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28
The multiplicative weights update method: a meta algorithm and applications
, 2005
"... Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies ..."
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Cited by 147 (13 self)
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Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1
Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization
, 2008
"... Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rankrevealing QR, which seeks a wellconditioned collection of colu ..."
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Cited by 20 (1 self)
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Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called rankrevealing QR, which seeks a wellconditioned collection of columns that spans the (numerical) range of the matrix. The functional analysis literature contains another strand of work on column selection whose algorithmic implications have not been explored. In particular, a celebrated result of Bourgain and Tzafriri demonstrates that each matrix with normalized columns contains a large column submatrix that is exceptionally well conditioned. Unfortunately, standard proofs of this result cannot be regarded as algorithmic. This paper presents
ROBUST COMPUTATION OF LINEAR MODELS, OR HOW TO FIND A NEEDLE IN A HAYSTACK
"... Abstract. Consider a dataset of vectorvalued observations that consists of a modest number of noisy inliers, which are explained well by a lowdimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reape ..."
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Cited by 18 (5 self)
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Abstract. Consider a dataset of vectorvalued observations that consists of a modest number of noisy inliers, which are explained well by a lowdimensional subspace, along with a large number of outliers, which have no linear structure. This work describes a convex optimization problem, called reaper, that can reliably fit a lowdimensional model to this type of data. The paper provides an efficient algorithm for solving the reaper problem, and it documents numerical experiments which confirm that reaper can dependably find linear structure in synthetic and natural data. In addition, when the inliers are contained in a lowdimensional subspace, there is a rigorous theory that describes when reaper can recover the subspace exactly. 1.
Parallel approximation of noninteractive zerosum quantum games
, 2008
"... This paper studies a simple class of zerosum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players ’ payoffs. We prove that an equilibrium point of any such game can be approxi ..."
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Cited by 13 (3 self)
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This paper studies a simple class of zerosum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players ’ payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that oneturn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space. 1
Approximating the exponential, the Lanczos method, and an Õ(m)time spectral algorithm for balanced separator
 IN: PROCEEDINGS OF THE 44TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC
, 2012
"... We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O ( √ γ) in G, or outputs a certificate that all bbalanced cuts in G have conductance at le ..."
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Cited by 10 (3 self)
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We give a novel spectral approximation algorithm for the balanced separator problem that, given a graph G, a constant balance b ∈ (0, 1/2], and a parameter γ, either finds an Ω(b)balanced cut of conductance O ( √ γ) in G, or outputs a certificate that all bbalanced cuts in G have conductance at least γ, and runs in time Õ(m). This settles the question of designing asymptotically optimal spectral algorithms for balanced separator. Our algorithm relies on a variant of the heat kernel random walk and requires, as a subroutine, an algorithm to compute exp(−L)v where L is the Laplacian of a graph related to G and v is a vector. Algorithms for computing the matrixexponentialvector product efficiently comprise our next set of results. Our main result here is a new algorithm which computes a good approximation to exp(−A)v for a class of symmetric positive semidefinite (PSD) matrices A and a given vector u, in time roughly Õ(m A), where m A is the number of nonzero entries of A. This uses, in a nontrivial way, the breakthrough result of Spielman and Teng on inverting symmetric and diagonallydominant matrices in Õ(m A) time. Finally, we prove that e −x can be uniformly approximated up to a small additive error, in a nonnegative interval [a, b] with a polynomial of
Parallel approximation of minmax problems with applications to classical and quantum zerosum games. Computational Complexity, 22(2):385428, 2013, the special issue of CCC 2012
 In Proceedings of the 27rd Annual IEEE Conference on Computational Complexity (CCC 2012
, 2012
"... Abstract This paper presents an efficient parallel algorithm for a new class of minmax problems based on the matrix multiplicative weight (MMW) update method. Our algorithm can be used to find nearoptimal strategies for competitive twoplayer classical or quantum games in which a referee exchange ..."
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Cited by 6 (3 self)
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Abstract This paper presents an efficient parallel algorithm for a new class of minmax problems based on the matrix multiplicative weight (MMW) update method. Our algorithm can be used to find nearoptimal strategies for competitive twoplayer classical or quantum games in which a referee exchanges any number of messages with one player followed by any number of additional messages with the other. This algorithm considerably extends the class of games which admit parallel solutions and demonstrates for the first time the existence of a parallel algorithm for any game (classical or quantum) in which one player reacts adaptively to the other. As a direct consequence, we prove that several competingprovers complexity classes collapse to PSPACE such as QRG Competitive multiturn twoplayer (say, Alice and Bob) games are often studied in the classical game theory either from the aspect of computing the game values or from the aspect of the complexity classes induced by those game models. For succinct games, exponentialtime algorithm exists for finding the exact value [KM92, KMvS94] and it is also EXPhard to approximate the game value [FIKU08, FKS95]. The situation is much different for shorter games, where succinct twoturn games admit polynomialspace approximation scheme and are also PSPACEhard to approximate Those game settings naturally extend to quantum case where provers and referees are allowed to exchange and process quantum information. It is known that the class of problems that admit quantum refereed games, denoted by QRG, coincide with its classical counterpart RG and henceforth EXP [GW07]. Also there exists a polynomialspace approximation scheme for quantum oneturn refereed games [JW09]. However, much more remains unknown about quantum refereed games of small number of turns. In this paper, we consider the following class of competitive twoplayer refereed games, either classical or quantum, that subsumes all the quantum refereed games of small number of turns studied so far [Gut05, GW05, GW07]. (i) The referee exchanges several messages only with Alice. (ii) After processing this interaction with Alice, the referee exchanges several additional messages only with Bob. After further processing, the referee declares a winner. * Full version available at arXiv:1011.2787 [quantph]. Please refer to the most recent version as a new version will be posted very shortly.
