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Meta TitleRegression: objectives and metrics | CatBoost
Meta DescriptionObjectives and metrics. Used for optimization. Objectives and metrics MAE.
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Objectives and metrics Used for optimization Objectives and metrics MAE ∑ i = 1 N w i ∣ a i − t i ∣ ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} | a_{i} - t_{i}| }{\sum\limits_{i=1}^{N} w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true MAPE ∑ i = 1 N w i ∣ a i − t i ∣ max ⁡ ( 1 , ∣ t i ∣ ) ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \displaystyle\frac{|a_{i}- t_{i}|}{\max(1, |t_{i}|)}}{\sum\limits_{i=1}^{N}w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true Poisson ∑ i = 1 N w i ( e a i − a i t i ) ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \left(e^{a_{i}} - a_{i}t_{i}\right)}{\sum\limits_{i=1}^{N}w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true Quantile ∑ i = 1 N ( α − I ( t i ≤ a i ) ) ( t i − a i ) w i ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} (\alpha - I(t_{i} \leq a_{i}))(t_{i} - a_{i}) w_{i} }{\sum\limits_{i=1}^{N} w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true alpha The coefficient used in quantile-based losses. Default: 0.5 MultiQuantile ∑ i = 1 N w i ∑ q = 1 Q ( α q − I ( t i ≤ a i , q ) ) ( t i − a i , q ) ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \sum\limits_{q=1}^{Q} (\alpha_{q} - I(t_{i} \leq a_{i,q}))(t_{i} - a_{i,q}) }{\sum\limits_{i=1}^{N} w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true alpha The vector of coefficients used in multi-quantile loss. Default: 0.5 RMSE ∑ i = 1 N ( a i − t i ) 2 w i ∑ i = 1 N w i \displaystyle\sqrt{\displaystyle\frac{\sum\limits_{i=1}^N (a_{i}-t_{i})^2 w_{i}}{\sum\limits_{i=1}^{N}w_{i}}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true RMSEWithUncertainty − ∑ i = 1 N w i log ⁡ N ( t i ∣ a i , 0 , e 2 a i , 1 ) ∑ i = 1 N w i = 1 2 log ⁡ ( 2 π ) + ∑ i = 1 N w i ( a i , 1 + 1 2 e − 2 a i , 1 ( t i − a i , 0 ) 2 ) ∑ i = 1 N w i \displaystyle-\frac{\sum_{i=1}^N w_i \log N(t_{i} \vert a_{i,0}, e^{2a_{i,1}})}{\sum_{i=1}^{N}w_{i}} = \frac{1}{2}\log(2\pi) +\frac{\sum_{i=1}^N w_i\left(a_{i,1} + \frac{1}{2} e^{-2a_{i,1}}(t_i - a_{i, 0})^2 \right)}{\sum_{i=1}^{N}w_{i}} , where t t is target, a 2-dimensional approx a 0 a_0 is target predict, a 1 a_1 is log ⁡ σ \log \sigma predict, and N ( y ∣ μ , σ 2 ) = 1 2 π σ 2 exp ⁡ ( − ( y − μ ) 2 2 σ 2 ) N(y\vert \mu,\sigma^2) = \frac{1}{\sqrt{2 \pi\sigma^2}} \exp(-\frac{(y-\mu)^2}{2\sigma^2}) is the probability density function of the normal distribution. See the Uncertainty section for more details. Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true LogLinQuantile Depends on the condition for the ratio of the label value and the resulting value: { ∑ i = 1 N α ∣ t i − e a i ∣ w i ∑ i = 1 N w i t i > e a i ∑ i = 1 N ( 1 − α ) ∣ t i − e a i ∣ w i ∑ i = 1 N w i t i ≤ e a i \begin{cases} \displaystyle\frac{\sum\limits_{i=1}^{N} \alpha |t_{i} - e^{a_{i}} | w_{i}}{\sum\limits_{i=1}^{N} w_{i}} & t_{i} > e^{a_{i}} \\ \displaystyle\frac{\sum\limits_{i=1}^{N} (1 - \alpha) |t_{i} - e^{a_{i}} | w_{i}}{\sum\limits_{i=1}^{N} w_{i}} & t_{i} \leq e^{a_{i}} \end{cases} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true alpha The coefficient used in quantile-based losses. Default: 0.5 Lq ∑ i = 1 N ∣ a i − t i ∣ q w i ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^N |a_{i} - t_{i}|^q w_i}{\sum\limits_{i=1}^N w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true q The power coefficient. Valid values are real numbers in the following range:  [ 1 ; + ∞ ) [1; +\infty) Default: Obligatory parameter Huber L ( t , a ) = ∑ i = 0 N l ( t i , a i ) ⋅ w i L(t, a) = \sum\limits_{i=0}^N l(t_i, a_i) \cdot w_{i} l ( t , a ) = { 1 2 ( t − a ) 2 , ∣ t − a ∣ ≤ δ δ ∣ t − a ∣ − 1 2 δ 2 , ∣ t − a ∣ > δ l(t,a) = \begin{cases} \frac{1}{2} (t - a)^{2} { , } & |t -a| \leq \delta \\ \delta|t -a| - \frac{1}{2} \delta^{2} { , } & |t -a| > \delta \end{cases} User-defined parameters: delta The δ \delta parameter of the Huber metric. Default: Obligatory parameter Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true Expectile ∑ i = 1 N ∣ α − I ( t i ≤ a i ) ∣ ( t i − a i ) 2 w i ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} |\alpha - I(t_{i} \leq a_{i})|(t_{i} - a_{i})^2 w_{i} }{\sum\limits_{i=1}^{N} w_{i}} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true alpha The coefficient used in expectile-based losses. Default: 0.5 Tweedie ∑ i = 1 N ( e a i ( 2 − λ ) 2 − λ − t i e a i ( 1 − λ ) 1 − λ ) ⋅ w i ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N}\left(\displaystyle\frac{e^{a_{i}(2-\lambda)}}{2-\lambda} - t_{i}\frac{e^{a_{i}(1-\lambda)}}{1-\lambda} \right)\cdot w_{i}}{\sum\limits_{i=1}^{N} w_{i}} λ \lambda is the value of the variance_power parameter. Labels t i t_i should be non-negative. Large labels may cause numerical overflows and/or divergence when training a tweedie regression model. On CPU, it is recommended to scale labels to range [ 0 , 1000 ] [0,1000] . On GPU, it is recommended to scale lables to range [ 0 , 1 ] [0,1] . Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true variance_power The variance of the Tweedie distribution. Supported values are in the range (1;2). Default: Obligatory parameter LogCosh ∑ i = 1 N w i log ⁡ ( cosh ⁡ ( a i − t i ) ) ∑ i = 1 N w i \displaystyle\frac{\sum_{i=1}^N w_i \log(\cosh(a_i - t_i))}{\sum_{i=1}^N w_i} Usage information See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true FairLoss ∑ i = 1 N c 2 ( ∣ t i − a i ∣ c − log ⁡ ( ∣ t i − a i ∣ c + 1 ) ) w i ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} c^2\left(\frac{|t_{i} - a_{i} |}{c} - \log\left(\frac{|t_{i} - a_{i} |}{c} + 1\right)\right)w_{i}}{\sum\limits_{i=1}^{N} w_{i}} c c is the value of the smoothness parameter. Can't be used for optimization. See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true use_weights The smoothness coefficient. Valid values are real values in the following range ( 0 ; + ∞ ) (0; +\infty) . Default: 1.0 NumErrors The proportion of predictions, for which the difference from the label value exceeds the specified value  greater_than . ∑ i = 1 N I ( ∣ a i − t i ∣ ≥ greater_than ) w i ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} I(|a_{i} - t_{i}|\geq \text{greater\_than}) w_{i}}{\sum\limits_{i=1}^{N} w_{i}} User-defined parameters: greater_than Can't be used for optimization. See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true SMAPE 100 ∑ i = 1 N w i ∣ a i − t i ∣ ( ∣ t i ∣ + ∣ a i ∣ ) / 2 ∑ i = 1 N w i \displaystyle\frac{100 \sum\limits_{i=1}^{N}\displaystyle\frac{w_{i} |a_{i} - t_{i} |}{(| t_{i} | + | a_{i} |) / 2}}{\sum\limits_{i=1}^{N} w_{i}} Can't be used for optimization. See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true R2 1 − ∑ i = 1 N w i ( a i − t i ) 2 ∑ i = 1 N w i ( t ˉ − t i ) 2 1 - \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} (a_{i} - t_{i})^{2}}{\sum\limits_{i=1}^{N} w_{i} (\bar{t} - t_{i})^{2}} t ˉ \bar{t} is the average label value: t ˉ = 1 N ∑ i = 1 N t i \bar{t} = \frac{1}{N}\sum\limits_{i=1}^{N}t_{i} Can't be used for optimization. See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true MSLE ∑ i = 1 N w i ( log ⁡ ( 1 + t i ) − log ⁡ ( 1 + a i ) ) 2 ∑ i = 1 N w i \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} (\log (1 + t_{i}) - \log (1 + a_{i}))^{2}}{\sum\limits_{i=1}^{N} w_{i}} Can't be used for optimization. See more . User-defined parameters use_weights Use object/group weights to calculate metrics if the specified value is  true and set all weights to 1 regardless of the input data if the specified value is false . Default: true MedianAbsoluteError median ( ∣ t 1 − a 1 ∣ , . . . , ∣ t N − a N ∣ ) \displaystyle\text{median}(|t_{1} - a_{1}|, ..., |t_{N} - a_{N}|) Can't be used for optimization. See more . User-defined parameters No. Cox ∑ t i > 0 ( a i − log ⁡ ∑ ∣ t j ∣ ≥ t i exp ⁡ ( a j ) ) \displaystyle\sum\limits_{t_i > 0}\left( a_i - \log\sum\limits_{|t_j| \ge t_i} \exp(a_j)\right) Labels t i > 0 t_i > 0 mean occurence of the event at time t i t_i , and labels t i < 0 t_i < 0 mean absence of the event at time ∣ t i ∣ |t_i| . Predictions a i a_i are hazard rates. Usage information See more . User-defined parameters No. SurvivalAft ∑ t i , 0 = t i , 1 log ⁡ ( f ( ϵ ( t i , 0 , a i ) ) + ∑ t i , 0 ≠ t i , 1 log ⁡ ( F ( ϵ ( t i , 1 , a i ) ) − F ( ϵ ( t i , 0 , a i ) ) ) \displaystyle\sum\limits_{t_{i,0} = t_{i,1}} \log\left(f(\epsilon(t_{i,0}, a_i)\right) + \sum\limits_{t_{i,0} \ne t_{i,1}} \log \left(F(\epsilon(t_{i,1}, a_i)) - F(\epsilon(t_{i,0}, a_i))\right) Observation interval is [ t i , 0 , t i , 1 ] [t_{i,0}, t_{i,1}] for t i , 1 ≠ − 1 t_{i,1} \ne -1 , and [ t i , 0 , ∞ ) [t_{i,0}, \infty) for t i , 1 = − 1 t_{i,1} = -1 . Predictions a i a_i are hazard rates. Helper ϵ ( t , a ) = ( log ⁡ t − a ) / σ \epsilon(t, a) = (\log t - a)/\sigma for t ≠ − 1 t \ne -1 , and ϵ ( − 1 , a ) = ∞ \epsilon(-1, a) = \infty , is hazard prediction error. Coefficient σ \sigma is scale of hazard prediction error, specified by scale parameter. Functions f f and F F are probability density and cumulative distribution, specified by dist parameter. dist Guessed distribution of hazard prediction error. Possible values: Normal , Extreme , Logistic . dist F F f f Normal 1 2 ( 1 + erf ( z 2 ) ) \displaystyle\frac{1}{2}\left(1+\text{erf}\left( \frac{z}{\sqrt{2}}\right)\right) e − z 2 / 2 2 π \displaystyle\frac{e^{-z^2/2}}{\sqrt{2\pi}} Logistic e z 1 + e z \displaystyle\frac{e^z}{1+e^z} e z ( 1 + e z ) 2 \displaystyle\frac{e^z}{(1+e^z)^2} Extreme 1 − e − e z \displaystyle 1-e^{-e^z} e z e − e z \displaystyle e^ze^{-e^z} Default: Normal scale Scale of hazard prediction error. Default: 1.0 Usage information See more . User-defined parameters No. Used for optimization Name Optimization GPU Support MAE + + MAPE + + Poisson + + Quantile + + MultiQuantile + - RMSE + + RMSEWithUncertainty + + LogLinQuantile + + Lq + + Huber + + Expectile + + Tweedie + + LogCosh + - Cox + - SurvivalAft + - FairLoss - - NumErrors - + SMAPE - - R2 - - MSLE - - MedianAbsoluteError - -
Markdown
