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 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); 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).

通过解析关于在线预测问题的与矩阵相关的比较类，提出了一种(beta,tau)-decomposability的性质，进而导出了一个有效的在线学习算法，证明了其对所有(beta,tau)-可分解矩阵的问题都有较优的regret界限，并对三个问题得出了更佳的regret界限和下界。

在线矩阵预测的近优算法