either $0$ or $2$. For every width $d$ and length $n$, our pseudorandom generator uses a seed of length $O((log d + loglog n + log(1/epsilon))log n)$ to produce $n$ bits that cannot be distinguished from a uniformly random string by any regular width $d$ length $n$ read-once branching program, except with probability $epsilon$. We also give a result for general read-once branching programs, in the case that there are no vertices that are reached with small probability. We show that if a (possibly non-regular) branching program of length $n$ and width $d$ has the property that every vertex in the program is traversed with probability at least $gamma$ on a uniformly random input, then the error of the generator above is at most $2 epsilon/gamma^2$.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115941436","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 present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no sub graph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost linear-time algorithm for deciding FOL properties in classes of graphs with locally bounded expansion, those include classes of graphs with locally bounded tree-width or locally excluding a minor. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low tree-depth colorings.
给出了一种确定有界展开图类一阶逻辑(FOL)性质的线性时间算法。许多自然类别的图都有有界展开:有界树宽度的图，所有适当的小闭类图，有界度的图，没有子图同构于固定图的细分的图，以及可以在固定表面上绘制的图，这样每条边最多穿过常数个其他边。我们还开发了一种几乎线性时间的算法，用于确定具有局部有界展开的图类的FOL属性，这些图类包括具有局部有界树宽度或局部不含次元的图类。更一般地说，我们设计了一种动态数据结构的图属于一类固定的有界展开图。在线性时间初始化后，数据结构允许我们在恒定时间内测试FOL属性，并且在添加/删除边后可以在恒定时间内更新数据结构，前提是要添加的可能边的列表是已知的，并且它们的添加结果在类中的图中。此外，我们还设计了一个动态的数据结构，用于测试这类图中指定顶点之间的存在性或短路径的存在性。我们所有的结果也适用于关系结构，并基于Nesetril和Ossona de Mendez关于低树深着色存在性的开创性结果。
{"title":"Deciding First-Order Properties for Sparse Graphs","authors":"Z. Dvořák, D. Král, R. Thomas","doi":"10.1109/FOCS.2010.20","DOIUrl":"https://doi.org/10.1109/FOCS.2010.20","url":null,"abstract":"We present a linear-time algorithm for deciding first-order logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no sub graph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost linear-time algorithm for deciding FOL properties in classes of graphs with locally bounded expansion, those include classes of graphs with locally bounded tree-width or locally excluding a minor. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low tree-depth colorings.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122194548","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}
Y. Dodis, Kristiyan Haralambiev, Adriana López-Alt, D. Wichs
We say that a cryptographic scheme is Continuous Leakage-Resilient (CLR), if it allows users to refresh their secret keys, using only fresh local randomness, such that: 1. The scheme remains functional after any number of key refreshes, although the public key never changes. Thus, the “outside world'' is neither affected by these key refreshes, nor needs to know about their frequency. 2. The scheme remains secure even if the adversary can continuously leak arbitrary information about the current secret-key, as long as the amount of leaked information is bounded in between any two successive key refreshes. There is no bound on the total amount of information that can be leaked during the lifetime of the system. In this work, we construct a variety of practical CLR schemes, including CLR one-way relations, CLR signatures, CLR identification schemes, and CLR authenticated key agreement protocols. For each of the above, we give general constructions, and then show how to instantiate them efficiently using a well established assumption on bilinear groups, called the K-Linear assumption (for any constant K greater than or equal to 1). Our constructions are highly modular, and we develop many interesting techniques and building-blocks along the way, including: leakage-indistinguishable re-randomizable relations, homomorphic NIZKs, and leakage-of-cipher text non-malleable encryption schemes.
