Regression: objectives and metrics

• Objectives and metrics
• Used for optimization

Objectives and metricsObjectives and metrics

MAEMAE

$\frac{\sum\limits_{i=1}^{N} w_{i} | a_{i} - t_{i}| }{\sum\limits_{i=1}^{N} w_{i}}$

Usage information See more.

User-defined parameters

MAPEMAPE

$\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \displaystyle\frac{|a_{i}- t_{i}|}{Max(1, |t_{i}|)}}{\sum\limits_{i=1}^{N}w_{i}}$

Usage information See more.

User-defined parameters

PoissonPoisson

$\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \left(e^{a_{i}} - a_{i}t_{i}\right)}{\sum\limits_{i=1}^{N}w_{i}}$

Usage information See more.

User-defined parameters

QuantileQuantile

$\displaystyle\frac{\sum\limits_{i=1}^{N} (\alpha - I(t_{i} \leq a_{i}))(t_{i} - a_{i}) w_{i} }{\sum\limits_{i=1}^{N} w_{i}}$

Usage information See more.

User-defined parameters

MultiQuantileMultiQuantile

$\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} \sum\limits_{q=1}^{Q} (\alpha_{q} - I(t_{i} \leq a_{i,q}))(t_{i} - a_{i,q}) }{\sum\limits_{i=1}^{N} w_{i}}$

Usage information See more.

User-defined parameters

RMSERMSE

$\displaystyle\sqrt{\displaystyle\frac{\sum\limits_{i=1}^N (a_{i}-t_{i})^2 w_{i}}{\sum\limits_{i=1}^{N}w_{i}}}$

Usage information See more.

User-defined parameters

RMSEWithUncertaintyRMSEWithUncertainty

$\displaystyle-\frac{\sum_{i=1}^N w_i \log N(t_{i} \vert a_{i,0}, e^{2a_{i,1}})}{\sum_{i=1}^{N}w_{i}} = \frac{1}{2}\log(2\pi) +\frac{\sum_{i=1}^N w_i\left(a_{i,1} + \frac{1}{2} e^{-2a_{i,1}}(t_i - a_{i, 0})^2 \right)}{\sum_{i=1}^{N}w_{i}}$,
where $t$ is target, a 2-dimensional approx $a_0$ is target predict, $a_1$ is $\log \sigma$ predict, and $N(y\vert \mu,\sigma^2) = \frac{1}{\sqrt{2 \pi\sigma^2}} \exp(-\frac{(y-\mu)^2}{2\sigma^2})$ is the probability density function of the normal distribution.

See the Uncertainty section for more details.

Usage information See more.

User-defined parameters

LogLinQuantileLogLinQuantile

Depends on the condition for the ratio of the label value and the resulting value:
$\begin{cases} \displaystyle\frac{\sum\limits_{i=1}^{N} \alpha |t_{i} - e^{a_{i}} | w_{i}}{\sum\limits_{i=1}^{N} w_{i}} & t_{i} > e^{a_{i}} \\ \displaystyle\frac{\sum\limits_{i=1}^{N} (1 - \alpha) |t_{i} - e^{a_{i}} | w_{i}}{\sum\limits_{i=1}^{N} w_{i}} & t_{i} \leq e^{a_{i}} \end{cases}$

Usage information See more.

User-defined parameters

LqLq

$\displaystyle\frac{\sum\limits_{i=1}^N |a_{i} - t_{i}|^q w_i}{\sum\limits_{i=1}^N w_{i}}$

Usage information See more.

User-defined parameters

HuberHuber

$L(t, a) = \sum\limits_{i=0}^N l(t_i, a_i) \cdot w_{i} { , where}$

$l(t,a) = \begin{cases} \frac{1}{2} (t - a)^{2} { , } & |t -a| \leq \delta \\ \delta|t -a| - \frac{1}{2} \delta^{2} { , } & |t -a| > \delta \end{cases}$

User-defined parameters:

Usage information See more.

User-defined parameters

ExpectileExpectile

$\displaystyle\frac{\sum\limits_{i=1}^{N} |\alpha - 1(t_{i} \leq a_{i})|(t_{i} - a_{i})^2 w_{i} }{\sum\limits_{i=1}^{N} w_{i}}$

Usage information See more.

User-defined parameters

TweedieTweedie

$\displaystyle\frac{\sum\limits_{i=1}^{N}\left(\displaystyle\frac{e^{a_{i}(2-\lambda)}}{2-\lambda} - t_{i}\frac{e^{a_{i}(1-\lambda)}}{1-\lambda} \right)\cdot w_{i}}{\sum\limits_{i=1}^{N} w_{i}} { , where}$

$\lambda$ is the value of the variance_power parameter.

Usage information See more.

User-defined parameters

LogCoshLogCosh

$\frac{\sum_{i=1}^N w_i \log(\cosh(a_i - t_i))}{\sum_{i=1}^N w_i}$

Usage information See more.

User-defined parameters

FairLossFairLoss

$\displaystyle\frac{\sum\limits_{i=1}^{N} c^2(\frac{|t_{i} - a_{i} |}{c} - \ln(\frac{|t_{i} - a_{i} |}{c} + 1))w_{i}}{\sum\limits_{i=1}^{N} w_{i}} { , where}$

$c$ is the value of the smoothness parameter.

Can't be used for optimization. See more.

User-defined parameters

NumErrorsNumErrors

The proportion of predictions, for which the difference from the label value exceeds the specified value greater_than.

$\displaystyle\frac{\sum\limits_{i=1}^{N} I\{x\} w_{i}}{\sum\limits_{i=1}^{N} w_{i}} { , where}$

$I\{x\} = \begin{cases} 1 { , } & |a_{i} - t_{i}| > greater\_than \\ 0 { , } & |a_{i} - t_{i}| \leq greater\_than \end{cases}$

User-defined parameters: greater_than

Increase the numerator of the formula if the following inequality is met:

$|prediction - label|>value$

Can't be used for optimization. See more.

User-defined parameters

SMAPESMAPE

$\displaystyle\frac{100 \sum\limits_{i=1}^{N}\displaystyle\frac{w_{i} |a_{i} - t_{i} |}{(| t_{i} | + | a_{i} |) / 2}}{\sum\limits_{i=1}^{N} w_{i}}$

Can't be used for optimization. See more.

User-defined parameters

R2R2

$1 - \displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} (a_{i} - t_{i})^{2}}{\sum\limits_{i=1}^{N} w_{i} (\bar{t} - t_{i})^{2}}$
$\bar{t}$ is the average label value:
$\bar{t} = \frac{1}{N}\sum\limits_{i=1}^{N}t_{i}$

Can't be used for optimization. See more.

User-defined parameters

MSLEMSLE

$\displaystyle\frac{\sum\limits_{i=1}^{N} w_{i} (\log_{e} (1 + t_{i}) - \log_{e} (1 + a_{i}))^{2}}{\sum\limits_{i=1}^{N} w_{i}}$

Can't be used for optimization. See more.

User-defined parameters

MedianAbsoluteErrorMedianAbsoluteError

$median(|t_{1} - a_{1}|, ..., |t_{i} - a_{i}|)$

Can't be used for optimization. See more.

User-defined parameters

No.

Used for optimizationUsed for optimization

Name	Optimization	GPU Support
MAE	+	-
MAPE	+	+
Poisson	+	+
Quantile	+	+
MultiQuantile	+	-
RMSE	+	+
RMSEWithUncertainty	+	-
LogLinQuantile	+	+
Lq	+	+
Huber	+	+
Expectile	+	+
Tweedie	+	+
LogCosh	+	-
FairLoss	-	-
NumErrors	-	+
SMAPE	-	-
R2	-	-
MSLE	-	-
MedianAbsoluteError	-	-