Universal Switching Linear Least Squares Prediction

In this paper, we consider sequential regression of individual sequences under the square-error loss. We focus on the class of switching linear predictors that can segment a given individual sequence into an arbitrary number of blocks within each of which a fixed linear regressor is applied. Using a competitive algorithm framework, we construct sequential algorithms that are competitive with the best linear regression algorithms for any segmenting of the data as well as the best partitioning of the data into any fixed number of segments, where both the segmenting of the data and the linear predictors within each segment can be tuned to the underlying individual sequence. The algorithms do not require knowledge of the data length or the number of piecewise linear segments used by the members of the competing class, yet can achieve the performance of the best member that can choose both the partitioning of the sequence as well as the best regressor within each segment. We use a transition diagram (F. M. J. Willems, 1996) to compete with an exponential number of algorithms in the class, using complexity that is linear in the data length. The regret with respect to the best member is O(ln(n)) per transition for not knowing the best transition times and O(ln(n)) for not knowing the best regressor within each segment, where n is the data length. We construct lower bounds on the performance of any sequential algorithm, demonstrating a form of min-max optimality under certain settings. We also consider the case where the members are restricted to choose the best algorithm in each segment from a finite collection of candidate algorithms. Performance on synthetic and real data are given along with a Matlab implementation of the universal switching linear predictor.