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An error in the derivative of the softplus function #4

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3Dori opened this issue Mar 9, 2021 · 0 comments

3Dori commented Mar 9, 2021

The derivative of the softplus function $\frac{d}{dx} (\log(1 + \exp(x)))$ is indeed $\frac{1}{1 + \exp(-x)}.$

But as a function with respect to a layer's output, it should actually be $1 - \exp(-x).$

To see this, denote the input of the activation function by y, and its corresponding output by x. Notice that
$\frac{d}{dx} \mathrm{softplus}(y) = \frac{1}{1 + \exp(-y)} = \frac{\exp(y)}{1 + \exp(y)} = \frac{\exp(\mathrm{softplus}(y)) - 1}{\exp(\mathrm{softplus}(y))} = 1 - \exp(-\mathrm{softplus}(y)) = 1 - \exp(-x).$

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