{"title":"Cryptography against Continuous Memory Attacks","authors":"Y. Dodis, Kristiyan Haralambiev, Adriana López-Alt, D. Wichs","doi":"10.1109/FOCS.2010.56","DOIUrl":"https://doi.org/10.1109/FOCS.2010.56","url":null,"abstract":"We say that a cryptographic scheme is Continuous Leakage-Resilient (CLR), if it allows users to refresh their secret keys, using only fresh local randomness, such that: 1. The scheme remains functional after any number of key refreshes, although the public key never changes. Thus, the “outside world'' is neither affected by these key refreshes, nor needs to know about their frequency. 2. The scheme remains secure even if the adversary can continuously leak arbitrary information about the current secret-key, as long as the amount of leaked information is bounded in between any two successive key refreshes. There is no bound on the total amount of information that can be leaked during the lifetime of the system. In this work, we construct a variety of practical CLR schemes, including CLR one-way relations, CLR signatures, CLR identification schemes, and CLR authenticated key agreement protocols. For each of the above, we give general constructions, and then show how to instantiate them efficiently using a well established assumption on bilinear groups, called the K-Linear assumption (for any constant K greater than or equal to 1). Our constructions are highly modular, and we develop many interesting techniques and building-blocks along the way, including: leakage-indistinguishable re-randomizable relations, homomorphic NIZKs, and leakage-of-cipher text non-malleable encryption schemes.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133326547","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}
There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F_2-linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan. We obtain the following results: 1. We show that every linear-invariant property that can be characterized by forbidding induced solutions to a (possibly infinite) set of linear equations can be tested with one-sided error. 2. We show that every linear-invariant property that can be tested with one-sided error can be characterized by forbidding induced solutions to a (possibly infinite) set of {em systems} of linear equations. We conjecture that our result from item (1) can be extended to cover systems of linear equations. We further show that the validity of this conjecture would have the following implications: 1. It would imply that every linear-invariant property that is closed under restrictions to linear subspaces is testable with one-sided error. Such a result would unify several previous results on testing Boolean functions, such as the testability of low-degree polynomials and of Fourier dimensionality. 2. It would imply that a linear-invariant property ${cal P}$ is testable with one-sided error {bf if and only if} ${cal P}$ is closed under restrictions to linear subspaces, thus resolving Sudan's problem.
{"title":"A Unified Framework for Testing Linear-Invariant Properties","authors":"Arnab Bhattacharyya, Elena Grigorescu, A. Shapira","doi":"10.1002/rsa.20507","DOIUrl":"https://doi.org/10.1002/rsa.20507","url":null,"abstract":"There has been a sequence of recent papers devoted to understanding the relation between the testability of properties of Boolean functions and the invariance of the properties with respect to transformations of the domain. Invariance with respect to F_2-linear transformations is arguably the most common such symmetry for natural properties of Boolean functions on the hypercube. Hence, it is an important goal to find necessary and sufficient conditions for testability of linear-invariant properties. This is explicitly posed as an open problem in a recent survey of Sudan. We obtain the following results: 1. We show that every linear-invariant property that can be characterized by forbidding induced solutions to a (possibly infinite) set of linear equations can be tested with one-sided error. 2. We show that every linear-invariant property that can be tested with one-sided error can be characterized by forbidding induced solutions to a (possibly infinite) set of {em systems} of linear equations. We conjecture that our result from item (1) can be extended to cover systems of linear equations. We further show that the validity of this conjecture would have the following implications: 1. It would imply that every linear-invariant property that is closed under restrictions to linear subspaces is testable with one-sided error. Such a result would unify several previous results on testing Boolean functions, such as the testability of low-degree polynomials and of Fourier dimensionality. 2. It would imply that a linear-invariant property ${cal P}$ is testable with one-sided error {bf if and only if} ${cal P}$ is closed under restrictions to linear subspaces, thus resolving Sudan's problem.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133565976","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}
Boosting is a general method for improving the accuracy of learning algorithms. We use boosting to construct improved {em privacy-preserving synopses} of an input database. These are data structures that yield, for a given set $Q$ of queries over an input database, reasonably accurate estimates of the responses to every query in~$Q$, even when the number of queries is much larger than the number of rows in the database. Given a {em base synopsis generator} that takes a distribution on $Q$ and produces a ``weak'' synopsis that yields ``good'' answers for a majority of the weight in $Q$, our {em Boosting for Queries} algorithm obtains a synopsis that is good for all of~$Q$. We ensure privacy for the rows of the database, but the boosting is performed on the {em queries}. We also provide the first synopsis generators for arbitrary sets of arbitrary low-sensitivity queries, {it i.e.}, queries whose answers do not vary much under the addition or deletion of a single row. In the execution of our algorithm certain tasks, each incurring some privacy loss, are performed many times. To analyze the cumulative privacy loss, we obtain an $O(eps^2)$ bound on the {em expected} privacy loss from a single $eps$-dfp{} mechanism. Combining this with evolution of confidence arguments from the literature, we get stronger bounds on the expected cumulative privacy loss due to multiple mechanisms, each of which provides $eps$-differential privacy or one of its relaxations, and each of which operates on (potentially) different, adaptively chosen, databases.
