Multiplicative Weights

In this lecture, we will study various applications of the theory of Multiplicative Weights (MW). In this section, we briefly review the general version of the MW algorithm that we studied in the previous lecture. The following sections then show how the theory can be applied to approximately solve zero-sum games and linear programs, and how it connects with the theory of boosting and approximation algorithms. We have n experts who predict the outcome of an event in consecutive rounds. Suppose that in each round there are P different outcomes for the event. If outcome j realizes, expert i pays a penalty of M(i, j). An important paremeter will prove to be the maximum allowable magnitude for the penalty. For that, let M(i, j) ∈ [−�, ρ], with 0 ≤ � ≤ ρ, where ρ is the width. Our goal is to devise a strategy that dictates which expert’s reccomendation to follow, in order to achieve an expected avegare penalty that is not much worse than that of the best expert (in hindsight). The strategy that we analyzed in the previous lecture is as follows. We maintain for each expert a scalar weight, which can be thought of as a quality score. Then, at each round we choose to follow the recommendation of a specific expert with probability that is proportional to her weight. After the outcome is realized, we update the weights of each expert accordingly. In mathematical terms, let wi t be the weight of the ith epxert at the beginning of round t. Then, the MW algorithm is