Drifting games

We introduce and study a general, abstract game played between two players called the shepherd and the adversary. The game is played in a series of rounds using a finite set of "chips" which are moved about in Rⁿ. On each round, the shepherd assigns a desired direction of movement and an importance weight to each of the chips. The adversary then moves the chips in any way that need only be weakly correlated with the desired directions assigned by the shepherd. The shepherd's goal is to cause the chips to be moved to low-loss positions, where the loss of each chip at its final position is measured by a given loss function. We present a shepherd algorithm for this game and prove an upper bound on its performance. We also prove a lower bound showing that the algorithm is essentially optimal for a large number of chips. We discuss computational methods for efficiently implementing our algorithm. We show that our general drifting-game algorithm subsumes some well studied boosting and on-line learning algorithms whose analyses follow as easy corollaries of our general result.