Switching Strategies for Sequential Decision Problems With Multiplicative Loss With Application to Portfolios

A wide variety of problems in signal processing can be formulated such that decisions are made by sequentially taking convex combinations of vector-valued observations and these convex combinations are then multiplicatively compounded over time. A ldquouniversalrdquo approach to such problems might attempt to sequentially achieve the performance of the best fixed convex combination, as might be achievable noncausally, by observing all of the outcomes in advance. By permitting different piecewise-fixed strategies within contiguous regions of time, the best algorithm in this broader class would be able to switch between different fixed strategies to optimize performance to the changing behavior of each individual sequence of outcomes. Without knowledge of the data length or the number of switches necessary, the algorithms developed in this paper can achieve the performance of the best piecewise-fixed strategy that can choose both the partitioning of the sequence of outcomes in time as well as the best strategy within each time segment. We compete with an exponential number of such partitions, using only complexity linear in the data length and demonstrate that the regret with respect to the best such algorithm is at most O(ln(n)) in the exponent, where n is the data length. Finally, we extend these results to include finite collections of candidate algorithms, rather than convex combinations and further investigate the use of an arbitrary side-information sequence.