We will deal with the multiplicative weigts alhorithm Imgain

We will deal with the multiplicative weigts alhorithm. Imgaine there is an authority who can predict your plan about how you pick your expert on each day and also able to determine the costs of each expert where cost ranges from 0 to 1 inclusively. Find the \"max expected regret\" ( = E[R]) that the authority can ensure for following cases.

note: p1, p2, p3 ... is probability of choosing expert 1, 2, 3 respectively.

(a)version 1: every day we choose only first expert from n experts (order of experts does not change) - equivalently, what is max regret when probability of choosing expert 1 is always 1?

(b)version 2: \"any-deterministic\" alhorism (in this algorithm, when we choose an expert we have ability to see how each expert performed in the past) - equivalently, what is max regret when probability of choosing some expert i in day t is always 1 for all days?

(c)version 3: until algorithm terminates, in every iteration we use just same probability distribution to pick the expert.

(d) version 4: find regret at the worstcase interms of the p_i\'s where p_i is the chance to pick an expert i, and its sum over all experts is 1. Also, what is the optimal distribution?

Solution

Algorithms in various fields use the idea of keeping a distribution over a sure set and use the multiplicative update rule to iteratively change these weights. Their analyses are usually very similar and rely on an exponential capability feature.

in this survey we present a easy meta-algorithm that unifies lots of those disparate algorithms and derives them as easy instantiations of the meta-algorithm. We feel that in view that this meta-set of rules and its evaluation are so easy, and its packages so vast, it have to be a general part of algorithms courses, like “divide and triumph over