Label optimal regret bounds for online local learning

Pranjal Awasthi, Moses Charikar, Kevin A Lai, Andrej Risteski
Proceedings of The 28th Conference on Learning Theory, PMLR 40:150-166, 2015.

Abstract

We resolve an open question from Christiano (2014b) posed in COLT’14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework, the algorithm is shown a pair of items at each step, chosen from a set of n items. The learner then predicts a label for each item, from a label set of size L and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as online gambling and online max cut. Christiano (2014a) designed an efficient online learning algorithm for this problem achieving a regret of O(\sqrtnL^3 T), where T is the number of rounds. Information theoretically, one can achieve a regret of O(\sqrtn \log L T). One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of Christiano (2014a) in fact achieves a regret of O(\sqrtnLT). Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.

Cite this Paper

BibTeX
@InProceedings{pmlr-v40-Awasthi15a, title = {Label optimal regret bounds for online local learning}, author = {Awasthi, Pranjal and Charikar, Moses and Lai, Kevin A and Risteski, Andrej}, booktitle = {Proceedings of The 28th Conference on Learning Theory}, pages = {150--166}, year = {2015}, editor = {Grünwald, Peter and Hazan, Elad and Kale, Satyen}, volume = {40}, series = {Proceedings of Machine Learning Research}, address = {Paris, France}, month = {03--06 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v40/Awasthi15a.pdf}, url = {https://proceedings.mlr.press/v40/Awasthi15a.html}, abstract = {We resolve an open question from Christiano (2014b) posed in COLT’14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework, the algorithm is shown a pair of items at each step, chosen from a set of n items. The learner then predicts a label for each item, from a label set of size L and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as online gambling and online max cut. Christiano (2014a) designed an efficient online learning algorithm for this problem achieving a regret of O(\sqrtnL^3 T), where T is the number of rounds. Information theoretically, one can achieve a regret of O(\sqrtn \log L T). One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of Christiano (2014a) in fact achieves a regret of O(\sqrtnLT). Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.} }
Endnote
%0 Conference Paper %T Label optimal regret bounds for online local learning %A Pranjal Awasthi %A Moses Charikar %A Kevin A Lai %A Andrej Risteski %B Proceedings of The 28th Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2015 %E Peter Grünwald %E Elad Hazan %E Satyen Kale %F pmlr-v40-Awasthi15a %I PMLR %P 150--166 %U https://proceedings.mlr.press/v40/Awasthi15a.html %V 40 %X We resolve an open question from Christiano (2014b) posed in COLT’14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework, the algorithm is shown a pair of items at each step, chosen from a set of n items. The learner then predicts a label for each item, from a label set of size L and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as online gambling and online max cut. Christiano (2014a) designed an efficient online learning algorithm for this problem achieving a regret of O(\sqrtnL^3 T), where T is the number of rounds. Information theoretically, one can achieve a regret of O(\sqrtn \log L T). One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of Christiano (2014a) in fact achieves a regret of O(\sqrtnLT). Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well.
RIS
TY - CPAPER TI - Label optimal regret bounds for online local learning AU - Pranjal Awasthi AU - Moses Charikar AU - Kevin A Lai AU - Andrej Risteski BT - Proceedings of The 28th Conference on Learning Theory DA - 2015/06/26 ED - Peter Grünwald ED - Elad Hazan ED - Satyen Kale ID - pmlr-v40-Awasthi15a PB - PMLR DP - Proceedings of Machine Learning Research VL - 40 SP - 150 EP - 166 L1 - http://proceedings.mlr.press/v40/Awasthi15a.pdf UR - https://proceedings.mlr.press/v40/Awasthi15a.html AB - We resolve an open question from Christiano (2014b) posed in COLT’14 regarding the optimal dependency of the regret achievable for online local learning on the size of the label set. In this framework, the algorithm is shown a pair of items at each step, chosen from a set of n items. The learner then predicts a label for each item, from a label set of size L and receives a real valued payoff. This is a natural framework which captures many interesting scenarios such as online gambling and online max cut. Christiano (2014a) designed an efficient online learning algorithm for this problem achieving a regret of O(\sqrtnL^3 T), where T is the number of rounds. Information theoretically, one can achieve a regret of O(\sqrtn \log L T). One of the main open questions left in this framework concerns closing the above gap. In this work, we provide a complete answer to the question above via two main results. We show, via a tighter analysis, that the semi-definite programming based algorithm of Christiano (2014a) in fact achieves a regret of O(\sqrtnLT). Second, we show a matching computational lower bound. Namely, we show that a polynomial time algorithm for online local learning with lower regret would imply a polynomial time algorithm for the planted clique problem which is widely believed to be hard. We prove a similar hardness result under a related conjecture concerning planted dense subgraphs that we put forth. Unlike planted clique, the planted dense subgraph problem does not have any known quasi-polynomial time algorithms. Computational lower bounds for online learning are relatively rare, and we hope that the ideas developed in this work will lead to lower bounds for other online learning scenarios as well. ER -
APA
Awasthi, P., Charikar, M., Lai, K.A. & Risteski, A.. (2015). Label optimal regret bounds for online local learning. Proceedings of The 28th Conference on Learning Theory, in Proceedings of Machine Learning Research 40:150-166 Available from https://proceedings.mlr.press/v40/Awasthi15a.html.

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