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What is the name of this function:

$E = \frac{1}{2}\ \sum\limits_{n=1}^N \{ output_n - target_n\}^2$

Which is used in machine learning (e.g. error back prop).

I have sometimes seen the name "MSE" for this, but there is nothing "Mean" here because MSE is calculated differently,

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Without the $\frac{1}{2}$, it's the residual sum of squares.

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If your aim is minimization of MSE, you can minimize any strictly monotone function $g$ of MSE, e.g. $$ E := g(\text{MSE}) = n \text{MSE}/2. $$

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