import numpy as np
def sigmoid(x):
"""
Compute the sigmoid function for the input here.
Arguments:
x -- A scalar or numpy array.
Return:
s -- sigmoid(x)
"""
### YOUR CODE HERE
s = 1 / (1 + np.exp(-x))
### END YOUR CODE
return s
$\sigma(-x) = \frac{1}{1 + e^{x}} = \frac{e^x+1}{e^x + 1} - \frac{e^x}{e^x + 1}=1-\sigma(x)$
$\sigma' = \sigma(x) \times (1 - \sigma(x))$
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 -- Your computed gradient.
"""
### YOUR CODE HERE
ds = s * (1-s)
### END YOUR CODE
return ds
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)
g = sigmoid_grad(f)
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!