# numpy : calculate the derivative of the softmax function

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Problem :

I am trying to understand `backpropagation` in a simple 3 layered neural network with `MNIST`.

There is the input layer with `weights` and a `bias`. The labels are `MNIST` so it’s a `10` class vector.

The second layer is a `linear tranform`. The third layer is the `softmax activation` to get the output as probabilities.

`Backpropagation` calculates the derivative at each step and call this the gradient.

Previous layers appends the `global` or `previous` gradient to the `local gradient`. I am having trouble calculating the `local gradient` of the `softmax`

Several resources online go through the explanation of the softmax and its derivatives and even give code samples of the softmax itself

``````def softmax(x):
"""Compute the softmax of vector x."""
exps = np.exp(x)
return exps / np.sum(exps)
``````

The derivative is explained with respect to when `i = j` and when `i != j`. This is a simple code snippet I’ve come up with and was hoping to verify my understanding:

``````def softmax(self, x):
"""Compute the softmax of vector x."""
exps = np.exp(x)
return exps / np.sum(exps)

def forward(self):
# self.input is a vector of length 10
# and is the output of
# (w * x) + b
self.value = self.softmax(self.input)

def backward(self):
for i in range(len(self.value)):
for j in range(len(self.input)):
if i == j:
else:
``````

Then `self.gradient` is the `local gradient` which is a vector. Is this correct? Is there a better way to write this?

Solution :

I am assuming you have a 3-layer NN with `W1`, `b1` for is associated with the linear transformation from input layer to hidden layer and `W2`, `b2` is associated with linear transformation from hidden layer to output layer. `Z1` and `Z2` are the input vector to the hidden layer and output layer. `a1` and `a2` represents the output of the hidden layer and output layer. `a2` is your predicted output. `delta3` and `delta2` are the errors (backpropagated) and you can see the gradients of the loss function with respect to model parameters.

This is a general scenario for a 3-layer NN (input layer, only one hidden layer and one output layer). You can follow the procedure described above to compute gradients which should be easy to compute! Since another answer to this post already pointed to the problem in your code, i am not repeating the same.

As I said, you have `n^2` partial derivatives.

If you do the math, you find that `dSM[i]/dx[k]` is `SM[i] * (dx[i]/dx[k] - SM[i])` so you should have:

``````if i == j:
else:
``````

``````if i == j:
else:
``````

By the way, this may be computed more concisely like so (vectorized):

``````SM = self.value.reshape((-1,1))
jac = np.diagflat(self.value) - np.dot(SM, SM.T)
``````

`np.exp` is not stable because it has Inf.
So you should subtract maximum in `x`.

``````def softmax(x):
"""Compute the softmax of vector x."""
exps = np.exp(x - x.max())
return exps / np.sum(exps)
``````

If `x` is matrix, please check the softmax function in this notebook.