What is the relationship between "square loss" and "Mean squared error"? Loss functions are important part of machine learning.

Square loss is one of the most popular loss functions.

Mean squared error (MSE) measures  the average squared difference between the estimated values and what
 is estimated

The estimated value is denoted as $\hat y$, what is estimated is denoted as $y$.

$${\displaystyle \operatorname {MSE} ={\frac {1}{m}}\sum _{i=1}^{m}(y_{i}-{\hat y_{i}})^{2}.}$$

in the context of distance/difference, it seems that MSE and square loss do the same job.

what is the relationship between "square loss" and "Mean squared error"?
$$V(f(x), y) = (1-yf(x))^2$$

In the context of classification where $y_i$ is $1$ or $-1$:

\begin{align} 
MSE &amp;= \frac1m \sum_{i=1}^m (y_i-f(x_i))^2 \\
&amp;= \frac1m \sum_{i=1}^m y_i^2 (1-y_if(x_i))^2\\
&amp;= \frac1m \sum_{i=1}^m (1-y_if(x_i))^2\\
&amp;= \frac1m \sum_{i=1}^m V(f(x_i), y_i)
\end{align}