Need help implementing ZILN custom loss function in lightGBM

Im trying to implement this zero-inflated log normal loss function based on this paper in lightGBM (https://arxiv.org/pdf/1912.07753.pdf) (page 5). But, admittedly, I just don’t know how. I don’t understand how to get the gradient and hessian of this function in order to implement it in LGBM and I’ve never needed to implement a custom loss function in the past.

The authors of this paper have open sourced their code, and the function is available in tensorflow (https://github.com/google/lifetime_value/blob/master/lifetime_value/zero_inflated_lognormal.py), but I’m unable to translate this to fit the parameters required for a custom loss function in LightGBM. An example of how LGBM accepts custom loss functions— loglikelihood loss would be written as:

def loglikelihood(preds, train_data):
    labels = train_data.get_label()
    preds = 1. / (1. + np.exp(-preds))
    grad = preds - labels
    hess = preds * (1. - preds)
    return grad, hess

Similarly, I would need to define a custom eval metric to accompany it, such as:

def binary_error(preds, train_data):
    labels = train_data.get_label()
    preds = 1. / (1. + np.exp(-preds))
    return 'error', np.mean(labels != (preds > 0.5)), False

Both of the above two examples are taken from the following repository:

https://github.com/microsoft/LightGBM/blob/e83042f20633d7f74dda0d18624721447a610c8b/examples/python-guide/advanced_example.py#L136

Would appreciate any help on this, and especially detailed guidance to help me learn how to do this on my own.