Correct theoretical regularized objective function for XGB/LGBM (regression task)
For question (a), it should be as simple as adding the usual L1 penalty:
$$\begin{align*}
\Omega(f) &amp;= \gamma T + \alpha \|w\|_1 + \frac12\lambda\|w\|_2^2 \\
&amp;= \gamma T + \alpha \sum |w_i| + \frac12\lambda \sum w_i^2
\end{align*}$$
For (b), see LightGBM vs XGBoost
For (c), see L1 &amp; L2 Regularization in Light GBM