增强是提高学习算法准确性的一种通用方法。我们使用增强来构造一个输入数据库的改进的{em隐私保护概要}。对于输入数据库上给定的查询集$ $Q$，这些数据结构可以产生对~$ $Q$中每个查询的响应的合理准确的估计，即使查询的数量远远大于数据库中的行数。给定一个{em基础概要生成器}，它在$Q$上取一个分布，并产生一个“弱”概要，该概要对$Q$中的大部分权重产生“好”答案，我们的{em Boosting for Queries}算法获得一个对所有~$Q$都好的概要。我们确保数据库行的私密性，但增强是在{em查询}上执行的。我们还为任意低灵敏度查询的任意集提供了第一个概要生成器，{it i.e.}，这些查询的答案在添加或删除单行时不会发生太大变化。在我们算法的执行过程中，某些会导致隐私丢失的任务会被执行多次。为了分析累积隐私损失，我们从单个$eps$-dfp{}机制中获得了{em期望}隐私损失的$O(eps^2)$界。将其与文献中可信度论证的演变相结合，我们得到了由于多种机制而导致的预期累积隐私损失的更强界限，每种机制都提供了$eps$差分隐私或其松弛之一，并且每种机制都在(可能)不同的，自适应选择的数据库上运行。
{"title":"Boosting and Differential Privacy","authors":"C. Dwork, G. Rothblum, S. Vadhan","doi":"10.1109/FOCS.2010.12","DOIUrl":"https://doi.org/10.1109/FOCS.2010.12","url":null,"abstract":"Boosting is a general method for improving the accuracy of learning algorithms. We use boosting to construct improved {em privacy-preserving synopses} of an input database. These are data structures that yield, for a given set $Q$ of queries over an input database, reasonably accurate estimates of the responses to every query in~$Q$, even when the number of queries is much larger than the number of rows in the database. Given a {em base synopsis generator} that takes a distribution on $Q$ and produces a ``weak'' synopsis that yields ``good'' answers for a majority of the weight in $Q$, our {em Boosting for Queries} algorithm obtains a synopsis that is good for all of~$Q$. We ensure privacy for the rows of the database, but the boosting is performed on the {em queries}. We also provide the first synopsis generators for arbitrary sets of arbitrary low-sensitivity queries, {it i.e.}, queries whose answers do not vary much under the addition or deletion of a single row. In the execution of our algorithm certain tasks, each incurring some privacy loss, are performed many times. To analyze the cumulative privacy loss, we obtain an $O(eps^2)$ bound on the {em expected} privacy loss from a single $eps$-dfp{} mechanism. Combining this with evolution of confidence arguments from the literature, we get stronger bounds on the expected cumulative privacy loss due to multiple mechanisms, each of which provides $eps$-differential privacy or one of its relaxations, and each of which operates on (potentially) different, adaptively chosen, databases.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122246253","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 construct the first general secure computation protocols that require no trusted infrastructure other than authenticated communication, and that satisfy a meaningful notion of security that is preserved under universal composition—{em assuming only the existence of enhanced trapdoor permutations.} The notion of security fits within a generalization of the ``angel-based'' framework of Prabhakaran and Sahai (STOC'04) and implies super-polynomial time simulation security. Security notions of this kind are currently known to be realizable only under strong and specific hardness assumptions. A key element in our construction is a commitment scheme that satisfies a new and strong notion of security. The notion, security against chosen-commitment-attacks (CCA security), means that security holds even if the attacker has access to a {em extraction oracle} that gives the adversary decommitment information to commitments of the adversary's choice. This notion is stronger than concurrent non-malleability and is of independent interest. We construct CCA-secure commitments based on standard one-way functions, and with no trusted set-up. To the best of our knowledge, this provides the first construction of a natural cryptographic primitive requiring emph{adaptive hardness} from standard hardness assumptions, using no trusted set-up or public keys.
