Score functions
The common approach to solve supervised learning tasks is to minimize the loss function 
:
is the value of the loss function at the point 
is the weight of the 
-th object
is the regularization term.
is the value of the loss function at the point 
is the weight of the 
-th object
is the regularization term.
(mean squared error)
(L2 regularization)
(mean squared error)
(L2 regularization)Gradient boosting
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 
.
Types of score functions
let 
denote 
, 
— the weight of 
-th object, and 
– the corresponding gradient of 
. Let’s consider the following score functions:
Finding the optimal tree structure
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 
.
and 
are the weighted averages:
After expanding brackets and removing terms, which are constant in the optimization:
The latter argmax can be calculated by brute force search.
- L2 score function: S is converted into a sum over leaves
. The next step is to find 
, where 
are the optimal values in leaves after the 
split. - Depthwise and Lossguide methods:
are sets of 
. 
stands for the index of the leaf, therefore the score function 
takes the following form: 
. Since 
is a convex function, different 
and 
(splits for different leaves) can be searched separately by finding the optimal 
. - SymmetricTree method: The same
are attempted to be found for each leaf, thus it's required to optimize the total sum over all leaves 
.
. The next step is to find 
, where 
are the optimal values in leaves after the 
split.
are sets of 
. 
stands for the index of the leaf, therefore the score function 
takes the following form: 
. Since 
is a convex function, different 
and 
(splits for different leaves) can be searched separately by finding the optimal 
.
are attempted to be found for each leaf, thus it's required to optimize the total sum over all leaves 
.Second-order score functions
Let's apply the Taylor expansion to the loss function at the point 
:
is the l2 regularization parameter
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 score functions
CatBoost provides the following score functions:
| Score function | Description |
|---|---|
| L2 | Use the first derivatives during the calculation. |
| Cosine (can not be used with the Lossguide tree growing policy) | |
| NewtonL2 | Use the second derivatives during the calculation. This may improve the resulting quality of the model. |
| NewtonCosine (can not be used with the Lossguide tree growing policy) |
| Score function | Description |
|---|---|
| L2 | Use the first derivatives during the calculation. |
| Cosine (can not be used with the Lossguide tree growing policy) | |
| NewtonL2 | Use the second derivatives during the calculation. This may improve the resulting quality of the model. |
| NewtonCosine (can not be used with the Lossguide tree growing policy) |
Usage
Use the corresponding parameter to set the score function during the training:
The supported score functions vary depending on the processing unit type:
GPU — All score types
CPU — Cosine, L2
| Python package | R package | Command-line interface | Description |
|---|---|---|---|
| score_function | score_function | --score-function | The score type used to select the next split during the tree construction. Possible values:
|
| Python package | R package | Command-line interface | Description |
|---|---|---|---|
| score_function | score_function | --score-function | The score type used to select the next split during the tree construction. Possible values:
|