Towards an SDPbased Approach to Spectral Methods A NearlyLinearTime Algorithm for Graph Partitioning and Decomposition
"... In this paper, we consider the following graph partitioning problem: The input is an undirected graph G = (V, E), a balance parameter b ∈ (0, 1/2] and a target conductance value γ ∈ (0, 1). The output is a cut which, if nonempty, is of conductance at most O ( f), for some function f (G, γ), and whi ..."
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Cited by 5 (2 self)
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In this paper, we consider the following graph partitioning problem: The input is an undirected graph G = (V, E), a balance parameter b ∈ (0, 1/2] and a target conductance value γ ∈ (0, 1). The output is a cut which, if nonempty, is of conductance at most O ( f), for some function f (G, γ), and which is either balanced or well correlated with all cuts of conductance at most γ. In a seminal paper, Spielman and Teng γ log 3 V [16] gave an Õ(E/γ2)time algorithm for f = and used it to decompose graphs into a collection of nearexpanders [18]. We present a new spectral algorithm for this problem which runs in time Õ(E/γ) for f = √ γ. Our result yields the first nearlylinear time algorithm for the classic Balanced Separator problem that achieves the asymptotically optimal approximation guarantee for spectral methods. Our method has the advantage of being conceptually simple and relies on a primaldual semidefiniteprogramming (SDP) approach. We first consider a natural SDP relaxation for the Balanced Separator problem. While it is easy to obtain from this SDP a certificate of the fact that the graph has no balanced cut of conductance less than γ, somewhat surprisingly, we can obtain a certificate for the stronger correlation condition. This is achieved via a novel separation oracle for our SDP and by appealing to Arora and Kale’s [3] framework to bound the running time. Our result contains technical ingredients that may be of independent interest.
A parallel approximation algorithm for positive semidefinite programming
 In The 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011
, 2011
"... ar ..."
Parallelized Solution to Semidefinite Programmings in Quantum Complexity Theory
, 2010
"... In this paper we present an equilibrium value based framework for solving SDPs via the multiplicative weight update method which is different from the one in Kale’s thesis [Kal07]. One of the main advantages of the new framework is that we can guarantee the convertibility from approximate to exact ..."
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Cited by 4 (4 self)
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In this paper we present an equilibrium value based framework for solving SDPs via the multiplicative weight update method which is different from the one in Kale’s thesis [Kal07]. One of the main advantages of the new framework is that we can guarantee the convertibility from approximate to exact feasibility in a much more general class of SDPs than previous result. Another advantage is the design of the oracle which is necessary for applying the multiplicative weight update method is much simplified in general cases. This leads to an alternative and easier solutions to the SDPs used in the previous results QIP(2)⊆PSPACE [JUW09] and QMAM=PSPACE [JJUW09]. Furthermore, we provide a generic form of SDPs which can be solved in the similar way. By parallelizing every step in our solution, we are able to solve a class of SDPs in NC. Although our motivation is from quantum computing, our result will also apply directly to any SDP which satisfies our conditions. In addition to the new framework for solving SDPs, we also provide a novel framework which improves the range of equilibrium value problems that can be solved via the multiplicative weight update method. Before this work we are only able to calculate the equilibrium value where one of the two convex sets needs to be the set of density operators. Our work demonstrates that in the case when one set is the set of density operators with further linear constraints, we are still able to approximate the equilibrium value to high precision via the multiplicative weight update method.
Fast Approximation Algorithms for Graph Partitioning Using Spectral and SemidefiniteProgramming Techniques
, 2011
"... Graphpartitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their farreaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamenta ..."
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Cited by 3 (1 self)
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Graphpartitioning problems are a central topic of research in the study of approximation algorithms. They are of interest to both theoreticians, for their farreaching connections to different areas of mathematics, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications, such as image segmentation and clustering. While many theoretical approximation algorithms exist for graph partitioning, they often rely on multicommodityflow computations that run in quadratic time in the worst case and are too timeconsuming for the massive graphs that are prevalent in today’s applications. In this dissertation, we study the design of approximation algorithms that yield strong approximation guarantees, while running in subquadratic time and relying on computational procedures that are often fast in practice. The results that we describe encompass two different approaches to the construction of such fast algorithms. Our first result exploits the CutMatching game of Khandekar, Rao and Vazirani [41], an elegant framework for designing graphpartitioning algorithms that rely on singlecommodity, rather than multicommodity, maximum flow. Within this framework, we give two novel algorithms that achieve an O(log n)approximation for the problem of finding the cut of minimum