{"title":"Adaptive Hardness and Composable Security in the Plain Model from Standard Assumptions","authors":"R. Canetti, Huijia Lin, R. Pass","doi":"10.1137/110847196","DOIUrl":"https://doi.org/10.1137/110847196","url":null,"abstract":"We construct the first general secure computation protocols that require no trusted infrastructure other than authenticated communication, and that satisfy a meaningful notion of security that is preserved under universal composition—{em assuming only the existence of enhanced trapdoor permutations.} The notion of security fits within a generalization of the ``angel-based'' framework of Prabhakaran and Sahai (STOC'04) and implies super-polynomial time simulation security. Security notions of this kind are currently known to be realizable only under strong and specific hardness assumptions. A key element in our construction is a commitment scheme that satisfies a new and strong notion of security. The notion, security against chosen-commitment-attacks (CCA security), means that security holds even if the attacker has access to a {em extraction oracle} that gives the adversary decommitment information to commitments of the adversary's choice. This notion is stronger than concurrent non-malleability and is of independent interest. We construct CCA-secure commitments based on standard one-way functions, and with no trusted set-up. To the best of our knowledge, this provides the first construction of a natural cryptographic primitive requiring emph{adaptive hardness} from standard hardness assumptions, using no trusted set-up or public keys.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123594941","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 correct a claim from cite{Vempala97} and provide a status update.
我们更正来自cite{Vempala97}的声明并提供状态更新。
{"title":"Corrigendum: A Random Sampling Algorithm for Learning an Intersection of Halfspaces","authors":"S. Vempala","doi":"10.1109/FOCS.2010.18","DOIUrl":"https://doi.org/10.1109/FOCS.2010.18","url":null,"abstract":"We correct a claim from cite{Vempala97} and provide a status update.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128499383","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 give sub linear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions of these problems, such as SVDD, hard margin SVM, and $L_2$-SVM, for which sub linear-time algorithms were not known before. These new algorithms use a combination of a novel sampling techniques and a new multiplicative update algorithm. We give lower bounds which show the running times of many of our algorithms to be nearly best possible in the unit-cost RAM model. We also give implementations of our algorithms in the semi-streaming setting, obtaining the first low pass polylogarithmic space and sub linear time algorithms achieving arbitrary approximation factor.
{"title":"Sublinear Optimization for Machine Learning","authors":"K. Clarkson, Elad Hazan, David P. Woodruff","doi":"10.1145/2371656.2371658","DOIUrl":"https://doi.org/10.1145/2371656.2371658","url":null,"abstract":"We give sub linear-time approximation algorithms for some optimization problems arising in machine learning, such as training linear classifiers and finding minimum enclosing balls. Our algorithms can be extended to some kernelized versions of these problems, such as SVDD, hard margin SVM, and $L_2$-SVM, for which sub linear-time algorithms were not known before. These new algorithms use a combination of a novel sampling techniques and a new multiplicative update algorithm. We give lower bounds which show the running times of many of our algorithms to be nearly best possible in the unit-cost RAM model. We also give implementations of our algorithms in the semi-streaming setting, obtaining the first low pass polylogarithmic space and sub linear time algorithms achieving arbitrary approximation factor.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127809186","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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