Score functions
The common approach to solve supervised learning tasks is to minimize the loss function $L$:
$L\left(f(x), y\right) = \sum\limits_{i} w_{i} \cdot l \left(f(x_{i}), y_{i}\right) + J(f){ , where}$
 $l\left( f(x), y\right)$ is the value of the loss function at the point $(x, y)$
 $w_{i}$ is the weight of the $i$th object
 $J(f)$ is the regularization term.
For example, these formulas take the following form for linear regression:
 $l\left( f(x), y\right) = w_{i} \left( (\theta, x)  y \right)^{2}$ (mean squared error)
 $J(f) = \lambda \left  \theta  \right_{l2}$ (L2 regularization)
Gradient boosting
Boosting is a method which builds a prediction model $F^{T}$ as an ensemble of weak learners $F^{T} = \sum\limits_{t=1}^{T} f^{t}$.
In our case, $f^{t}$ is a decision tree. Trees are built sequentially and each next tree is built to approximate negative gradients $g_{i}$ of the loss function $l$ at predictions of the current ensemble:
$g_{i} = \frac{\partial l(a, y_{i})}{\partial a} \Bigr_{a = F^{T1}(x_{i})}$
Thus, it performs a gradient descent optimization of the function $L$. The quality of the gradient approximation is measured by a score function $Score(a, g) = S(a, g)$.
Types of score functions
Let's suppose that it is required to add a new tree to the ensemble. A score function is required in order to choose between candidate trees. Given a candidate tree $f$ let $a_{i}$ denote $f(x_{i})$, $w_{i}$ — the weight of $i$th object, and $g_{i}$ – the corresponding gradient of $l$. Let’s consider the following score functions:
 $L2 =  \sum\limits_{i} w_{i} \cdot (a_{i}  g_{i})^{2}$
 $Cosine = \displaystyle\frac{\sum w_{i} \cdot a_{i} \cdot g_{i}}{\sqrt{\sum w_{i}a_{i}^{2}} \cdot \sqrt{\sum w_{i}g_{i}^{2}}}$
Finding the optimal tree structure
Let's suppose that it is required to find the structure for the tree $f$ of depth 1. The structure of such tree is determined by the index $j$ of some feature and a border value $c$. Let $x_{i, j}$ be the value of the $j$th feature on the $i$th object and $a_{left}$ and $a_{right}$ be the values at leafs of $f$. Then, $f(x_{i})$ equals to $a_{left}$ if $x_{i,j} \leq c$ and $a_{right}$ if $x_{i,j} > c$. Now the goal is to find the best $j$ and $c$ in terms of the chosen score function.
For the L2 score function the formula takes the following form:
$S(a, g) = \sum\limits_{i} w_{i} (a_{i}  g_{i})^{2} =  \left( \displaystyle\sum\limits_{i:x_{i,j}\leq c} w_{i}(a_{left}  g_{i})^{2} + \sum\limits_{i: x_{i,j}>c} w_{i}(a_{right}  g_{i})^{2} \right)$
Let's denote $W_{left} = \displaystyle\sum_{i: x_{I,j} \leq c} w_{i}$ and $W_{right} = \displaystyle\sum_{i: x_{i,j} >c} w_{i}$.
The optimal values for $a_{left}$ and $a_{right}$ are the weighted averages:
 $a^{*}_{left} =\displaystyle\frac{\sum\limits_{i: x_{i,j} \leq c} w_{i} g_{i}}{W_{left}}$
 $a^{*}_{right} =\displaystyle\frac{\sum\limits_{i: x_{i,j} > c} w_{i} g_{i}}{W_{right}}$
After expanding brackets and removing terms, which are constant in the optimization:
$j^{*}, c^{*} = argmax_{j, c} W_{left} \cdot (a^{*}_{left})^{2} + W_{right} \cdot (a^{*}_{right})^{2}$
The latter argmax can be calculated by brute force search.
The situation is slightly more complex when the tree depth is bigger than 1:
 L2 score function: S is converted into a sum over leaves $S(a,g) = \sum_{leaf} S(a_{leaf}, g_{leaf})$. The next step is to find $j*, c* = argmax_{j,c}{S(\bar a, g)}$, where $\bar a$ are the optimal values in leaves after the $j*, c*$ split.
 Depthwise and Lossguide methods: $j, c$ are sets of $\{j_k\}, \{c_k\}$. $k$ stands for the index of the leaf, therefore the score function $S$ takes the following form: $S(\bar a, g) = \sum_{l = leaf}S(\bar a(j_l, c_l), g_l)$. Since $S(leaf)$ is a convex function, different $j_{k1}, c_{k1}$ and $j_{k2}, c_{k2}$ (splits for different leaves) can be searched separately by finding the optimal $j*, c* = argmax_{j,c}\{S(leaf_{left}) + S(leaf_{right})  S(leaf_{before\_split})\}$.
 SymmetricTree method: The same $j, c$ are attempted to be found for each leaf, thus it's required to optimize the total sum over all leaves $S(a,g) = \sum_{leaf} S(leaf)$.
Secondorder score functions
Let's apply the Taylor expansion to the loss function at the point $a^{t1} = F^{t1}(x)$:
$L(a^{t1}_{i} + \phi , y) \approx \displaystyle\sum w_{i} \left[ l_{i} + l^{'}_{i} \phi + \frac{1}{2} l^{''}_{i} \phi^{2} \right] + \frac{1}{2} \lambda \phi_{2}{ , where:}$
 $l_{i} = l(a^{t1}_{i}, y_{i})$
 $l'_{i} = \frac{\partial l(a, y_{i})}{\partial a}\Bigr_{a=a^{t1}_{i}}$
 $l''_{i} = \frac{\partial^{2} l(a, y_{i})}{\partial a^{2}}\displaystyle\Bigr_{a=a^{t1}_{i}}$
 $\lambda$ is the l2 regularization parameter
Since the first term is constant in optimization, the formula takes the following form after regrouping by leaves:
$\sum\limits_{leaf=1}^{L} \left( \sum\limits_{i \in leaf} w_{i} \left[ l_{i} + l^{'}_{i} \phi_{leaf} + \frac{1}{2} l^{''}_{i} \phi^{2} \right] + \frac{1}{2} \lambda \phi_{leaf}^{2} \right) \to min$
Then let's minimize this expression for each leaf independently:
$\sum\limits_{i \in leaf} w_{i} \left[ l_{i} + l^{'}_{i} \phi_{leaf} + \frac{1}{2} l^{''}_{i} \phi^{2}_{leaf} \right] + \frac{1}{2} \lambda \phi_{leaf}^2 \to min$
Differentiate by leaf value $\phi_{leaf}$:
$\sum\limits_{i \in leaf} w_{i} \left[ l^{'}_{i} + l^{''}_{i} \phi_{leaf} \right] + \lambda \phi_{leaf} = 0$
So, the optimal value of $\phi_{leaf}$ is:
$ \displaystyle\frac{\sum_{i}w_{i}l^{'}_{i}}{\sum_{i}w_{i}l^{''}_{i}+\lambda}$
The summation is over $i$ such that the object $x_{i}$ gets to the considered leaf. Then these optimal values of $\phi_{leaf}$ can be used instead of weighted averages of gradients ($a^{*}_{left}$ and $a^{*}_{right}$ in the example above) in the same score functions.
CatBoost score functions
CatBoost provides the following score functions:
Score function: L2
Description
Use the first derivatives during the calculation.
Score function: Cosine (can not be used with the Lossguide tree growing policy)
Score function: NewtonL2
Description
Use the second derivatives during the calculation. This may improve the resulting quality of the model.
Score function: NewtonCosine (can not be used with the Lossguide tree growing policy)
Perobject and perfeature penalties
CatBoost provides the following methods to affect the score with penalties:

