Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates

Allen-Zhu, Zeyuan; Liao, Zhenyu; Orecchia, Lorenzo

doi:10.1145/2746539.2746610

Cited by 42 publications

(97 citation statements)

References 42 publications

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“…This lemma was first proved in [BSS12] for the case of q = 1, and was extended in [AZLO15] to general values of q. For completeness, we include the proof of the lemma in the appendix.…”

Section: Analysis Of a Single Iterationmentioning

confidence: 96%

“…For fast approximation of the sampling probabilities, we adopt the idea proposed in [AZLO15]: Instead of defining the potential function by (2.2), we define the potential function by…”

Section: Overview Of Our Approachmentioning

confidence: 99%

“…We will assume that the following assumption holds on A. We remark that an almost-linear time algorithm for computing similar quantities was shown in [AZLO15].…”

Section: Implementation Of the Algorithmmentioning

confidence: 99%

“…Part of this work was done while both authors were visiting the Simons Institute for the Theory of Computing, UC Berkeley, and the second author was affiliated with the Max Planck Institute for Informatics, Germany. We thank Zeyuan Allen-Zhu, Zhenyu Liao, and Lorenzo Orecchia for sending us their manuscript of [AZLO15] and the inspiring talk Zeyuan Allen-Zhu gave at the Simons Institute for the Theory of Computing. Finally, we thank Michael Cohen for pointing out a gap in a previous version of the paper and his fixes for the gap, as well as Lap-Chi Lau for many insightful comments on improving the presentation of the paper.…”

Section: Acknowledgmentmentioning

confidence: 99%

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Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Lee

Sun

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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We present the first almost-linear time algorithm for constructing linear-sized spectral sparsification for graphs. This improves all previous constructions of linear-sized spectral sparsification, which requires Ω(n 2 ) time [BSS12, Zou12, AZLO15]. A key ingredient in our algorithm is a novel combination of two techniques used in literature for constructing spectral sparsification: Random sampling by effective resistance [SS11], and adaptive construction based on barrier functions [BSS12, AZLO15].

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“…This lemma was first proved in [BSS12] for the case of q = 1, and was extended in [AZLO15] to general values of q. For completeness, we include the proof of the lemma in the appendix.…”

Section: Analysis Of a Single Iterationmentioning

confidence: 96%

“…For fast approximation of the sampling probabilities, we adopt the idea proposed in [AZLO15]: Instead of defining the potential function by (2.2), we define the potential function by…”

Section: Overview Of Our Approachmentioning

confidence: 99%

“…We will assume that the following assumption holds on A. We remark that an almost-linear time algorithm for computing similar quantities was shown in [AZLO15].…”

Section: Implementation Of the Algorithmmentioning

confidence: 99%

Section: Acknowledgmentmentioning

confidence: 99%

See 2 more Smart Citations

Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Lee

Sun

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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“…Wang et al [32] building on [6] give a (1 + )-approximation if k ≥ m 2 . Recently, Allen-Zhu et al [2] use the connection of the problem to sparsification [7,29] and use regret minimization methods [3] and gave O(1)-approximation algorithm if k ≥ 2m and (1 + )-approximation when k ≥ O m 2 . We also remark that their results also apply to other optimality criteria.…”

Section: Our Results and Contributionsmentioning

confidence: 99%

Approximate Positive Correlated Distributions and Approximation Algorithms for D-optimal Design

Singh

Xie

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Experimental design is a classical area in statistics [21] and has also found new applications in machine learning [2]. In the combinatorial experimental design problem, the aim is to estimate an unknown m-dimensional vector x from linear measurements where a Gaussian noise is introduced in each measurement. The goal is to pick k out of the given n experiments so as to make the most accurate estimate of the unknown parameter x. Given a set S of chosen experiments, the most likelihood estimate x can be obtained by a least squares computation. One of the robust measures of error estimation is the D-optimality criterion [27] which aims to minimize the generalized variance of the estimator. This corresponds to minimizing the volume of the standard confidence ellipsoid for the estimation error x − x . The problem gives rise to two natural variants depending on whether repetitions of experiments is allowed or not. The latter variant, while being more general, has also found applications in geographical location of sensors [19].We show a close connection between approximation algorithms for the D-optimal design problem and constructions of approximately m-wise positively correlated distributions.This connection allows us to obtain a 1 e -approximation for the D-optimal design problem with and without repetitions giving the first constant factor approximation for the problem. We then consider the case when the number of experiments chosen is much larger than the dimension m and show one can obtain (1 − )-approximation if k ≥ 2m when repetitions are allowed andwhen no repetitions are allowed improving on previous work.

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Nearly linear-time packing and covering LP solvers

Allen-Zhu

Orecchia

2018

Math. Program.

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Packing and covering linear programs (PC-LPs) form an important class of linear programs (LPs) across computer science, operations research, and optimization. In 1993, Luby and Nisan [25] constructed an iterative algorithm for approximately solving PC-LPs in nearly linear time, where the time complexity scales nearly linearly in N , the number of nonzero entries of the matrix, and polynomially in ε, the (multiplicative) approximation error. Unfortunately, existing nearly linear-time algorithms [2, 11, 24, 32, 36, 37] for solving PC-LPs require time at least proportional to ε −2. In this paper, we break this longstanding barrier by designing a packing solver that runs in time O(N ε −1) and covering LP solver that runs in time O(N ε −1.5). Our packing solver can be extended to run in time O(N ε −1) for a class of well-behaved covering programs. In a follow-up work, Wang et al. [35] showed that all covering LPs can be converted into well-behaved ones by a reduction that blows up the problem size only logarithmically. At high level, these two algorithms can be described as linear couplings of several first-order descent steps. This is an application of our linear coupling technique (see [3]) to problems that are not amenable to blackbox applications known iterative algorithms in convex optimization.

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Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates

Cited by 42 publications

References 42 publications

Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Constructing Linear-Sized Spectral Sparsification in Almost-Linear Time

Approximate Positive Correlated Distributions and Approximation Algorithms for D-optimal Design

Nearly linear-time packing and covering LP solvers

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