6.883: Online Methods in Machine Learning

For the experts problem (and its close relative B1/B∞ linear game) we have developed a deterministic method (Exponential Weights) and a randomized method (Follow the Perturbed Leader). Both the deterministic and randomized methods extend to other online prediction problems where the loss is linear in the decision of the learner and linear in the outcome. We have seen that the geometry of the two sets (decisions and outcomes) plays a crucial role, both for computational tractability and for attainable regret guarantees. (Recall the online shortest path problem with its flow polytope, the online ranking problem with the Birkhoff polytope or the semidefinite representation) In this lecture, we present techniques for deriving deterministic methods that take advantage of the geometry of the decision/outcome sets. We illustrate several closely related techniques on the B2/B2 analysis, since the solution is very clean and immediately suggests generalization to other norms.