[![Logo icon](https://yastatic.net/s3/locdoc/daas-static/catboost/71b237a322eec6f2889af0dae2a9c549.svg)](https://catboost.ai/ "CatBoost") - Installation - [Overview](https://catboost.ai/docs/en/concepts/en/concepts/installation) - Python package installation - CatBoost for Apache Spark installation - R package installation - Command-line version binary - Build from source - Key Features - Training parameters - Python package - CatBoost for Apache Spark - R package - Command-line version - Applying models - Objectives and metrics - [Overview](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions) - [Variables used in formulas](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-variables-used) - [Regression](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression) - [Multiregression](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-multiregression) - [Classification](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-classification) - [Multiclassification](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-multiclassification) - [Multilabel Classification](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-multilabel-classification) - [Ranking](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-ranking) - Model analysis - Data format description - [Parameter tuning](https://catboost.ai/docs/en/concepts/en/concepts/parameter-tuning) - [Speeding up the training](https://catboost.ai/docs/en/concepts/en/concepts/speed-up-training) - Data visualization - Algorithm details - [FAQ](https://catboost.ai/docs/en/concepts/en/concepts/faq) - Educational materials - [Development and contributions](https://catboost.ai/docs/en/concepts/en/concepts/development-and-contributions) - [Contacts](https://catboost.ai/docs/en/concepts/en/concepts/contacts) Objectives and metrics ## In this article: - [Objectives and metrics](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#objectives-and-metrics) - [MAE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MAE) - [MAPE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MAPE) - [Poisson](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Poisson) - [Quantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Quantile) - [MultiQuantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MultiQuantile) - [RMSE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#RMSE) - [RMSEWithUncertainty](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#RMSEWithUncertainty) - [LogLinQuantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#LogLinQuantile) - [Lq](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#lq) - [Huber](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Huber) - [Expectile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Expectile) - [Tweedie](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Tweedie) - [LogCosh](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#LogCosh) - [FairLoss](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#FairLoss) - [NumErrors](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#NumErrors) - [SMAPE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#SMAPE) - [R2](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#R2) - [MSLE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MSLE) - [MedianAbsoluteError](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MedianAbsoluteError) - [Cox](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Cox) - [SurvivalAft](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#SurvivalAft) - [Used for optimization](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information) 1. [Objectives and metrics](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions) 2. Regression # Regression: objectives and metrics - [Objectives and metrics](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#objectives-and-metrics) - [MAE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MAE) - [MAPE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MAPE) - [Poisson](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Poisson) - [Quantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Quantile) - [MultiQuantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MultiQuantile) - [RMSE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#RMSE) - [RMSEWithUncertainty](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#RMSEWithUncertainty) - [LogLinQuantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#LogLinQuantile) - [Lq](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#lq) - [Huber](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Huber) - [Expectile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Expectile) - [Tweedie](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Tweedie) - [LogCosh](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#LogCosh) - [FairLoss](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#FairLoss) - [NumErrors](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#NumErrors) - [SMAPE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#SMAPE) - [R2](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#R2) - [MSLE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MSLE) - [MedianAbsoluteError](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MedianAbsoluteError) - [Cox](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Cox) - [SurvivalAft](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#SurvivalAft) - [Used for optimization](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information) - [Objectives and