Computational Learning Theory Lecture 8: Expert Advice & Randomized Weighted Majority

For the past few lectures we have been discussing online classification in scenarios in which there is a perfect target function. In this lecture we study a more general learning framework, learning from expert advice, which removes the assumption of a perfect target and applies to a wide range of problems beyond just classification. Suppose that you are interested in making a sequence of decisions based on the advice of a set of n “experts.” In this context, an expert could be a human being, such as a weather forecaster or a financial analyst. More generally, experts could represent simple decision rules based on features of a learning problem, or a collection of different learning algorithms. On each round, you must choose an expert to follow and each expert suffers a loss for its prediction. Your loss is then the loss of the expert you chose. For example, in the case of binary classification, an expert might suffer a loss of 0 if its prediction is correct and 1 if its prediction is incorrect. For convenience, we assume that losses are bounded in [0, 1], but it is easy to relax this assumption to any bounded range. We could formalize this setting as follows. Note that we have abstracted away the particular decisions that are being made, and look only at the loss of each decision.