Universal Randomized Switching

In this paper, we consider a competitive approach to sequential decision problems, suitable for a variety of signal processing applications where at each of a succession of times, a selection must be made from among a fixed set of strategies (or outcomes). For each such decision and outcome pair, loss is incurred, and it is the time-accumulation of these losses that is sought to be minimized. Rather than using a statistical performance measure, our goal in this pursuit is to sequentially accumulate loss that is no larger than that of the best loss that could be obtained through a partitioning of the sequence of observations into an arbitrary fixed number of segments and independently selecting a different strategy for each segment. For this purpose, we introduce a randomized sequential algorithm built upon that of Kozat and Singer that asymptotically achieves the performance of a noncausal algorithm that would be able to choose the number of segments and the best algorithm for each segment, based on observing the whole observation process a priori. In addition to improving upon the bounds of Kozat and Singer as well as Gyorgy, the results we provide hold for more general loss functions than the square-error loss studied therein.