metrics](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#objectives-and-metrics) - [Used for optimization](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-inforimation) ## Objectives and metrics ### MAE ∑ i \= 1 N w i ∣ a i − t i ∣ ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \| a\_{i} - t\_{i}\| }{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​wi​∣ai​−ti​∣​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### MAPE ∑ i \= 1 N w i ∣ a i − t i ∣ max ⁡ ( 1 , ∣ t i ∣ ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \\displaystyle\\frac{\|a\_{i}- t\_{i}\|}{\\max(1, \|t\_{i}\|)}}{\\sum\\limits\_{i=1}^{N}w\_{i}} i\=1∑N​wi​i\=1∑N​wi​max(1,∣ti​∣)∣ai​−ti​∣​​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### Poisson ∑ i \= 1 N w i ( e a i − a i t i ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \\left(e^{a\_{i}} - a\_{i}t\_{i}\\right)}{\\sum\\limits\_{i=1}^{N}w\_{i}} i\=1∑N​wi​i\=1∑N​wi​(eai​−ai​ti​)​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### Quantile ∑ i \= 1 N ( α − I ( t i ≤ a i ) ) ( t i − a i ) w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} (\\alpha - I(t\_{i} \\leq a\_{i}))(t\_{i} - a\_{i}) w\_{i} }{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​(α−I(ti​≤ai​))(ti​−ai​)wi​​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The coefficient used in quantile-based losses. *Default:* 0.5 ### MultiQuantile ∑ i \= 1 N w i ∑ q \= 1 Q ( α q − I ( t i ≤ a i , q ) ) ( t i − a i , q ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \\sum\\limits\_{q=1}^{Q} (\\alpha\_{q} - I(t\_{i} \\leq a\_{i,q}))(t\_{i} - a\_{i,q}) }{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​wi​q\=1∑Q​(αq​−I(ti​≤ai,q​))(ti​−ai,q​)​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The vector of coefficients used in multi-quantile loss. *Default:* 0.5 ### RMSE ∑ i \= 1 N ( a i − t i ) 2 w i ∑ i \= 1 N w i \\displaystyle\\sqrt{\\displaystyle\\frac{\\sum\\limits\_{i=1}^N (a\_{i}-t\_{i})^2 w\_{i}}{\\sum\\limits\_{i=1}^{N}w\_{i}}} i\=1∑N​wi​i\=1∑N​(ai​−ti​)2wi​​ ​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### RMSEWithUncertainty − ∑ i \= 1 N w i log ⁡ N ( t i ∣ a i , 0 , e 2 a i , 1 ) ∑ i \= 1 N w i \= 1 2 log ⁡ ( 2 π ) \+ ∑ i \= 1 N w i ( a i , 1 \+ 1 2 e − 2 a i , 1 ( t i − a i , 0 ) 2 ) ∑ i \= 1 N w i \\displaystyle-\\frac{\\sum\_{i=1}^N w\_i \\log N(t\_{i} \\vert a\_{i,0}, e^{2a\_{i,1}})}{\\sum\_{i=1}^{N}w\_{i}} = \\frac{1}{2}\\log(2\\pi) +\\frac{\\sum\_{i=1}^N w\_i\\left(a\_{i,1} + \\frac{1}{2} e^{-2a\_{i,1}}(t\_i - a\_{i, 0})^2 \\right)}{\\sum\_{i=1}^{N}w\_{i}} −∑i\=1N​wi​∑i\=1N​wi​logN(ti​∣ai,0​,e2ai,1​)​\=21​log(2π)\+∑i\=1N​wi​∑i\=1N​wi​(ai,1​\+21​e−2ai,1​(ti​−ai,0​)2)​, where t t t is target, a 2-dimensional approx a 0 a\_0 a0​ is target predict, a 1 a\_1 a1​ is log ⁡ σ \\log \\sigma logσ predict, and N ( y ∣ μ , σ 2 ) \= 1 2 π σ 2 exp ⁡ ( − ( y − μ ) 2 2 σ 2 ) N(y\\vert \\mu,\\sigma^2) = \\frac{1}{\\sqrt{2 \\pi\\sigma^2}} \\exp(-\\frac{(y-\\mu)^2}{2\\sigma^2}) N(y∣μ,σ2)\= 2πσ2 ​ 1 ​ exp(−2σ2(y−μ)2​) is the probability density function of the normal distribution. See the [Uncertainty section](https://catboost.ai/docs/en/concepts/en/references/uncertainty) for more details. **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### LogLinQuantile Depends on the condition for the ratio of the label value and the resulting value: { ∑ i \= 1 N α ∣ t i − e a i ∣ w i ∑ i \= 1 N w i t i \> e a i ∑ i \= 1 N ( 1 − α ) ∣ t i − e a i ∣ w i ∑ i \= 1 N w i t i ≤ e a i \\begin{cases} \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} \\alpha \|t\_{i} - e^{a\_{i}} \| w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} & t\_{i} \> e^{a\_{i}} \\\\ \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} (1 - \\alpha) \|t\_{i} - e^{a\_{i}} \| w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} & t\_{i} \\leq e^{a\_{i}} \\end{cases} ⎩ ⎨ ⎧ ​ i\=1∑N​wi​i\=1∑N​α∣ti​−eai​∣wi​​i\=1∑N​wi​i\=1∑N​(1−α)∣ti​−eai​∣wi​​​ti​\>eai​ti​≤eai​​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The coefficient used in quantile-based losses. *Default:* 0.5 ### Lq ∑ i \= 1 N ∣ a i − t i ∣ q w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^N \|a\_{i} - t\_{i}\|^q w\_i}{\\sum\\limits\_{i=1}^N w\_{i}} i\=1∑N​wi​i\=1∑N​∣ai​−ti​∣qwi​​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true q The power coefficient. Valid values are real numbers in the following range: \[ 1 ; \+ ∞ ) \[1; +\\infty) \[1;\+∞) *Default:* Obligatory parameter ### Huber L ( t , a ) \= ∑ i \= 0 N l ( t i , a i ) ⋅ w i L(t, a) = \\sum\\limits\_{i=0}^N l(t\_i, a\_i) \\cdot w\_{i} L(t,a)\=i\=0∑N​l(ti​,ai​)⋅wi​ l ( t , a ) \= { 1 2 ( t − a ) 2 , ∣ t − a ∣ ≤ δ δ ∣ t − a ∣ − 1 2 δ 2 , ∣ t − a ∣ \> δ l(t,a) = \\begin{cases} \\frac{1}{2} (t - a)^{2} { , } & \|t -a\| \\leq \\delta \\\\ \\delta\|t -a\| - \\frac{1}{2} \\delta^{2} { , } & \|t -a\| \> \\delta \\end{cases} l(t,a)\={21​(t−a)2,δ∣t−a∣−21​δ2,​∣t−a∣≤δ∣t−a∣\>δ​ User-defined parameters: delta The δ \\delta δ parameter of the Huber metric. *Default:* Obligatory parameter **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### Expectile ∑ i \= 1 N ∣ α − I ( t i ≤ a i ) ∣ ( t i − a i ) 2 w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} \|\\alpha - I(t\_{i} \\leq a\_{i})\|(t\_{i} - a\_{i})^2 