In [1]:
import numpy as np


# $sigmoid(x) = \frac{1}{1+e^{(-x)}}$¶

In [2]:
def sigmoid(x):
"""
Compute the sigmoid function for the input here.

Arguments:
x -- A scalar or numpy array.

Return:
s -- sigmoid(x)
"""

s = 1 / (1 + np.exp(-x))

return s


• $\sigma(-x) = \frac{1}{1 + e^{x}} = \frac{e^x+1}{e^x + 1} - \frac{e^x}{e^x + 1}=1-\sigma(x)$

• $\sigma' = \sigma(x) \times (1 - \sigma(x))$

In [3]:
def sigmoid_grad(s):
"""
Compute the gradient for the sigmoid function here. Note that
for this implementation, the input s should be the sigmoid
function value of your original input x.

Arguments:
s -- A scalar or numpy array.

Return:
"""

ds = s * (1-s)

return ds

In [4]:
def test_sigmoid_basic():
"""
Some simple tests to get you started.
Warning: these are not exhaustive.
"""
print("Running basic tests...")
x = np.array([[1, 2], [-1, -2]])
f = sigmoid(x)
print(f)
f_ans = np.array([
[0.73105858, 0.88079708],
[0.26894142, 0.11920292]])
assert np.allclose(f, f_ans, rtol=1e-05, atol=1e-06)
print(g)
g_ans = np.array([
[0.19661193, 0.10499359],
[0.19661193, 0.10499359]])
assert np.allclose(g, g_ans, rtol=1e-05, atol=1e-06)
print("You should verify these results by hand!\n")

if __name__ == "__main__":
test_sigmoid_basic();

Running basic tests...
[[0.73105858 0.88079708]
[0.26894142 0.11920292]]
[[0.19661193 0.10499359]
[0.19661193 0.10499359]]
You should verify these results by hand!