Perfeature penalties for the first occurrence of the feature in the model. The given value is subtracted from the score if the current candidate is the first one to include the feature in the model.

Perobject penalties for the first use of the feature for the object. The given value is multiplied by the number of objects that are divided by the current split and use the feature for the first time.
The final score is calculated as follows:
$Score' = Score \cdot \prod_{f\in S}W_{f}  \sum_{f\in S}P_{f} \cdot U(f)  \sum_{f\in S}\sum_{x \in L}EP_{f} \cdot U(f, x)$
 $W_{f}$ is the feature weight
 $P_{f}$ is the perfeature penalty
 $EP_{f}$ is the perobject penalty
 $S$ is the current split
 $L$ is the current leaf
 $U(f) = \begin{cases} 0,& \text{if } f \text{ was used in model already}\\ 1,& \text{otherwise} \end{cases}$
 $U(f, x) = \begin{cases} 0,& \text{if } f \text{ was used already for object } x\\ 1,& \text{otherwise} \end{cases}$
Usage
Use the corresponding parameter to set the score function during the training:
Alert
The supported score functions vary depending on the processing unit type:

GPU — All score types

CPU — Cosine, L2
Python package: score_function
R package: score_function
Commandline interface: scorefunction
Description
The score type used to select the next split during the tree construction.
Possible values:
 Cosine (do not use this score type with the Lossguide tree growing policy)
 L2
 NewtonCosine (do not use this score type with the Lossguide tree growing policy)
 NewtonL2