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In the back-propogation algorithm, the error term is:

$$ E=\frac{1}{2}\sum_k(\hat{y}_k - y_k)^2, $$

where $\hat{y}_k$ is a vector of outputs from the network, $y_k$ is the vector of correct labels (and we work out the error by calculating predicted-observed, squaring the answer, and then summing the answers for each $k$ (and dividing by 2).

How do you prove that if this answer is $0$ (i.e., if $E=0$), then $\hat{y}_k=y_k$ for all $k$?

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This is very easy to prove.

Let's first prove that, if $\hat{y}_k = y_k$, then the $E = 0$. I will leave all steps, so that it's super clear.

\begin{align} E &=\frac{1}{2}\sum_k(\hat{y}_k - y_k)^2 \\ &=\frac{1}{2}\sum_k(y_k - y_k)^2\\ &=\frac{1}{2}\sum_k(0)^2\\ &=\frac{1}{2}\sum_k 0\\ &=\frac{1}{2} 0\\ &=0\\ \end{align}

To prove the other way around, i.e. if $E = 0$, then $\hat{y}_k = y_k$, you can do as follows

\begin{align} \frac{1}{2}\sum_k(\hat{y}_k - y_k)^2 &=E\\ &=0 \end{align} Recall now that any number squared is non-negative (i.e. positive or zero). Given that $(\hat{y}_k - y_k)^2 $ is non-negative, then $\sum_k(\hat{y}_k - y_k)^2$ is a sum of non-negative numbers. The only way that a sum of non-negative numbers is equal to zero is if all numbers are zero, so we must have $\hat{y}_k = y_k$ (because any non-zero number squared is non-zero).

(Note that $E$ is the mean squared error, i.e. a loss function, and it's not the back-propagation algorithm, which is just the algorithm that you use to compute partial derivatives of $E$ with respect to the parameters of the model, which are not even visible in the way you wrote $E$).

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