  We study nonlinear regression of real valued data in an individual sequence
manner, where we provide results that are guaranteed to hold without any
statistical assumptions. We address the convergence and undertraining issues of
conventional nonlinear regression methods and introduce an algorithm that
elegantly mitigates these issues via an incremental hierarchical structure,
(i.e., via an incremental decision tree). Particularly, we present a piecewise
linear (or nonlinear) regression algorithm that partitions the regressor space
in a data driven manner and learns a linear model at each region. Unlike the
conventional approaches, our algorithm gradually increases the number of
disjoint partitions on the regressor space in a sequential manner according to
the observed data. Through this data driven approach, our algorithm
sequentially and asymptotically achieves the performance of the optimal twice
differentiable regression function for any data sequence with an unknown and
arbitrary length. The computational complexity of the introduced algorithm is
only logarithmic in the data length under certain regularity conditions. We
provide the explicit description of the algorithm and demonstrate the
significant gains for the well-known benchmark real data sets and chaotic
signals.
