Square Mean Error vs. Log Loss

Overview of cost functions

These two functions are used to calculate the loss and then the cost of our models.

Loss being the difference between our predictions and ground labels for an instance while cost being the average sum of differences from our predictions.

The following are the few over arching differences

Formula's of each function

Square Mean Error

$$loss = \frac{1}{2}(\hat{y}-y)^2$$

Log Loss

$$loss = - ( (y)\log{\hat{y}} + (1-y)\log{(1-\hat{y})} )$$

Let's see the two functions in action:

y	y_hat	LOG LOSS	SME
0.1	0.9	2.08	0.64
0.2	0.8	1.33	0.36
0.3	0.7	0.95	0.16
0.4	0.6	0.75	0.04
0.5	0.5	0.69	0.0
0.6	0.4	0.75	0.04
0.7	0.3	0.95	0.16
0.8	0.2	1.33	0.36
0.9	0.1	2.08	0.64