Abstract
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 $(\beta,\tau)$-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, 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 (i...
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