Near-optimal algorithms for online matrix prediction

In several online prediction problems of recent interest the comparison class is composed of matrices. For example, in the online max-cut problem, the comparison class is matrices which represent cuts of a given graph, and in online gambling the comparison class is matrices which represent permutations over n teams. Another important example is online collaborative filtering, in which a widely used comparison class is the set of matrices with a small trace norm. In this paper we isolate a property of matrices, which we call (β τ)-decomposability, and derive an efficient online learning algorithm that enjoys a regret bound of ∼O( √ β τ T) for all problems in which the comparison class is composed of (β τ)-decomposable matrices. By analyzing the decomposability of cut matrices, low trace-norm matrices, and triangular matrices, we derive near-optimal regret bounds for online max-cut, online collaborative filtering, and online gambling. In particular, this resolves (in the affirmative) an open problem posed by Abernethy [Proceedings of the 23rd Annual Conference on Learning Theory (COLT 2010), pp. 318-319] and Kleinberg, Niculescu-Mizil, and Sharma [Machine Learning, 80 (2010), pp. 245-272]. Finally, we derive lower bounds for the three problems and show that our upper bounds are optimal up to logarithmic factors. In particular, our lower bound for the online collaborative filtering problem resolves another open problem posed by Shamir and Srebro [Proceedings of the 24th Annual Conference on Learning Theory (COLT 1011), pp. 661-678].