How is the gradient and hessian of logarithmic loss computed in the custom objective function example script in xgboost's github repository?

Question

I would like to understand how the gradient and hessian of the logloss function are computed in an xgboost sample script.

I've simplified the function to take numpy arrays, and generated y_hat and y_true which are a sample of the values used in the script.

Here is a simplified example:

import numpy as np


def loglikelihoodloss(y_hat, y_true):
    prob = 1.0 / (1.0 + np.exp(-y_hat))
    grad = prob - y_true
    hess = prob * (1.0 - prob)
    return grad, hess

y_hat = np.array([1.80087972, -1.82414818, -1.82414818,  1.80087972, -2.08465433,
                  -1.82414818, -1.82414818,  1.80087972, -1.82414818, -1.82414818])
y_true = np.array([1.,  0.,  0.,  1.,  0.,  0.,  0.,  1.,  0.,  0.])

loglikelihoodloss(y_hat, y_true)

The log loss function is the sum of where .

The gradient (with respect to p) is then however in the code its .

Likewise the second derivative (with respect to p) is however in the code it is .

How are the equations equal?

Answer 1

The log loss function is given as:

where

Taking the partial derivative we get the gradient as

Thus we get the negative of gradient as p-y.

Similar calculations can be done to obtain the hessian.