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I am trying to wrap my head around back-propagation in a neural network with a Softmax classifier, which uses the Softmax function:

\begin{equation} p_j = \frac{e^{o_j}}{\sum_k e^{o_k}} \end{equation}

This is used in a loss function of the form

\begin{equation}L = -\sum_j y_j \log p_j,\end{equation}

where $o$ is a vector. I need the derivative of $L$ with respect to $o$. Now if my derivatives are right,

\begin{equation} \frac{\partial p_j}{\partial o_i} = p_i(1 - p_i),\quad i = j \end{equation}

and

\begin{equation} \frac{\partial p_j}{\partial o_i} = -p_i p_j,\quad i \neq j. \end{equation}

Using this result we obtain

\begin{eqnarray} \frac{\partial L}{\partial o_i} &=& - \left (y_i (1 - p_i) + \sum_{k\neq i}-p_k y_k \right )\\ &=&p_i y_i - y_i + \sum_{k\neq i} p_k y_k\\ &=& \left (\sum_i p_i y_i \right ) - y_i \end{eqnarray}

According to slides I'm using, however, the result should be

\begin{equation} \frac{\partial L}{\partial o_i} = p_i - y_i. \end{equation}

Can someone please tell me where I'm going wrong?

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    $\begingroup$ For others who end up here, this thread is about computing the derivative of the cross-entropy function, which is the cost function often used with a softmax layer (though the derivative of the cross-entropy function uses the derivative of the softmax, -p_k * y_k, in the equation above). Eli Bendersky has an awesome derivation of the softmax and its associated cost function here: eli.thegreenplace.net/2016/… $\endgroup$
    – duhaime
    Jan 1, 2018 at 17:52

1 Answer 1

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Your derivatives $\large \frac{\partial p_j}{\partial o_i}$ are indeed correct, however there is an error when you differentiate the loss function $L$ with respect to $o_i$.

We have the following (where I have highlighted in $\color{red}{red}$ where you have gone wrong) $$\frac{\partial L}{\partial o_i}=-\sum_ky_k\frac{\partial \log p_k}{\partial o_i}=-\sum_ky_k\frac{1}{p_k}\frac{\partial p_k}{\partial o_i}\\=-y_i(1-p_i)-\sum_{k\neq i}y_k\frac{1}{p_k}({\color{red}{-p_kp_i}})\\=-y_i(1-p_i)+\sum_{k\neq i}y_k({\color{red}{p_i}})\\=-y_i+\color{blue}{y_ip_i+\sum_{k\neq i}y_k({p_i})}\\=\color{blue}{p_i\left(\sum_ky_k\right)}-y_i=p_i-y_i$$ given that $\sum_ky_k=1$ from the slides (as $y$ is a vector with only one non-zero element, which is $1$).

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    $\begingroup$ I am unsure how to get to the last line from the previous line in this answer. Would it be possible to post more information? $\endgroup$
    – FatalMojo
    Sep 19, 2015 at 5:21
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    $\begingroup$ @FatalMojo I have added an extra line between the last and the penultimate lines, and highlighted some terms in blue. $\endgroup$ Sep 19, 2015 at 9:10
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    $\begingroup$ @AlijahAhmed Can i ask how did you get the first line. And how did it go to the second line $\endgroup$
    – aceminer
    May 25, 2017 at 14:20
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    $\begingroup$ @aceminer For the first line, the $y_k$ do not depend on $o_j$, so they are constants. This leads to $\frac{\partial L}{\partial o_i}=-\sum_k\color{red}{y_k}\frac{\partial \log p_k}{\partial o_i}$. Then, you use the differential identity $\frac{\partial \log{f(x)}}{\partial x}=\frac{1}{f(x)}\frac{\partial f(x)}{\partial x}$, leading to result $-\sum_ky_k\frac{1}{p_k}\frac{\partial p_k}{\partial o_i}$, at the end of the first line. For the second line, we use result $\frac{\partial p_j}{\partial o_i} = p_i(1 - p_i),\quad i = j$ and $\frac{\partial p_j}{\partial o_i} = -p_i p_j,\quad i \neq j$. $\endgroup$ Jun 4, 2017 at 11:51
  • $\begingroup$ Can someone explain as to how this result be generalized to $\frac{\partial L}{\partial o}$ i.e., derivative of the loss with respect to the vector instead of the single entry of a vector. $\endgroup$
    – amj
    Nov 28, 2019 at 21:10

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