Vectorizing across multiple examples 

University of Michigan - Ann Arbor

To compute logistic regression and its gradient descent on $m$ examples, we can use for loop to accumulate errors and derivatives and then average them, but it will take a long time to run on a big data set. So vectorization is a good way to get rid of explicit for loop in your code.
First, stack the $m$ examples horizontally into vectors of $X$ and $Y$, so the shape of $X$ is $(n_x, m)$, where $n_x$ is the number of features, and the shape of $Y$ is (1, m). 
Then, compute the $Z$ and $A$,
$$[z^{(1)}z^{(2)}...z^{(m)}]=Z=w^TX+b$$
$$=[(w^Tx{(1)}+b)(w^Tx{(2)}+b)...(w^Tx{(m)}+b)]$$
$$[a{(1)} a{(2)} ... a{(m)}] = A=\sigma (Z)$$
, the shape of $Z$ and $A$ should be $(1, m)$.
The derivatives of $L$ with respect to $Z$, $w$, and $b$ are
$$\frac{d\mathcal L}{dz} = A - Y$$
$$\frac{d\mathcal L}{dw} = \frac{1}{m} X {(\frac{d\mathcal L}{dz})}^T$$
$$\frac{d\mathcal L}{db} = \frac{1}{m} \sum \frac{d\mathcal L}{dz}$$

Vectorizing logistic regression on *m* examples

https://www.coursera.org/learn/neural-networks-deep-learning?specialization=deep-learning

Neural Networks and Deep Learning

```python
for iter in range(10000): # Iterations of Gradient Descent
   Z = np.dot(w.T, X) + b   # Broadcasting
   A = sigma(Z)
   dLdZ = A - Y
   dLdW = (1/m) * X * dLdZ.T
   dLdb = (1/m) * np.sum(dLdZ)
   w = w - a * dLdW
   b = b - a * dLdb
```

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