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Suppose I have $N$ training examples and there are $K$ classes and the targets have a $1$ of $K$ encoding

Let $t_k^n$ denote the kth component of the nth training target

Let $x^n$ denote the the input to the final hidden layer for the nth training example

Let $a^n = \theta^Tx^n$ denote the activation of the final hidden layer for the nth training example

Let $z^n = \sigma(a^n)$ denote the output of the final hidden layer for the nth training example

Let $y_k^n = \frac{exp(z_k^n)}{\sum_{j=1}^k exp(z_j^n)}$ denote the output of the net for the nth training example

Let $E = -\sum_{n=1}^N \sum_{k=1}^{K} t_k^n \log y_k^n$ denote the loss function

Then: \begin{eqnarray} \frac{\partial}{\partial \theta_{ij}}t_k \log y_k = t_k \frac{d\log y_k}{dy_k} \frac{\partial y_k}{\partial z_j} \frac{dz_j}{da_j} \frac{\partial a_j}{\partial \theta_{ij}} \end{eqnarray} Where:

\begin{eqnarray} \frac{d\log y_k}{dy_k} = \frac{1}{y_k} \end{eqnarray}

\begin{eqnarray} \frac{\partial y_k}{\partial z_j} &=& \frac{\delta_{jk}e^{z_k}(\sum_{l=1}^K e^{z_l}) - e^{z_k}e^{z_j}}{(\sum_{l=1}^K e^{z_l})^2} \\ &=& \frac{\delta_{jk}e^{z_k}}{\sum_{l=1}^K e^{z_l}} - \frac{e^{z_k}}{\sum_{l=1}^K e^{z_l}}\frac{e^{z_j}}{\sum_{l=1}^K e^{z_l}} \\ &=& \delta_{jk}y_k - y_ky_j \\ &=& y_k(\delta_{jk} - y_j) \end{eqnarray}

\begin{eqnarray} \frac{d z_j}{da_j} = \frac{d}{da_j} \sigma(a_j)) = \frac{d}{da_j} \frac{1}{1+e^{-a_j}} = \frac{e^{a_j}}{(1+e^{-a_j})^2} \end{eqnarray}

\begin{eqnarray} \frac{\partial a_j}{\partial \theta_{ij}} = x_i \end{eqnarray}

So:

\begin{eqnarray} \frac{\partial E}{\partial \theta_{ij}} = -\sum_{n=1}^N \sum_{k=1}^{K} t_k^n(\delta_{jk} - y_j^n)\frac{e^{a_j^n}}{(1+e^{-a_j^n})^2}x_i^n \end{eqnarray}

Does this seem right?

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Look like you have error in the last formula.

\begin{eqnarray} \frac{\partial E}{\partial \theta_{ij}} = -\sum_{n=1}^N \sum_{k=1}^{K} t_k^n(\delta_{jk} - y_j^n)\frac{e^{a_j^n}}{(1+e^{-a_j^n})^2}x_i^n \end{eqnarray}

You try get derivative for some special weight in error function and in result you get sum of all network outputs. If you differentiate by some weight you must get in result only answer for output which in result contains that weight. There must be one output which contains weight $\theta_{ij}$. (another outputs which don't contains that weight must be a constants for this differentation)

Also you make differentiation for the final layer, for hidden layers this formula must be different.

Note:

For sigmoid function you can use easier formula:

\begin{eqnarray} \frac{d z_j}{da_j} = \frac{d}{da_j} \sigma(a_j)) = \frac{d}{da_j} \frac{1}{1+e^{-a_j}} = \frac{e^{a_j}}{(1+e^{-a_j})^2} = \frac{1}{(1+e^{-a_j})} \frac{e^{a_j}}{(1+e^{-a_j})} = \frac{1}{(1+e^{-a_j})} (1 - \frac{1}{(1+e^{-a_j})}) = \sigma(a_j) (1 - \sigma(a_j)) \end{eqnarray}

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