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I'm currently trying to create a neural network with 2 hidden layers from scratch.

  • The input layer has 784 dimensions (MNIST dataset)
  • The first hidden layer has 100 neurons using sigmoid activator
  • The second hidden layer has 10 neurons using sigmoid activator
  • The output layer has 10 possible outcomes (digit 0-9) using softmax activator

I easily computed the output (third) layer derivative with respect to the final (third) weight matrix: $$\frac{\partial L}{\partial W_3} = \frac{\partial L}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial z_3} \frac{\partial z_3}{\partial W_3} = [1\ H_2]^T(\hat{y} - y)$$

$\frac{\partial L}{\partial z_3} = (\hat{y} - y)$ is the partial derivative of the loss function with respect to the softmax input.

$\frac{\partial z3}{\partial z_3} = \frac{\partial}{\partial z_3} [1\ H_2]W_3 = [1\ H_2]^T$ is the partial derivative of the softmax input with respect to the weight matrix.

I checked the dimension, and since $[1\ H_2]^T \in \mathbb{R}^{11\times1}$ and $(\hat{y} - y) \in \mathbb{R}^{1\times10}$, $\frac{\partial L}{\partial W_3}\in \mathbb{R}^{11\times10}$, which has the same dimension as $W_3$. Thus, it would be possible to perform gradient descent.

Next up, I want to compute the derivative of loss function with respect to the second layer's weight matrix:

$$\frac{\partial L}{\partial W_2} = \frac{\partial L}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial z_3} \frac{\partial z_3}{\partial H_2} \frac{\partial H_2}{\partial z_2} \frac{\partial z_2}{\partial W_2}$$

  • $\frac{\partial L}{\partial z_3} = (\hat{y} - y) \in \mathbb{R}^{1\times10}$
  • $\frac{\partial z_3}{\partial H_2} = \frac{\partial}{\partial H_2} [1\ H_2]W_3=W_3^T \in \mathbb{R}^{10\times10}$
  • $\frac{\partial H_2}{\partial z_2} = \frac{\partial}{\partial z_2} \sigma(z_2)=\sigma(z_2)(1-\sigma(z_2))=H_2(1-H_2) \in \mathbb{R}^{10\times1}$
  • $\frac{\partial z_2}{\partial W_2} = \frac{\partial}{\partial W_2} [1\ H_1]W_2=[1\ H_1] \in \mathbb{R}^{1\times101}$

I can't merge these derivatives using chain rule into $Dim(W_2) \in \mathbb{R}^{101\times10}$ no matter what, which is needed for gradient descent. I feel like I'm missing something in my $\frac{\partial L}{\partial W_2}$. Could someone please give me some insights?

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I found the answer, turns out the derivative can be solved as shown below:

$[1\ H_1]^T(\hat{y}-y)W_3^TH_2(1-H_2) \in \mathbb{R}^{101\times1}$

This is the same dimension as $W_2$, so it is possible to compute gradient descent.

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Björn Friedrich Oct 27 '18 at 20:25
  • $\begingroup$ @BjörnFriedrich Hence he is in fact the author of the OP I would say it is "okay" in some way to post an answer of this type; but nevertheless I would suggest you tkhu to just add such information - like when you figured out an important detail concerning your question which might answer the question itself - within your question as an marked edit for clarification since an actual answer should be more comprehensive than just one line :) $\endgroup$ – mrtaurho Oct 27 '18 at 20:59
  • $\begingroup$ Umm okay this is my first time answering my own question. I'll add more information here. Let me know if that's fine now? $\endgroup$ – tkhu Oct 28 '18 at 0:02

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