## Label optimal regret bounds for online local learning

published: Aug. 20, 2015, recorded: July 2015, views: 2401

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# Description

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 collaborative ﬁltering, online gambling, and online max cut among others. Christiano (2014a) designed an efﬁcient online learning algorithm for this problem achieving a regret of O(\sqrt{nL^3T}), where T is the number of rounds. Information theoretically, one can achieve a regret of O(\sqrt{n log LT}). 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(\sqrt{n L T}). 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 in regimes 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 dense subgraph version does not have 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.

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