Softmax Regression from Scratch

[mli]

http://d2l.ai/chapter_linear-networks/softmax-regression-scratch.html

[gpk2000]

The cross entropy loss is given by the below formula according to this

But in the code below is the same thing happening?

def cross_entropy(y_hat, y):
    return - np.log(y_hat[range(len(y_hat)), y])

Am I missing something?

[StevenJokes]

@gpk2000
You can test by your own next time…
Just try to input some numbers to see the outputs…

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And use the search button as possible.

http://preview.d2l.ai/d2l-en/master/chapter_appendix-mathematics-for-deep-learning/information-theory.html?highlight=cross%20entropy%20loss

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.mean() is for n*1 matrix?
@goldpiggy

[osamaGkhafagy]

What this snippet of code is doing is exactly the same as the formula above. This code uses the index of the true y to fetch the predicted value y_hat and then taking the log to those predicted values for all examples in a minibatch

[katduecker]

I am also a bit confused about that. I get that we select the y_hat for each y, in train_epoch_ch3 we sum over the cross-entropy loss, but where do we multiply each y with it’s corresponding y_hat as per the equation?

[goldpiggy]

Hi @katduecker, great question! I guess you were referring to the “loss” function. Here we usedcross entropy loss rather than simply multiplying y and y_hat in the function cross_entropy.

[Nisar_Khan]

Hi,
I think there is a typo in the sentence : “Before looking at the code, let us recall how this looks expressed as an equation:”
It should be : “Before looking at the code, let us recall how this looks when expressed as an equation:”
Thank you.
Nisar

[gudzpoz]

I might be able to explain the derivation of the cross_entropy function a little bit.
Indeed, 𝓁(y,ŷ)=-Σy_j log(ŷ_j).
However, we see that y is in reality (0, … 0, 1, 0, …, 0) so we have:
𝓁(y,ŷ)=-Σy_j log(ŷ_j)=-y_i log(ŷ_i), where y_i=1,
as all other terms are just zero.
Since y_i=1, we can further simplify this into:
𝓁(y,ŷ)=-y_i log(ŷ_i)=-log(ŷ_i), where y_i=1.

def cross_entropy(y_hat, y):
    return - np.log(y_hat[range(len(y_hat)), y])

In the code, y is (i⁽¹⁾, i⁽²⁾, …, i⁽ⁿ⁾)
and y_hat[range(len(y_hat)), y] is just (ŷ⁽¹⁾_i⁽¹⁾, ŷ⁽²⁾_i⁽²⁾, …, ŷ⁽ⁿ⁾_i⁽ⁿ⁾)
with -np.log getting the logarithm of them all at once.