MediumCalculusca-244

Second Derivative of Cross-Entropy

Problem

For the binary cross-entropy as a function of the predicted probability p, L(p) = -y log(p) - (1 - y) log(1 - p) with y in {0, 1}, the second derivative is y / p^2 + (1 - y) / (1 - p)^2.

Given y and p with 0 < p < 1, return L''(p).

Hint

The curvature is positive everywhere, so cross-entropy is convex in p.

Test cases

case 1

in: [1,0.5]

expected: 4

case 2

in: [0,0.5]

expected: 4

case 3

in: [1,0.25]

expected: 16

case 4

in: [0,0.8]

expected: 25

Starter codePython

def second_derivative_cross_entropy(y, p):
    # Your code here
    pass

4 test cases · run your code against them in the editor