Elad Hazan, Satyen Kale and Shai Shalev-Shwartz
In SIAM Journal on Computing (SICOMP), 2017. Proceedings of 25th Conference on Learning Theory (COLT), 2012

In several online prediction problems of recent interest the comparison class is composed of matrices with bounded entries. 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 $$(\beta,\tau)$$-decomposability, and derive an efficient online learning algorithm, that enjoys a regret bound of $$\tilde{O}(\sqrt{\beta \tau T})$$ for all problems in which the comparison class is composed of $$(\beta,\tau)$$-decomposable matrices. By analyzing the decomposability of cut matrices, triangular matrices, and low trace-norm matrices, we derive near optimal regret bounds for online max-cut, online gambling, and online collaborative filtering. In particular, this resolves (in the affirmative) an open problem posed by Abernethy (2010) and Kleinberg et al (2010). 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 (2011).