research-article

Online local learning via semidefinite programming

Author:

Paul ChristianoAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 468 - 474

https://doi.org/10.1145/2591796.2591880

Published: 31 May 2014 Publication History

Abstract

In many online learning problems we are interested in predicting local information about some universe of items. For example, we may want to know whether two items are in the same cluster rather than computing an assignment of items to clusters; we may want to know which of two teams will win a game rather than computing a ranking of teams. Although finding the optimal clustering or ranking is typically intractable, it may be possible to predict the relationships between items as well as if you could solve the global optimization problem exactly.

Formally, we consider an online learning problem in which a learner repeatedly guesses a pair of labels (ℓ(x), ℓ(y)) and receives an adversarial payoff depending on those labels. The learner's goal is to receive a payoff as good as the best fixed labeling of the items. We show that a simple algorithm based on semidefinite programming can achieve asymptotically optimal regret in the case where the number of possible labels is constant, resolving an open problem posed by Hazan, Kale, and Shalev-Schwartz [10]. Our main technical contribution is a novel use and analysis of the log det regularizer, exploiting the observation that log det (Σ + I) upper bounds the entropy of any distribution with covariance matrix Σ.

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References

[1]

Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(1):121--164, 2012.

[2]

Blum, Chawla, and Kalai. Static optimality and dynamic search-optimality in lists and trees. Algorithmica, 36, 2003.

[3]

A. Blum. On-line algorithms in machine learning. Lecture Notes in Computer Science, 1442, 1998.

Digital Library

[4]

Cover. Universal data compression and portfolio selection. In FOCS: IEEE Symposium on Foundations of Computer Science (FOCS), 1996.

Digital Library

[5]

T. Cover and J. Thomas. Elements of Information Theory. Wiley, 1991.

Digital Library

[6]

Dean P. Foster and Rakesh Vohra. Regret in the on-line decision problem, July 11 1997.

[7]

Freund and Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. JCSS: Journal of Computer and System Sciences, 55, 1997.

Digital Library

[8]

Freund, Schapire, Singer, and Warmuth. Using and combining predictors that specialize. In STOC: ACM Symposium on Theory of Computing (STOC), 1997.

Digital Library

[9]

E. Hazan. The convex optimization approach to regret minimization. Technical report, Technion -- Israel Institute of Technology, Haifa, Israel, September 2009.

[10]

Elad Hazan, Satyen Kale, and Shai Shalev-Shwartz. Near-optimal algorithms for online matrix prediction. CoRR, abs/1204.0136, 2012.

[11]

Helmbold and Schapire. Predicting nearly as well as the best pruning of a decision tree. Machine Learning, 27, 1997.

Digital Library

[12]

Prateek Jain, Brian Kulis, and Inderjit Dhillon. Online linear regression using burg entropy. Technical Report CS-TR-07-08, The University of Texas at Austin, Department of Computer Sciences, February 14 2007. Mon, 28 Jan 108 21:38:56 GMT.

[13]

Kalai and Vempala. Efficient algorithms for online decision problems. JCSS: Journal of Computer and System Sciences, 71, 2005.

Digital Library

[14]

Brian Kulis and Peter L. Bartlett. Implicit online learning. In Johannes Fürnkranz and Thorsten Joachims, editors, ICML, pages 575--582. Omnipress, 2010.

[15]

Gang Niu, Bo Dai, Makoto Yamada, and Masashi Sugiyama. Information-theoretic semi-supervised metric learning via entropy regularization. In ICML. icml.cc/Omnipress, 2012.

[16]

Takimoto and Warmuth. Predicting nearly as well as the best pruning of a planar decision graph. TCS: Theoretical Computer Science, 288, 2002.

Digital Library

[17]

Martin J. Wainwright and Michael I. Jordan. Log-determinant relaxation for approximate inference in discrete markov random fields. IEEE Transactions on Signal Processing, 54(6-1):2099--2109, 2006.

Cited By

Velicheti RBastopcu MEtesami SBaşar T(2024)Learning How to Strategically Disclose Information2024 American Control Conference (ACC)10.23919/ACC60939.2024.10644843(1604-1609)Online publication date: 10-Jul-2024
https://doi.org/10.23919/ACC60939.2024.10644843
Vitale FParotsidis NGentile C(2018)Online reciprocal recommendation with theoretical performance guaranteesProceedings of the 32nd International Conference on Neural Information Processing Systems10.5555/3327757.3327919(8267-8277)Online publication date: 3-Dec-2018
https://dl.acm.org/doi/10.5555/3327757.3327919
MORIDOMI KHATANO KTAKIMOTO E(2018)Online Linear Optimization with the Log-Determinant RegularizerIEICE Transactions on Information and Systems10.1587/transinf.2017EDP7317E101.D:6(1511-1520)Online publication date: 1-Jun-2018
https://doi.org/10.1587/transinf.2017EDP7317

Index Terms

Online local learning via semidefinite programming
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
2. Theory of computation

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 31 May 2014

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STOC '14: Symposium on Theory of Computing

May 31 - June 3, 2014

New York, New York

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STOC '14 Paper Acceptance Rate 91 of 319 submissions, 29%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Velicheti RBastopcu MEtesami SBaşar T(2024)Learning How to Strategically Disclose Information2024 American Control Conference (ACC)10.23919/ACC60939.2024.10644843(1604-1609)Online publication date: 10-Jul-2024
https://doi.org/10.23919/ACC60939.2024.10644843
Vitale FParotsidis NGentile C(2018)Online reciprocal recommendation with theoretical performance guaranteesProceedings of the 32nd International Conference on Neural Information Processing Systems10.5555/3327757.3327919(8267-8277)Online publication date: 3-Dec-2018
https://dl.acm.org/doi/10.5555/3327757.3327919
MORIDOMI KHATANO KTAKIMOTO E(2018)Online Linear Optimization with the Log-Determinant RegularizerIEICE Transactions on Information and Systems10.1587/transinf.2017EDP7317E101.D:6(1511-1520)Online publication date: 1-Jun-2018
https://doi.org/10.1587/transinf.2017EDP7317
van Stee R(2014)SIGACT news online algorithms column 24ACM SIGACT News10.1145/2670418.267044145:3(105-111)Online publication date: 17-Sep-2014
https://dl.acm.org/doi/10.1145/2670418.2670441

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