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What is the derivative of negative log MLE (MLE used as a negative cost) when the variables are passed through softmax activations?

The derivative of the cost function is needed for back-propagation.

When the output layer uses a MLE/cross-entropy cost and softmax activation, the derivative of the cost function combined with the activation function simplifies to a very simple expression. This note covers this computation.

Consider the last layer of a neural network shown below:


For the given training example \( x_1 \), the correct category is 2 (out of all categories 0,1 and 2). Only the output corresponding to category 2 is used in the calculation of the loss contributed by the example. The loss is given by:

\[ Loss_{ex_1} = -ln(p_3) = -ln(\frac{e^{f_3}}{e^{f_1} + e^{f_2} + e^{f_3}}) \]

While only the probability for the correct category is used to calculate the loss for the example, the other outputs, \( p_1 \) and \( p_2 \) still affect the loss, as \( p_3 \) depends on these values (all sum to 1).

The partial derivatives will be:

\[ \begin{align*} \frac{\partial L}{\partial f_1} &= p_1 \\ \frac{\partial L}{\partial f_2} &= p_2 \\ \frac{\partial L}{\partial f_3} &= -(1-p_3) = p_3 - 1\end{align*} \]

So if we have output probabilities [0.1, 0.3, 0.6], then the derivative of the loss with respect to last layer's output before applying the softmax activation will be [0.1, 0.3, -0.4].

Derivations

Formulating the derivative of loss wrt different variables is simplified if the log and exponential are canceled at the beginning:

\[ \begin{align*} Loss_{ex_1} = -ln(p_3) &= -ln(\frac{e^{f_3}}{e^{f_1} + e^{f_2} + e^{f_3}}) \\  &= ln(\frac{e^{f_1} + e^{f_2} + e^{f_3}}{e^{f_3}}) \\ &= ln(e^{f_1} + e^{f_2} + e^{f_3}) - ln(e^{f_3}) \\ &= ln(e^{f_1} + e^{f_2} + e^{f_3}) - f_3  \end{align*} \]

From here, the partial derivates are obvious. For example:

\[ \begin{align*} \frac{\partial L}{\partial f_1} &= \frac{\partial}{\partial f_1}(ln(e^{f_1} + e^{f_2} + e^{f_3}) - f_3) \\ &= \frac{\partial}{\partial x}(ln(x)) \frac{\partial}{\partial f_1}(e^{f_1} + e^{f_2} + e^{f_3}) \\ &= \frac{1}{x} e^{f_1} \\ &= \frac{e^{f_1}}{e^{f_1} + e^{f_2} + e^{f_3}} \\ &= p_1 \end{align*} \]

The below image is part of a separate card, but I'm including it here, as it is quite relevant.


Old image:


Some old stuff:

Given the loss (lets just call it L, with the example 1 being implicit), we wish to find: $\frac{\partial L}{\partial f_1} \text{, } \frac{\partial L}{\partial f_2} \text{, } \frac{\partial L}{\partial f_3}$. First consider $f_1$:

\[ \begin{align*} \frac{\partial L}{\partial f_1} = \frac{\partial}{\partial f_1}(-ln(p_3)) &= (\frac{\partial}{\partial x}(-ln(x)) (\frac{\partial}{\partial y}(\frac{e^{f_3}}{y + e^{f_2} + e^{f_3}}))(\frac{\partial}{\partial f_1}(e^{f_1})) \\ \text{where } x = \frac{e^{f_3}}{y + e^{f_2} + e^{f_3}}, \text{ and } y = e^{f_1} \\ &= (- \frac{1}{x})(\frac{-e^{f_3}}{(y + e^{f_2} + e^{f_3})^2})(e^{f_1}) \\ &= (-\frac{e^{f_1} + e^{f_2} + e^{f_3}}{e^{f_3}})(\frac{-e^{f_3}}{(e^{f_1} + e^{f_2} + e^{f_3})^2})(e^{f_1}) \\ &= \frac{e^{f_1}}{e^{f_1} + e^{f_2} + e^{f_3}} \\ &= p_1 \end{align*} \]

Similarly, $\frac{\partial L}{\partial f_2} = p_2$. $\frac{\partial L}{\partial f_3}$ is calculated as follows:

\[ \begin{align*} \frac{\partial L}{\partial f_3} = \frac{\partial}{\partial f_3}(-ln(p_3)) &= (\frac{\partial}{\partial x}(-ln(x)) (\frac{\partial}{\partial y}(\frac{e^{f_3}}{e^{f_1}+ e^{f_2} + y}))(\frac{\partial}{\partial f_3}(e^{f_3})) \\ \text{where } x = \frac{e^{f_3}}{e^{f_1} + e^{f_2} + y}, \text{ and } y = e^{f_3} \\ &= (- \frac{1}{x})(\frac{1}{e^{f_1} + e^{f_2} + y} - \frac{e^{f_3}}{e^{f_1} + e^{f_2} + y})(e^{f_3}) \\ &= (- \frac{e^{f_1} + e^{f_2} + e^{f_3}}{e^{f_3}})(\frac{1}{e^{f_1} + e^{f_2} + e^{f_3}} - \frac{e^{f_3}}{e^{f_1} + e^{f_2} + e^{f_3}})(e^{f_3}) \\ &= -(1 - \frac{e^{f_3}}{e^{f_1} + e^{f_2} + e^{f_3}}) \\ &= p_3 - 1 \end{align*} \]