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Objective function vs Evaluation function
The objective function is the target that your model tries to optimize when training on a dataset.
The Evaluation function (or evaluation metrics) is, as suggested by its name, a function to evaluate the performance of a Machine learning model on a dataset.
New researchers often get confused about objective functions and evaluation functions. Their difference is: the objective function is perceived by your model/algorithm, it targets to optimize the objective function. On the other hand, the evaluation function is only observed by the researchers themselves and is evaluated after the training complete.
A common question is: Why do we care about the evaluation function but have the model to optimize the objective function? If we want the evaluation function, why don’t we use it as the target for our model to optimize?
The answer is: yes, in the perfect world, a separate objective function should not exist, our model should optimize the evaluation function, which is the function we – researchers really care about. But in our world, it is not so easy. Some evaluation function is not optimizable by the machine, which is why we need an objective function to act as a proxy function to approximate the evaluation function. To go a bit farther, the reason for being not optimizable is because they are not differentiable, which is a needed condition for optimization algorithms like Gradient Descent.
On a side note, a model can have only 1 objective function but can have many evaluation functions. It is also advisable for researchers to evaluate model performance on various points of view (various evaluation functions).
Every objective function can work as an evaluation function, but not vice versa.
There are other terms that are closely related to Objective function, like Loss function or Cost function.
In highlevel usage, you can just assume that those terms have the same meaning and are just other names for Objective function.
But in some literature, the authors may use them a little bit differently:
The functions
Below, I will list out some of the most common objective/evaluation functions for regression models. Note that these functions measure the error of the whole dataset, not just an individual sample like loss functions.
Depending on the problem we want to solve that we choose a suitable objective function (and one or more evaluation functions).
Mean Absolute Error
Mean Absolute Error (MAE) is also called the L1 cost function.
MAE = ,
where:
n is the number of samples,
is the true response of the ith sample,
is the predicted response of the ith sample.
Value of MAE is in range .
For our model to be better, we should minimize MAE.
MAE can be used as an Objective function.
Mean Squared Error
Mean Squared Error (MSE) is also called the L2 cost function.
MSE = ,
where:
n is the number of samples,
is the true response of the ith sample,
is the predicted response of the ith sample.
Value of MSE is in range .
For our model to be better, we should minimize MSE.
MSE can be used as an Objective function.
Max Absolute Error
Max Absolute Error (MaxAE) is also called the L cost function.
MaxAE = ,
where:
n is the number of samples,
is the true response of the ith sample,
is the predicted response of the ith sample.
Value of MaxAE is in range .
For our model to be better, we should minimize the MaxAE.
MaxAE cannot be used as an Objective function.
Root Mean Squared Error
RMSE = ,
where:
n is the number of samples,
is the true response of the ith sample,
is the predicted response of the ith sample.
Value of RMSE is in range .
For our model to be better, we should minimize RMSE.
RMSE can be used as an Objective function.
RSquared
Rsquared (or or ) represents how much variance of the response variable is predictable from the predictor variables.
Suppose:
is the mean of all observed responses, we have:
The total sum of squares, which measure the variance of responses:
The sum of squares of residual (residual is the difference between true response and predicted response, sometimes this term can be used interchangeably with error):
The Rsquared is computed by:
.
Normally, value of is in range [0, 1]. can only be negative if the model you use is worse than a simple model that gives the output as the mean of the responses for any sample.
A value of , for e.g. 0.78, means that using our model, 78% of the difference in the response variable can be explained by the predictor variables.
For our model to be better, we should maximize .
Note that should not be used for nonlinear models since those models are extremely complex and can perfectlyfit almost any input data, using will overrate those models’ strength. Even for linear models, is not enough to determine if a model is acceptable or not.
cannot be used as an Objective function.
Adjusted RSquared
A disadvantage of Rsquared is that it prefers models with a higher number of predictor variables. Notice the more the predictors, the more your linear model will overfit the training data, which results in a higher . In other words, when we want a measurement that gives higher value if the model is better (not just on the training data, but in general), Rsquared gives higher value when the model fits the training data better, even if it overfits.
What should be emphasized here is that will only be higher or equal when we add more predictors to the model, which is not good because we cannot compare 2 models with a different number of predictors. Hence, the Adjusted was born to solve this problem.
For Adjusted , adding a new predictor will only increase Adjusted if that predictor increases the model’s performance more than expected by chance. That is to say, adding a good predictor will help the Adjusted increase, while adding a bad one will make Adjusted decrease.
Adjusted ,
where:
n is the number of samples,
k is the number of predictor variables in the model.
You may also noticed that:
Adjusted ,
where:
is the unbiased variance of residuals.
is the unbiased variance of responses.
Adjusted is usually in range [0, 1] and sometimes can be negative. Adjusted .
For our model to be better, we should maximize the Adjusted .
Adjusted cannot be used as an Objective function.
Predicted RSquared
Predicted is simply of the testing data. While using or Adjusted , we don’t separate our data to a training set and a testing set, yet, with Predicted , we do.
We can use many methods to split data into training and testing sets (e.g. random split, kfold crossvalidation). Kfold is preferred because it is stronger over overfitting. Here is a guideline for splitting.
The idea here is: we training our model on the training set, and Predicted is the measured on the testing set. By calculating on a separated set of data, we can be more confident with the generalization power of the model.
Note that how we split the data is very important. If any information on the testing data is leaked to the training data, it will ruin our measurement.
Predicted cannot be used as an Objective function.
Mean Squared Log Error
MSLE =,
where:
n is the number of samples,
is the true response of the ith sample,
is the predicted response of the ith sample.
MSLE is often used when the response can be exponentially large. E.g. when your problem is to predict house prices, the response variable can vary, from several thousand to some millions. The error will be proportional to the ratio of predicted response over true response, rather than absolute difference.
Given an example below:
sample 1  sample 2  
true response  20  2000 
predicted response  30  3000 
Squared Log Error  0.164  0.164 
Using MSLE, the errors of 2 samples are the same, even if the absolute differences are not so.
A note is that MSLE penalizes underestimation more than overestimation. Hence, the final model will more likely overestimate the samples rather than underestimate. Look at the following example.
sample 1  sample 2  
true response  20  20 
predicted response  15  25 
Squared Log Error  0.083  0.05 
One more note: to use MSLE, the responses must be positive as cannot take zero as its argument ( is undefined). However, there is quite a high chance that some values or be zero, so in practice, we often add 1 to and when computing MSLE. In other words, we often use this modified version of MSLE:
MSLE =
Value of MSLE is in range .
For our model to be better, we should minimize MSLE.
MSLE can be used as an Objective function.
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Summary
In this blog, we introduced the Objective function and the Evaluation function along with their differences. The Objective function is the target that your model tries to optimize, while the Evaluation function is what we – humans see and care about (and want to optimize).
We also presented some of the most common objective/evaluation functions for regression problems. Each of them has its own advantages and disadvantages.
MAE is simple and easily interpreted.
MSE is an alternative for MAE if you want to emphasize on penalizing higher error.
MaxAE is less common than the 2 above. Use it if you only care about the worstcase scenario.
RMSE is quite similar to MSE. Use it instead of MSE if you want the error to have the same unit as the response.
, Adjusted and Predicted are used only for linear regression. is the original version, so it is the simplest among the 3. Adjusted penalizes models that have useless predictors. Predicted is the most robust one against overfitting.
MSLE should be used when your response is nonnegative and is exponential, and you want the error to be proportional to ratio rather than absolute difference. Notice that MSLE penalizes underestimation more than overestimation.
So, to close this topic, I would say that choosing which objective/evaluation function to use depends on your specific problem and how you want your outcome to be. You can choose one from above, or just create your own custom function.
References: