MediumCalculusca-200

Optimal Baseline Control Variate

Problem

For the control variate (R - b) * s with score s, the baseline that minimizes the estimator variance is b* = E[R * s^2] / E[s^2].

Given reward samples and score samples, return b* computed as sum(R_i * s_i^2) / sum(s_i^2). Assume the scores are not all zero.

Hint

Minimize the variance as a quadratic in b and solve for b.

case 1

in: [[1,2],[1,1]]

expected: 1.5

case 2

in: [[1,2,3],[1,-1,0]]

expected: 1.5

case 3

in: [[2,4],[0.5,-0.5]]

expected: 3

Starter codePython

def optimal_baseline(rewards, scores):
    # Your code here
    pass

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