Multi-parameter optimization with custom loss function for probabilistic forecasting

Dear community,

I am currently working in a probabilistic extension of XGBoost called [XGBoostLSS ](https://github.com/StatMixedML/XGBoostLSS) that models all parameters of a distribution. This allows to create probabilistic forecasts from which prediction intervals and quantiles of interest can be derived.

The problem is that XGBoost doesn`t permit do optimize over several parameters. Assume we have a Normal distribution y ~ N(µ, sigma). So far, my approach is a two-step procedure, where I first optimize µ with sigma fixed, and then optimize sigma with µ fixed and then iterate between these two.

Since this is inefficient, are there any ways of **simultaneously optimize both µ and sigma** using a custom loss function?