w\_{i} }{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​∣α−I(ti​≤ai​)∣(ti​−ai​)2wi​​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The coefficient used in expectile-based losses. *Default:* 0.5 ### Tweedie ∑ i \= 1 N ( e a i ( 2 − λ ) 2 − λ − t i e a i ( 1 − λ ) 1 − λ ) ⋅ w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N}\\left(\\displaystyle\\frac{e^{a\_{i}(2-\\lambda)}}{2-\\lambda} - t\_{i}\\frac{e^{a\_{i}(1-\\lambda)}}{1-\\lambda} \\right)\\cdot w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​(2−λeai​(2−λ)​−ti​1−λeai​(1−λ)​)⋅wi​​ λ \\lambda λ is the value of the variance\_power parameter. Labels t i t\_i ti​ should be non-negative. Large labels may cause numerical overflows and/or divergence when training a tweedie regression model. On CPU, it is recommended to scale labels to range \[ 0 , 1000 \] \[0,1000\] \[0,1000\]. On GPU, it is recommended to scale lables to range \[ 0 , 1 \] \[0,1\] \[0,1\]. **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true variance\_power The variance of the Tweedie distribution. Supported values are in the range (1;2). *Default:* Obligatory parameter ### LogCosh ∑ i \= 1 N w i log ⁡ ( cosh ⁡ ( a i − t i ) ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\_{i=1}^N w\_i \\log(\\cosh(a\_i - t\_i))}{\\sum\_{i=1}^N w\_i} ∑i\=1N​wi​∑i\=1N​wi​log(cosh(ai​−ti​))​ **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### FairLoss ∑ i \= 1 N c 2 ( ∣ t i − a i ∣ c − log ⁡ ( ∣ t i − a i ∣ c \+ 1 ) ) w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} c^2\\left(\\frac{\|t\_{i} - a\_{i} \|}{c} - \\log\\left(\\frac{\|t\_{i} - a\_{i} \|}{c} + 1\\right)\\right)w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​c2(c∣ti​−ai​∣​−log(c∣ti​−ai​∣​\+1))wi​​ c c c is the value of the smoothness parameter. **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true use\_weights The smoothness coefficient. Valid values are real values in the following range ( 0 ; \+ ∞ ) (0; +\\infty) (0;\+∞). *Default:* 1.0 ### NumErrors The proportion of predictions, for which the difference from the label value exceeds the specified value `greater_than`. ∑ i \= 1 N I ( ∣ a i − t i ∣ ≥ greater\_than ) w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} I(\|a\_{i} - t\_{i}\|\\geq \\text{greater\\\_than}) w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​I(∣ai​−ti​∣≥greater\_than)wi​​ User-defined parameters: greater\_than **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### SMAPE 100 ∑ i \= 1 N w i ∣ a i − t i ∣ ( ∣ t i ∣ \+ ∣ a i ∣ ) / 2 ∑ i \= 1 N w i \\displaystyle\\frac{100 \\sum\\limits\_{i=1}^{N}\\displaystyle\\frac{w\_{i} \|a\_{i} - t\_{i} \|}{(\| t\_{i} \| + \| a\_{i} \|) / 2}}{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​100i\=1∑N​(∣ti​∣\+∣ai​∣)/2wi​∣ai​−ti​∣​​ **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### R2 1 − ∑ i \= 1 N w i ( a i − t i ) 2 ∑ i \= 1 N w i ( t ˉ − t i ) 2 1 - \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} (a\_{i} - t\_{i})^{2}}{\\sum\\limits\_{i=1}^{N} w\_{i} (\\bar{t} - t\_{i})^{2}} 1−i\=1∑N​wi​(tˉ−ti​)2i\=1∑N​wi​(ai​−ti​)2​ t ˉ \\bar{t} tˉ is the average label value: t ˉ \= 1 N ∑ i \= 1 N t i \\bar{t} = \\frac{1}{N}\\sum\\limits\_{i=1}^{N}t\_{i} tˉ\=N1​i\=1∑N​ti​ **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### MSLE ∑ i \= 1 N w i ( log ⁡ ( 1 \+ t i ) − log ⁡ ( 1 \+ a i ) ) 2 ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} (\\log (1 + t\_{i}) - \\log (1 + a\_{i}))^{2}}{\\sum\\limits\_{i=1}^{N} w\_{i}} i\=1∑N​wi​i\=1∑N​wi​(log(1\+ti​)−log(1\+ai​))2​ **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### MedianAbsoluteError median ( ∣ t 1 − a 1 ∣ , . . . , ∣ t N − a N ∣ ) \\displaystyle\\text{median}(\|t\_{1} - a\_{1}\|, ..., \|t\_{N} - a\_{N}\|) median(∣t1​−a1​∣,...,∣tN​−aN​∣) **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** No. ### Cox ∑ t i \> 0 ( a i − log ⁡ ∑ ∣ t j ∣ ≥ t i exp ⁡ ( a j ) ) \\displaystyle\\sum\\limits\_{t\_i \> 0}\\left( a\_i - \\log\\sum\\limits\_{\|t\_j\| \\ge t\_i} \\exp(a\_j)\\right) ti​\>0∑​ ​ ai​−log∣tj​∣≥ti​∑​exp(aj​) ​ Labels t i \> 0 t\_i \> 0 ti​\>0 mean occurence of the event at time t i t\_i ti​, and labels t i \< 0 t\_i \< 0 ti​\<0 mean absence of the event at time ∣ t i ∣ \|t\_i\| ∣ti​∣. Predictions a i a\_i ai​ are hazard rates. **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** No. ### SurvivalAft ∑ t i , 0 \= t i , 1 log ⁡ ( f ( ϵ ( t i , 0 , a i ) ) \+ ∑ t i , 0 ≠ t i , 1 log ⁡ ( F ( ϵ ( t i , 1 , a i ) ) − F ( ϵ ( t i , 0 , a i ) ) ) \\displaystyle\\sum\\limits\_{t\_{i,0} = t\_{i,1}} \\log\\left(f(\\epsilon(t\_{i,0}, a\_i)\\right) + \\sum\\limits\_{t\_{i,0} \\ne t\_{i,1}} \\log \\left(F(\\epsilon(t\_{i,1}, a\_i)) - F(\\epsilon(t\_{i,0}, a\_i))\\right) ti,0​\=ti,1​∑​log(f(ϵ(ti,0​,ai​))\+ti,0​\=ti,1​∑​log(F(ϵ(ti,1​,ai​))−F(ϵ(ti,0​,ai​))) Observation interval is \[ t i , 0 , t i , 1 \] \[t\_{i,0}, t\_{i,1}\] \[ti,0​,ti,1​\] for t i , 1 ≠ − 1 t\_{i,1} \\ne -1 ti,1​\=−1, and \[ t i , 0 , ∞ ) \[t\_{i,0}, \\infty) \[ti,0​,∞) for t i , 1 \= − 1 t\_{i,1} = -1 ti,1​\=−1. Predictions a i a\_i ai​ are hazard rates. Helper ϵ ( t , a ) \= ( log ⁡ t − a ) / σ \\epsilon(t, a) = (\\log t - a)/\\sigma ϵ(t,a)\=(logt−a)/σ for t ≠ − 1 t \\ne -1 t\=−1, and ϵ ( − 1 , a ) \= ∞ \\epsilon(-1, a) = \\infty