XGBoost Split Gain

Compute the XGBoost split gain from the gradient and Hessian sums of the left child (GL, HL), the right child (GR, HR) and the parent (GP, HP), with regularisation lam: $\frac{1}{2}[\frac{G_L^2}{H_L+\lambda }+\frac{G_R^2}{H_R+\lambda }-\frac{G_P^2}{H_P+\lambda }]\backslash tfrac12\backslash big[\backslash tfrac\{G\_L^2\}\{H\_L+\backslash lambda\}+\backslash tfrac\{G\_R^2\}\{H\_R+\backslash lambda\}-\backslash tfrac\{G\_P^2\}\{H\_P+\backslash lambda\}\backslash big]$. There is no extra per-leaf penalty.

Implement solve(GL,HL,GR,HR,GP,HP,lam).

Returns. Return the gain as a float. The three sums are used as given, so the parent sums are not forced to equal the sum of the children, and the gain can be negative.

Examples

Example 1

Input:

solve(-2.0, 2.0, 1.0, 1.0, -1.0, 3.0, 1.0)

Output:

0.791667

Example 2

Input:

solve(3.0, 2.0, -3.0, 2.0, 0.0, 4.0, 0.0)

Output:

4.5