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)
    """

    ### YOUR CODE HERE
    s = 1 / (1 + np.exp(-x))
    ### END YOUR CODE

    return s

sigmoid_grad

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

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

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 -- Your computed gradient.
    """

    ### YOUR CODE HERE
    ds = s * (1-s)
    ### END YOUR CODE

    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)
    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!