L2 loss vs. mean squared loss I see some literature consider L2 loss (least squared error) and mean squared error loss are two different kinds of loss functions.

However, it seems to me these two loss functions essentially compute the same thing (with a 1/n factor difference).

So I am wondering if I have missed anything? Is there any scenario that one should use one of the two loss functions?
Function $L_2(x):=\left \|x \right \|_2$ is a norm, it is not a loss by itself. It is called a "loss" when it is used in a loss function to measure a distance between two vectors, $\left \| y_1 - y_2 \right \|^2_2$, or to measure the size of a vector, $\left \| \theta \right \|^2_2$. This goes with a loss minimization that tries to bring these quantities to the "least" possible value.

These are some illustrations:


$L_p$ norm: $L_p(x) := \left \|x \right \|_p = (\sum_{i=1}^{D} |x_i|^p)^{1/p}$,
where $D$ is the dimension of vector $x$,
Squared error: $\mbox{SE}(A, \theta) =\sum_{n=1}^{N} \left \| y_n - f_{\theta}(x_n) \right \|^2_2$,
where $A=\{(x_n, y_n)_{n=1}^{N}\}$ is a set of data points, and $f_{\theta}(x_n)$ is model's estimation of $y_n$,
Mean squared error: $\mbox{MSE}(A, \theta) =\mbox{SE}(A, \theta)/N$,   
Least squares optimization: $\theta^*=\mbox{argmin}_{\theta} \mbox{MSE}(A, \theta)$$=\mbox{argmin}_{\theta} \mbox{SE}(A, \theta)$,  
Ridge loss: $\mbox{R}(A, \theta, \lambda) = \mbox{MSE}(A, \theta) + \lambda\left \| \theta \right \|^2_2$
Ridge optimization (regression): $\theta^*=\mbox{argmin}_{\theta} \mbox{R}(A, \theta, \lambda)$.


In all of the above examples, $L_2$ norm can be replaced with $L_1$ norm or $L_\infty$ norm, etc.. However the names "squared error", "least squares",
 and "Ridge" are reserved for $L_2$ norm. For example for $L_1$, "squared error" becomes "absolute error":


Absolute error: $\mbox{AE}(A, \theta) =\sum_{n=1}^{N} \left \| y_n - f_{\theta}(x_n) \right \|_1$,