Score functions

Boosting is a method which builds a prediction model as an ensemble of weak learners .

In our case, is a decision tree. Trees are built sequentially and each next tree is built to approximate negative gradients of the loss function at predictions of the current ensemble:

Thus, it performs a gradient descent optimization of the function . The quality of the gradient approximation is measured by a score function .

Let's suppose that it is required to find the structure for the tree of depth 1. The structure of such tree is determined by the index of some feature and a border value . Let be the value of the -th feature on the -th object and and be the values at leafs of . Then, equals to if and if . Now the goal is to find the best and in terms of the chosen score function.

For the L2 score function the formula takes the following form:

Let's denote and .

The optimal values for and are the weighted averages:

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After expanding brackets and removing terms, which are constant in the optimization:

The latter argmax can be calculated by brute force search.

The L2 score function can be calculated for each leaf separately. This is not true for Cosine. The restrictions imposed on the growing policy are shown below.

Let's apply the Taylor expansion to the loss function at the point :

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• is the l2 regularization parameter

Since the first term is constant in optimization, the formula takes the following form after regrouping by leaves:

So, the optimal value of is:

The summation is over such that the object gets to the considered leaf. Then these optimal values of can be used instead of weighted averages of gradients ( and in the example above) in the same score functions.

CatBoost provides the following score functions:

Use the corresponding parameter to set the score function during the training: