Logarithmic loss (or cross-entropy) measures the performance of a classification model where the prediction input is a probability value between 0 and 1.
The goal of our machine learning models is to minimize this value. It is also heavily used in Kaggle competitions to estimate the score of submissions.
A perfect model would have a log loss of 0. Log loss increases as the predicted probability diverge from the actual label. So predicting a probability of .012 when the actual observation label is 1 would be bad and result in a high log loss.
We compute log loss by this formula for binary classifiers:
Lets take an example
| x(Input) | y(Actual label) | y^(Prediction or probability of class label=1) |
|---|---|---|
| x1 | 1 | 0.95 |
| x2 | 0 | 0.91 |
| x3 | 1 | 0.87 |
| x4 | 1 | 0.65 |
| x5 | 0 | 0.7 |
log loss = -[(log(0.95) + log(0.09)+log(0.87)+log(0.65)+log(0.3))]/5
Log loss can be defined as average negative log of probability of correct clas label.
here is a plot of -log(x)
As we can see form above figure, the smaller the log loss, the better it is.
Multiclass Log loss
The formula we discussed above only holds for binary classifiers. For multiclass log loss estimation we use:
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