Relax and Randomize : From Value to Algorithms

We show a principled way of deriving online learning algorithms from a minimaxanalysis. Various upper bounds on the minimax value, previously thought to benon-constructive, are shown to yield algorithms. This allows us to seamlessly re-cover known methods and to derive new ones, also capturing such “unorthodox”methods as Follow the Perturbed Leader and the R forecaster. Understandingthe inherent complexity of the learning problem thus leads to the development ofalgorithms. To illustrate our approach, we present several new algorithms, includ-ing a family of randomized methods that use the idea of a “random playout”. Newversions of the Follow-the-Perturbed-Leader algorithms are presented, as well asmethods based on the Littlestone’s dimension, efficient methods for matrix com-pletion with trace norm, and algorithms for the problems of transductive learningand prediction with static experts.