each of the leaves.

One characteristic of these and other approaches, is that they avoid using the final model (usually linear regression) as a criterion when optimizing the splits of the internal nodes. M5 for instance, optimizes the split for piecewise constant models in the leaves and only inserts the linear models in the leaves post-hoc.

However, it would be more desirable to build the tree using a split criterion that takes into account the models actually employed in the leaves: $$\begin{aligned}
\label{eq:lrt}
  \sum_{\bx_i \in A} \paren[2]{ y_i- \hat{f}_A(\bx_i;w_A) }^2
  + \sum_{\bx_j \in A^c} \paren[2]{ y_j- \hat{f}_{A^c}(\bx_j;w_{A^c}) }^2,\end{aligned}$$ where $A,A^c$ are the two split subsets and $\hat{f}_A(\cdot;w_A)$ is a linear regressor built on the corresponding subset. The discrepancy of using different models for optimizing a split and predicting at the leaves, was first pointed out by Karalic [@karalic1992employing], at the time however, solving the full optimization problem that results from using a piecewise linear model as part of the splitting criterion was neither tractable nor numerically stable. This intractability and instability was asserted many times in subsequent years [@dobra2002secret; @Potts2005; @Nata], and in the view of solving this optimization problem, multiple efforts were made to efficiently approximate the above split 