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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 \\ …

ReLU and sigmoid have different properties (i.e. range), as you already noticed. I've never seen the ReLU being used as the activation function of the output layer (but some people may use it for some …

If you look at the definition of the cross-entropy (e.g. here), you will see that it is defined for probability distributions (in fact, it comes from information theory). You can also show that the ma …

On page 5 of the VAE paper, it's clearly stated We let $p_{\boldsymbol{\theta}}(\mathbf{x} \mid \mathbf{z})$ be a multivariate Gaussian (in case of real-valued data) or Bernoulli (in case of binary d …

Positive only losses result in weights only incrementing in their pre-initialised sign No. In gradient descent, you use the (partial) derivative or the gradient (if you're using vector operations an …

As a rule of thumb, mean squared error (MSE) is more appropriate for regression problems, that is, problems where the output is a numerical value (i.e. a floating-point number or, in general, a real n …

$g(x) = x^2$ is indeed a parabola and thus has just one optimum. However, the $\text{MSE}(\boldsymbol{x}, \boldsymbol{y}) = \sum_i (y_i - f(x_i))^2$, where $\boldsymbol{x}$ are the inputs, $\boldsymbo …