Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced Separator in undirected and directed weighted graphs, and the Min UnCut problem, this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones. 1

We introduce a family of spectral partitioning methods. Edge separators of a graph are produced by iteratively reweighting the edges until the graph disconnects into the prescribed number of components. At each iteration a small number of eigenvectors with small eigenvalue are computed and used to determine the reweighting. In this way spectral rounding directly produces discrete solutions where as current spectral algorithms must map the continuous eigenvectors to discrete solutions by employing a heuristic geometric separator (e.g. k-means). We show that spectral rounding compares favorably to current spectral approximations on the Normalized Cut criterion (NCut). Results are given for natural image segmentation, medical image segmentation, and clustering. A practical version is shown to converge. 1.

We describe a simple random-sampling based procedure for producing sparse matrix approximations. Our procedure and analysis are extremely simple: the analysis uses nothing more than the Chernoff-Hoeffding bounds. Despite the simplicity, the approximation is comparable and sometimes better than previous work. Our algorithm computes the sparse matrix approximation in a single pass over the data. Further, most of the entries in the output matrix are quantized, and can be succinctly represented by a bit vector, thus leading to much savings in space. 1

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Minimum rank problems arise frequently in machine learning applications and are notoriously difficult to solve due to the non-convex nature of the rank objective. In this paper, we present the first online learning approach for the problem of rank minimization of matrices over polyhedral sets. In particular, we present two online learning algorithms for rank minimization- our first algorithm is a multiplicative update method based on a generalized experts framework, while our second algorithm is a novel application of the online convex programming framework (Zinkevich, 2003). In the latter, we flip the role of the decision maker by making the decision maker search over the constraint space instead of feasible points, as is usually the case in online convex programming. A salient feature of our online learning approach is that it allows us to give provable approximation guarantees for the rank minimization problem over polyhedral sets. We demonstrate the effectiveness of our methods on synthetic examples, and on the real-life application of low-rank kernel learning. 1.

We show how to compute O ( √ log n)-approximations to Sparsest Cut and Balanced Separator problems in Õ(n2) time, thus improving upon the recent algorithm of Arora, Rao and Vazirani (2004). Their algorithm uses semidefinite programming and required Õ(n4.5) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani. 1

We prove that QIP(2), the class of problems having two-message quantum interactive proof systems, is a subset of PSPACE. This relationship is obtained by means of an efficient parallel algorithm, based on the multiplicative weights update method, for approximately solving a certain class of semidefinite programs. 1

Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In this thesis, we explore applications of the Multiplicative Weights method in the design of efficient algorithms for various optimization problems. This method, which was repeatedly discovered in quite diverse fields, is an algorithmic technique which maintains a distribution on a certain set of interest, and updates it iteratively by multiplying the probability mass of elements by suitably chosen factors based on feedback obtained by running another algorithm on the distribution. We present a single meta-algorithm which unifies all known applications of this method in a common framework. Next, we generalize the method to the setting of symmetric matrices rather than real numbers. We derive the following applications of the resulting Matrix Multiplicative Weights algorithm: 1. The first truly general, combinatorial, primal-dual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for obtaining O(plog n) approximations to various graph partitioning problems, such as Sparsest Cut, Balanced Separator in both directed and undirected weighted graphs, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the Alon-Roichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3) time deterministic O(log n) approximation algorithm for the quantum hypergraph covering problem. 4. An alternative proof of a result of Aaronson that the fl-fat-shattering dimension of quantum states on n qubits is O ( nfl2).

Optimization problems with a nuclear norm regularization, such as e.g. low norm matrix factorizations, have seen many applications recently. We propose a new approximation algorithm building upon the recent sparse approximate SDP solver of (Hazan, 2008). The experimental efficiency of our method is demonstrated on large matrix completion problems such as the Netflix dataset. The algorithm comes with strong convergence guarantees, and can be interpreted as a first theoretically justified variant of Simon-Funk-type SVD heuristics. The method is free of tuning parameters, and very easy to parallelize. 1.

This paper studies a simple class of zero-sum 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 one-turn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space. 1