####Example: We have a group of people betting on a game. Our objective is to minimise the loss we make while betting.
- There are multiple rounds of betting.
- You must bet on every game.
- One person in that group is an expert and is always correct.
Solution: Find the expert as quickly as possible.
xi(t) = prediction of the ith person in game t wi(t) = (boolean value) whether we listen to the ith person's advice z(t) = the winner.
Idea: Bet on the side which most people bet on. Only consider the bets of those people who have always been correct up to this point.
y(t) = [ Σi (wi(t) xi(t)) > 1/2 Σi wi(t) ]
Update the weight associated with each person (only those who have always been right before this game). If they guessed incorrectly in this game set their weight to 0 else 1.
wi(t+1) = wi(t) [ xi(t) = z(t)]
Regret : How much we will lose before we find the expert. Regret <= log2n
Update Problem: No one is always correct but some people are better then others at guessing.
wi(t) = weight associated with the ith persons advice.
Idea: Reduce the weight an individual has by 1/2 if they bet incorrectly. Keep their weight the same if they guess correctly.
wi(t+1) = wi(t) ( 1/2 [xi(t) = z(t)] + 1/2)