A Note on Multi-Parametric Gradient Boosting Machines with Non-Life Insurance Applications

In the present note a general multi-parametric gradient boosting machine (GBM) approach is introduced. The starting point is a standard univariate GBM, which is generalised to higher dimensions by using cyclic coordinate descent. This allows for different covariate dependencies in different dimensions. The suggested approach is also easily extended to, e.g., multi-parametric versions of XGBoost. Given weak assumptions the method can be shown to converge for convex negative log-likelihood loss functions, which is the case, e.g., for d-parameter exponential families. Further, when having d-parametric distribution functions, it is important to design appropriate early stopping schemes. A simple alternative is introduced and more advanced schemes are discussed. The flexibility of the method is illustrated both on simulated and real insurance data examples using different multi-parametric distributions, with both convex and non-convex losses