I was wondering if anyone has tried to use a LightGBM to estimate the alpha and beta of a linear regression model. I am looking into this because I am seeking an interpretable model. A direct lgbm regression would be more of a blackbox than I would like.
In particular, I am thinking about a model $$y_t = \alpha_{W_t} + \beta^1_{W_t} X^1_t + \beta^2_{W_t} X^2_t$$
where $W_t$ is vector that governs the conditional loadings for the $X^i_t$. so the correlation between the two independent variables and the dependent variable $y$ varies with changes in the environmental variable $W$.
In my case $W$ would be some weather variables and the $X$ some timeseries that are correlated with $y$. But the correlation changes depending on the weather in a non-linear fashion.
My idea is to use a multiple regression lgbm with a low depths, a feature vector $W$, and a custom loss function that models the linear regression equation from above.
Does this idea sound reasonable? Are there other approaches that preserve a linear model structure (in python not R) which are more established?