Universal Piecewise Linear Regression of Individual Sequences: Lower Bound

We consider universal piecewise linear regression of real valued bounded sequences under the squared loss function. In this setting, we present a lower bound on the regret of a universal sequential piecewise linear regressor compared to the best piecewise linear regressor that has access to the entire sequence in advance. This lower bound is tight in that it achieves the corresponding upper bound, suggesting a minmax optimality of the sequential regressor, for every individual bounded sequence.