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I'm trying to perform backpropagation on a neural network using Softmax activation on the output layer and a cross-entropy cost function. Here are the steps I take:

  1. Calculate the error gradient with respect to each output neuron's input:

$$ \frac{\partial E} {\partial z_j} = {\frac{\partial E} {\partial o_j}}{\frac{\partial o_j} {\partial z_j}} $$

where $\frac{\partial E} {\partial o_j}$ is the derivative of the cost function with respect to the node's output and $\frac{\partial o_j} {\partial z_j}$ is the derivative of the activation function.

  1. Adjust the output layer's weights using the following formula:

$$ w_{ij} = w'_{ij} - r{\frac{\partial E} {\partial z_j}} {o_i} $$

where $r$ is some learning rate constant and $o_i$ is the $i$th output from the previous layer.

  1. Adjust the hidden layer's weights using the following formula:

$$ w_{ij} = w'_{ij} - r{\frac{\partial o_j} {\partial z_j}} {\sum_k (E_k w'_{jk})} {o_i} $$

where ${\frac{\partial o_j} {\partial z_j}}$ is the derivative of the hidden layer's activation function and $E$ is the vector of output layer error gradients computed in Step 1.

Question: The internet has told me that when using Softmax combined with cross entropy, Step 1 simply becomes

$$ \frac{\partial E} {\partial z_j} = o_j - t_j $$ where $t$ is a one-hot encoded target output vector. Is this correct?

For some reason, each round of backpropagation is causing my network to adjust itself heavily toward the provided label - so much that the network's predictions are always whatever the most recent backpropagation label was, regardless of input. I don't understand why this is happening, or how it can even be possible.

There must be something wrong with the method I'm using. Any ideas?

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The internet has told me that when using Softmax combined with cross entropy, Step 1 simply becomes $\frac{\partial E} {\partial z_j} = o_j - t_j$ where $t$ is a one-hot encoded target output vector. Is this correct?

Yes. Before going through the proof, let me change the notation to avoid careless mistakes in translation:

Notation:

I'll follow the notation in this made-up example of color classification:

enter image description here

whereby $j$ is the index denoting any of the $K$ output neurons - not necessarily the one corresponding to the true, ($t)$, value. Now,

$$\begin{align} o_j&=\sigma(j)=\sigma(z_j)=\text{softmax}(j)=\text{softmax (neuron }j)=\frac{e^{z_j}}{\displaystyle\sum_K e^{z_k}}\\[3ex] z_j &= \mathbf w_j^\top \mathbf x = \text{preactivation (neuron }j) \end{align}$$

The loss function is the negative log likelihood:

$$E = -\log \sigma(t) = -\log \left(\text{softmax}(t)\right)$$

The negative log likelihood is also known as the multiclass cross-entropy (ref: Pattern Recognition and Machine Learning Section 4.3.4), as they are in fact two different interpretations of the same formula.


Gradient of the loss function with respect to the pre-activation of an output neuron:

$$\begin{align} \frac{\partial E}{\partial z_j}&=\frac{\partial}{\partial z_j}\,-\log\left( \sigma(t)\right)\\[2ex] &= \frac{-1}{\sigma(t)}\quad\frac{\partial}{\partial z_j}\sigma(t)\\[2ex] &= \frac{-1}{\sigma(t)}\quad\frac{\partial}{\partial z_j}\sigma(z_j)\\[2ex] &= \frac{-1}{\sigma(t)}\quad\frac{\partial}{\partial z_j}\frac{e^{z_t}}{\displaystyle\sum_k e^{z_k}}\\[2ex] &= \frac{-1}{\sigma(t)}\quad\left[ \frac{\frac{\partial }{\partial z_j }e^{z_t}}{\displaystyle \sum_K e^{z_k}} \quad - \quad \frac{e^{z_t}\quad \frac{\partial}{\partial z_j}\displaystyle \sum_K e^{z_k}}{\left[\displaystyle\sum_K e^{z_k}\right]^2}\right]\\[2ex] &= \frac{-1}{\sigma(t)}\quad\left[ \frac{\delta_{jt}\;e^{z_t}}{\displaystyle \sum_K e^{z_k}} \quad - \quad \frac{e^{z_t}}{\displaystyle\sum_K e^{z_k}} \frac{e^{z_j}}{\displaystyle\sum_K e^{z_k}}\right]\\[2ex] &= \frac{-1}{\sigma(t)}\quad\left(\delta_{jt}\sigma(t) - \sigma(t)\sigma(j) \right)\\[2ex] &= - (\delta_{jt} - \sigma(j))\\[2ex] &= \sigma(j) - \delta_{jt} \end{align}$$