ϵ(−1,a)\=∞, is hazard prediction error. Coefficient σ \\sigma σ is scale of hazard prediction error, specified by `scale` parameter. Functions f f f and F F F are probability density and cumulative distribution, specified by `dist` parameter. dist Guessed distribution of hazard prediction error. Possible values: `Normal`, `Extreme`, `Logistic`. | `dist` | F F F | f f f | |---|---|---| | `Normal` | 1 2 ( 1 \+ erf ( z 2 ) ) \\displaystyle\\frac{1}{2}\\left(1+\\text{erf}\\left( \\frac{z}{\\sqrt{2}}\\right)\\right) 21​ (1\+erf ( 2 ​ z ​ ) ) | e − z 2 / 2 2 π \\displaystyle\\frac{e^{-z^2/2}}{\\sqrt{2\\pi}} 2π ​ e−z2/2 ​ | | `Logistic` | e z 1 \+ e z \\displaystyle\\frac{e^z}{1+e^z} 1\+ezez​ | e z ( 1 \+ e z ) 2 \\displaystyle\\frac{e^z}{(1+e^z)^2} (1\+ez)2ez​ | | `Extreme` | 1 − e − e z \\displaystyle 1-e^{-e^z} 1−e−ez | e z e − e z \\displaystyle e^ze^{-e^z} eze−ez | *Default:* `Normal` scale Scale of hazard prediction error. *Default:* 1.0 **Usage information** See [more](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** No. ## Used for optimization | Name | Optimization | GPU Support | |---|---|---| | [MAE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MAE) | \+ | \+ | | [MAPE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MAPE) | \+ | \+ | | [Poisson](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Poisson) | \+ | \+ | | [Quantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Quantile) | \+ | \+ | | [MultiQuantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MultiQuantile) | \+ | \- | | [RMSE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#RMSE) | \+ | \+ | | [RMSEWithUncertainty](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#RMSEWithUncertainty) | \+ | \+ | | [LogLinQuantile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#LogLinQuantile) | \+ | \+ | | [Lq](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#lq) | \+ | \+ | | [Huber](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Huber) | \+ | \+ | | [Expectile](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Expectile) | \+ | \+ | | [Tweedie](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Tweedie) | \+ | \+ | | [LogCosh](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#LogCosh) | \+ | \- | | [Cox](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#Cox) | \+ | \- | | [SurvivalAft](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#SurvivalAft) | \+ | \- | | [FairLoss](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#FairLoss) | \- | \- | | [NumErrors](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#NumErrors) | \- | \+ | | [SMAPE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#SMAPE) | \- | \- | | [R2](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#R2) | \- | \- | | [MSLE](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MSLE) | \- | \- | | [MedianAbsoluteError](https://catboost.ai/docs/en/concepts/en/concepts/loss-functions-regression#MedianAbsoluteError) | \- | \- | ### Was the article helpful? 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- [Objectives and metrics](https://catboost.ai/docs/en/concepts/loss-functions-regression#objectives-and-metrics) - [Used for optimization](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-inforimation) ## Objectives and metrics ### MAE ∑ i \= 1 N w i ∣ a i − t i ∣ ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \| a\_{i} - t\_{i}\| }{\\sum\\limits\_{i=1}^{N} w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### MAPE ∑ i \= 1 N w i ∣ a i − t i ∣ max ⁡ ( 1 , ∣ t i ∣ ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \\displaystyle\\frac{\|a\_{i}- t\_{i}\|}{\\max(1, \|t\_{i}\|)}}{\\sum\\limits\_{i=1}^{N}w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### Poisson ∑ i \= 1 N w i ( e a i − a i t i ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \\left(e^{a\_{i}} - a\_{i}t\_{i}\\right)}{\\sum\\limits\_{i=1}^{N}w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### Quantile ∑ i \= 1 N ( α − I ( t i ≤ a i ) ) ( t i − a i ) w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} (\\alpha - I(t\_{i} \\leq a\_{i}))(t\_{i} - a\_{i}) w\_{i} }{\\sum\\limits\_{i=1}^{N} w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The coefficient used in quantile-based losses. *Default:* 0.5 ### MultiQuantile ∑ i \= 1 N w i ∑ q \= 1 Q ( α q − I ( t i ≤ a i , q ) ) ( t i − a i , q ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} \\sum\\limits\_{q=1}^{Q} (\\alpha\_{q} - I(t\_{i} \\leq a\_{i,q}))(t\_{i} - a\_{i,q}) }{\\sum\\limits\_{i=1}^{N} w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The vector of coefficients used in multi-quantile loss. *Default:* 0.5 ### RMSE ∑ i \= 1 N ( a i − t i ) 2 w i ∑ i \= 1 N w i \\displaystyle\\sqrt{\\displaystyle\\frac{\\sum\\limits\_{i=1}^N (a\_{i}-t\_{i})^2 w\_{i}}{\\sum\\limits\_{i=1}^{N}w\_{i}}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### RMSEWithUncertainty − ∑ i \= 1 N w i log ⁡ N ( t i ∣ a i , 0 , e 2 a i , 1 ) ∑ i \= 1 N w i \= 1 2 log ⁡ ( 2 π ) \+ ∑ i \= 1 N w i ( a i , 1 \+ 1 2 e − 2 a i , 1 ( t i − a i , 0 ) 2 ) ∑ i \= 1 N w i \\displaystyle-\\frac{\\sum\_{i=1}^N w\_i \\log N(t\_{i} \\vert a\_{i,0}, e^{2a\_{i,1}})}{\\sum\_{i=1}^{N}w\_{i}} = \\frac{1}{2}\\log(2\\pi) +\\frac{\\sum\_{i=1}^N