Expected Improvement

Problem

Compute the expected improvement acquisition for minimization.

With z = (best - mu) / sigma, return (best - mu) * Phi(z) + sigma * phi(z), where Phi is the standard normal CDF and phi its PDF. When sigma <= 0 return max(0, best - mu).

Hint

The improvement is max(0, best - y), and its expectation has a closed form.

Test cases

case 1

in: [0,1,1]

expected: 1.0833154705876864

case 2

in: [2,0.5,1]

expected: 0.004245351308414823

case 3

in: [1,0,0.5]

expected: 0

case 4

in: [3,2,1]

expected: 0.16663094117537258

Starter codePython

def expected_improvement(mu, sigma, best):
    # Your code here
    pass

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