This is practically identical to $\frac{\partial E} {\partial z_j} = o_j - t_j$, and it does become identical if instead of focusing on $j$ as an individual output neuron, we transition to vectorial notation (as indicated in your question), and $t_j$ becomes the one-hot encoded vector of true values, which in my notation would be $\small \begin{bmatrix}0&0&0&\cdots&1&0&0&0_K\end{bmatrix}^\top$.

Then, with $\frac{\partial E} {\partial z_j} = o_j - t_j$ we are really calculating the gradient of the loss function with respect to the preactivation of all output neurons: the vector $t_j$ will contain a $1$ only in the neuron corresponding to the correct category, which is equivalent to the delta function $\delta_{jt}$, which is $1$ only when differentiating with respect to the pre-activation of the output neuron of the correct category.


In the Geoffrey Hinton's Coursera ML course the following chunk of code illustrates the implementation in Octave:

%% Compute derivative of cross-entropy loss function.
error_deriv = output_layer_state - expanded_target_batch;

The expanded_target_batch corresponds to the one-hot encoded sparse matrix with corresponding to the target of the training set. Hence, in the majority of the output neurons, the error_deriv = output_layer_state $(\sigma(j))$, because $\delta_{jt}$ is $0$, except for the neuron corresponding to the correct classification, in which case, a $1$ is going to be subtracted from $\sigma(j).$

The actual measurement of the cost is carried out with...

% MEASURE LOSS FUNCTION.
CE = -sum(sum(...
  expanded_target_batch .* log(output_layer_state + tiny))) / batchsize;

We see again the $\frac{\partial E}{\partial z_j}$ in the beginning of the backpropagation algorithm:

$$\small\frac{\partial E}{\partial W_{hidd-2-out}}=\frac{\partial \text{outer}_{input}}{\partial W_{hidd-2-out}}\, \frac{\partial E}{\partial \text{outer}_{input}}=\frac{\partial z_j}{\partial W_{hidd-2-out}}\, \frac{\partial E}{\partial z_j}$$

in

hid_to_output_weights_gradient =  hidden_layer_state * error_deriv';
output_bias_gradient = sum(error_deriv, 2);

since $z_j = \text{outer}_{in}= W_{hidd-2-out} \times \text{hidden}_{out}$


Observation re: OP additional questions:

  • The splitting of partials in the OP, $\frac{\partial E} {\partial z_j} = {\frac{\partial E} {\partial o_j}}{\frac{\partial o_j} {\partial z_j}}$, seems unwarranted.

  • The updating of the weights from hidden to output proceeds as...

    hid_to_output_weights_delta = ...
     momentum .* hid_to_output_weights_delta + ...
     hid_to_output_weights_gradient ./ batchsize;
    hid_to_output_weights = hid_to_output_weights...
     - learning_rate * hid_to_output_weights_delta;
    

which don't include the output $o_j$ in the OP formula: $w_{ij} = w'_{ij} - r{\frac{\partial E} {\partial z_j}} {o_i}.$ The formula would be more along the lines of...

$$W_{hidd-2-out}:=W_{hidd-2-out}-r\, \small \frac{\partial E}{\partial W_{hidd-2-out}}\, \Delta_{hidd-2-out}$$

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