  We consider the problem of estimating piecewise regular functions in an
online setting, i.e., the data arrive sequentially and at any round our task is
to predict the value of the true function at the next revealed point using the
available data from past predictions. We propose a suitably modified version of
a recently developed online learning algorithm called the sleeping experts
aggregation algorithm. We show that this estimator satisfies oracle risk bounds
simultaneously for all local regions of the domain. As concrete instantiations
of the expert aggregation algorithm proposed here, we study an online mean
aggregation and an online linear regression aggregation algorithm where experts
correspond to the set of dyadic subrectangles of the domain. The resulting
algorithms are near linear time computable in the sample size. We specifically
focus on the performance of these online algorithms in the context of
estimating piecewise polynomial and bounded variation function classes in the
fixed design setup. The simultaneous oracle risk bounds we obtain for these
estimators in this context provide new and improved (in certain aspects)
guarantees even in the batch setting and are not available for the state of the
art batch learning estimators.