w\_i\\left(a\_{i,1} + \\frac{1}{2} e^{-2a\_{i,1}}(t\_i - a\_{i, 0})^2 \\right)}{\\sum\_{i=1}^{N}w\_{i}}, where t t is target, a 2-dimensional approx a 0 a\_0 is target predict, a 1 a\_1 is log ⁡ σ \\log \\sigma predict, and N ( y ∣ μ , σ 2 ) \= 1 2 π σ 2 exp ⁡ ( − ( y − μ ) 2 2 σ 2 ) N(y\\vert \\mu,\\sigma^2) = \\frac{1}{\\sqrt{2 \\pi\\sigma^2}} \\exp(-\\frac{(y-\\mu)^2}{2\\sigma^2}) is the probability density function of the normal distribution. See the [Uncertainty section](https://catboost.ai/docs/en/references/uncertainty) for more details. **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### LogLinQuantile Depends on the condition for the ratio of the label value and the resulting value: { ∑ i \= 1 N α ∣ t i − e a i ∣ w i ∑ i \= 1 N w i t i \> e a i ∑ i \= 1 N ( 1 − α ) ∣ t i − e a i ∣ w i ∑ i \= 1 N w i t i ≤ e a i \\begin{cases} \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} \\alpha \|t\_{i} - e^{a\_{i}} \| w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} & t\_{i} \> e^{a\_{i}} \\\\ \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} (1 - \\alpha) \|t\_{i} - e^{a\_{i}} \| w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} & t\_{i} \\leq e^{a\_{i}} \\end{cases} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The coefficient used in quantile-based losses. *Default:* 0.5 ### Lq ∑ i \= 1 N ∣ a i − t i ∣ q w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^N \|a\_{i} - t\_{i}\|^q w\_i}{\\sum\\limits\_{i=1}^N w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true q The power coefficient. Valid values are real numbers in the following range: \[ 1 ; \+ ∞ ) \[1; +\\infty) *Default:* Obligatory parameter ### Huber L ( t , a ) \= ∑ i \= 0 N l ( t i , a i ) ⋅ w i L(t, a) = \\sum\\limits\_{i=0}^N l(t\_i, a\_i) \\cdot w\_{i} l ( t , a ) \= { 1 2 ( t − a ) 2 , ∣ t − a ∣ ≤ δ δ ∣ t − a ∣ − 1 2 δ 2 , ∣ t − a ∣ \> δ l(t,a) = \\begin{cases} \\frac{1}{2} (t - a)^{2} { , } & \|t -a\| \\leq \\delta \\\\ \\delta\|t -a\| - \\frac{1}{2} \\delta^{2} { , } & \|t -a\| \> \\delta \\end{cases} User-defined parameters: delta The δ \\delta parameter of the Huber metric. *Default:* Obligatory parameter **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### Expectile ∑ i \= 1 N ∣ α − I ( t i ≤ a i ) ∣ ( t i − a i ) 2 w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} \|\\alpha - I(t\_{i} \\leq a\_{i})\|(t\_{i} - a\_{i})^2 w\_{i} }{\\sum\\limits\_{i=1}^{N} w\_{i}} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true alpha The coefficient used in expectile-based losses. *Default:* 0.5 ### Tweedie ∑ i \= 1 N ( e a i ( 2 − λ ) 2 − λ − t i e a i ( 1 − λ ) 1 − λ ) ⋅ w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N}\\left(\\displaystyle\\frac{e^{a\_{i}(2-\\lambda)}}{2-\\lambda} - t\_{i}\\frac{e^{a\_{i}(1-\\lambda)}}{1-\\lambda} \\right)\\cdot w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} λ \\lambda is the value of the variance\_power parameter. Labels t i t\_i should be non-negative. Large labels may cause numerical overflows and/or divergence when training a tweedie regression model. On CPU, it is recommended to scale labels to range \[ 0 , 1000 \] \[0,1000\]. On GPU, it is recommended to scale lables to range \[ 0 , 1 \] \[0,1\]. **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true variance\_power The variance of the Tweedie distribution. Supported values are in the range (1;2). *Default:* Obligatory parameter ### LogCosh ∑ i \= 1 N w i log ⁡ ( cosh ⁡ ( a i − t i ) ) ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\_{i=1}^N w\_i \\log(\\cosh(a\_i - t\_i))}{\\sum\_{i=1}^N w\_i} **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### FairLoss ∑ i \= 1 N c 2 ( ∣ t i − a i ∣ c − log ⁡ ( ∣ t i − a i ∣ c \+ 1 ) ) w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} c^2\\left(\\frac{\|t\_{i} - a\_{i} \|}{c} - \\log\\left(\\frac{\|t\_{i} - a\_{i} \|}{c} + 1\\right)\\right)w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} c c is the value of the smoothness parameter. **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true use\_weights The smoothness coefficient. Valid values are real values in the following range ( 0 ; \+ ∞ ) (0; +\\infty). *Default:* 1.0 ### NumErrors The proportion of predictions, for which the difference from the label value exceeds the specified value `greater_than`. ∑ i \= 1 N I ( ∣ a i − t i ∣ ≥ greater\_than ) w i ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} I(\|a\_{i} - t\_{i}\|\\geq \\text{greater\\\_than}) w\_{i}}{\\sum\\limits\_{i=1}^{N} w\_{i}} User-defined parameters: greater\_than **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### SMAPE 100 ∑ i \= 1 N w i ∣ a i − t i ∣ ( ∣ t i ∣ \+ ∣ a i ∣ ) / 2 ∑ i \= 1 N w i \\displaystyle\\frac{100 \\sum\\limits\_{i=1}^{N}\\displaystyle\\frac{w\_{i} \|a\_{i} - t\_{i} \|}{(\| t\_{i} \| + \| a\_{i} \|) / 2}}{\\sum\\limits\_{i=1}^{N} w\_{i}} **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### R2 1 − ∑ i \= 1 N w i ( a i − t i ) 2 ∑ i \= 1 N w i ( t ˉ − t i ) 2 1 - \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} (a\_{i} - t\_{i})^{2}}{\\sum\\limits\_{i=1}^{N} w\_{i} (\\bar{t} - t\_{i})^{2}} t ˉ \\bar{t} is the average label value: t ˉ \= 1 N ∑ i \= 1 N t i \\bar{t} = \\frac{1}{N}\\sum\\limits\_{i=1}^{N}t\_{i} **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### MSLE ∑ i \= 1 N w i ( log ⁡ ( 1 \+ t i ) − log ⁡ ( 1 \+ a i ) ) 2 ∑ i \= 1 N w i \\displaystyle\\frac{\\sum\\limits\_{i=1}^{N} w\_{i} (\\log (1 + t\_{i}) - \\log (1 + a\_{i}))^{2}}{\\sum\\limits\_{i=1}^{N} w\_{i}} **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** use\_weights Use object/group weights to calculate metrics if the specified value is "true" and set all weights to "1" regardless of the input data if the specified value is "false". *Default:* true ### MedianAbsoluteError median ( ∣ t 1 − a 1 ∣ , . . . , ∣ t N − a N ∣ ) \\displaystyle\\text{median}(\|t\_{1} - a\_{1}\|, ..., \|t\_{N} - a\_{N}\|) **Can't be used for optimization.** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** No. ### Cox ∑ t i \> 0 ( a i − log ⁡ ∑ ∣ t j ∣ ≥ t i exp ⁡ ( a j ) ) \\displaystyle\\sum\\limits\_{t\_i \> 0}\\left( a\_i - \\log\\sum\\limits\_{\|t\_j\| \\ge t\_i} \\exp(a\_j)\\right) Labels t i \> 0 t\_i \> 0 mean occurence of the event at time t i t\_i, and labels t i \< 0 t\_i \< 0 mean absence of the event at time ∣ t i ∣ \|t\_i\|. Predictions a i a\_i are hazard rates. **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** No. ### SurvivalAft ∑ t i , 0 \= t i , 1 log ⁡ ( f ( ϵ ( t i , 0 , a i ) ) \+ ∑ t i , 0 ≠ t i , 1 log ⁡ ( F ( ϵ ( t i , 1 , a i ) ) − F ( ϵ ( t i , 0 , a i ) ) ) \\displaystyle\\sum\\limits\_{t\_{i,0} = t\_{i,1}} \\log\\left(f(\\epsilon(t\_{i,0}, a\_i)\\right) + \\sum\\limits\_{t\_{i,0} \\ne t\_{i,1}} \\log \\left(F(\\epsilon(t\_{i,1}, a\_i)) - F(\\epsilon(t\_{i,0}, a\_i))\\right) Observation interval is \[ t i , 0 , t i , 1 \] \[t\_{i,0}, t\_{i,1}\] for t i , 1 ≠ − 1 t\_{i,1} \\ne -1, and \[ t i , 0 , ∞ ) \[t\_{i,0}, \\infty) for t i , 1 \= − 1 t\_{i,1} = -1. Predictions a i a\_i are hazard rates. Helper ϵ ( t , a ) \= ( log ⁡ t − a ) / σ \\epsilon(t, a) = (\\log t - a)/\\sigma for t ≠ − 1 t \\ne -1, and ϵ ( − 1 , a ) \= ∞ \\epsilon(-1, a) = \\infty, is hazard prediction error. Coefficient σ \\sigma is scale of hazard prediction error, specified by `scale` parameter. Functions f f and F F are probability density and cumulative distribution, specified by `dist` parameter. dist Guessed distribution of hazard prediction error. Possible values: `Normal`, `Extreme`, `Logistic`. | `dist` | F F | f f | |---|---|---| | `Normal` | 1 2 ( 1 \+ erf ( z 2 ) ) \\displaystyle\\frac{1}{2}\\left(1+\\text{erf}\\left( \\frac{z}{\\sqrt{2}}\\right)\\right) | e − z 2 / 2 2 π \\displaystyle\\frac{e^{-z^2/2}}{\\sqrt{2\\pi}} | | `Logistic` | e z 1 \+ e z \\displaystyle\\frac{e^z}{1+e^z} | e z ( 1 \+ e z ) 2 \\displaystyle\\frac{e^z}{(1+e^z)^2} | | `Extreme` | 1 − e − e z \\displaystyle 1-e^{-e^z} | e z e − e z \\displaystyle e^ze^{-e^z} | *Default:* `Normal` scale Scale of hazard prediction error. *Default:* 1.0 **Usage information** See [more](https://catboost.ai/docs/en/concepts/loss-functions-regression#usage-information). **User-defined parameters** No. ## Used for optimization | Name | Optimization | GPU Support | |---|---|---| | [MAE](https://catboost.ai/docs/en/concepts/loss-functions-regression#MAE) | \+ | \+ | | [MAPE](https://catboost.ai/docs/en/concepts/loss-functions-regression#MAPE) | \+ | \+ | | [Poisson](https://catboost.ai/docs/en/concepts/loss-functions-regression#Poisson) | \+ | \+ | | [Quantile](https://catboost.ai/docs/en/concepts/loss-functions-regression#Quantile) | \+ | \+ | | [MultiQuantile](https://catboost.ai/docs/en/concepts/loss-functions-regression#MultiQuantile) | \+ | \- | | [RMSE](https://catboost.ai/docs/en/concepts/loss-functions-regression#RMSE) | \+ | \+ | | [RMSEWithUncertainty](https://catboost.ai/docs/en/concepts/loss-functions-regression#RMSEWithUncertainty) | \+ | \+ | | [LogLinQuantile](https://catboost.ai/docs/en/concepts/loss-functions-regression#LogLinQuantile) | \+ | \+ | | [Lq](https://catboost.ai/docs/en/concepts/loss-functions-regression#lq) | \+ | \+ | | [Huber](https://catboost.ai/docs/en/concepts/loss-functions-regression#Huber) | \+ | \+ | | [Expectile](https://catboost.ai/docs/en/concepts/loss-functions-regression#Expectile) | \+ | \+ | | [Tweedie](https://catboost.ai/docs/en/concepts/loss-functions-regression#Tweedie) | \+ | \+ | | [LogCosh](https://catboost.ai/docs/en/concepts/loss-functions-regression#LogCosh) | \+ | \- | | [Cox](https://catboost.ai/docs/en/concepts/loss-functions-regression#Cox) | \+ | \- | | [SurvivalAft](https://catboost.ai/docs/en/concepts/loss-functions-regression#SurvivalAft) | \+ | \- | | [FairLoss](https://catboost.ai/docs/en/concepts/loss-functions-regression#FairLoss) | \- | \- | | [NumErrors](https://catboost.ai/docs/en/concepts/loss-functions-regression#NumErrors) | \- | \+ | | [SMAPE](https://catboost.ai/docs/en/concepts/loss-functions-regression#SMAPE) | \- | \- | | [R2](https://catboost.ai/docs/en/concepts/loss-functions-regression#R2) | \- | \- | | [MSLE](https://catboost.ai/docs/en/concepts/loss-functions-regression#MSLE) | \- | \- | | [MedianAbsoluteError](https://catboost.ai/docs/en/concepts/loss-functions-regression#MedianAbsoluteError) | \- | \- |
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Unparsed URLai,catboost!/docs/en/concepts/loss-functions-regression s443