Smooth Approximation of Max

Problem

The log-sum-exp function smooths the maximum:

max_i x_i <= (1/tau) * log(sum exp(tau * x_i)) <= max_i x_i + log(n)/tau

Given values and the temperature parameter tau > 0, return the smooth maximum. Subtract the max before exponentiating for stability.

Hint

As tau grows the approximation approaches the exact maximum.

case 1

in: [[0,0],1]

expected: 0.6931471805599453

case 2

in: [[1,2],1]

expected: 2.313261687518223

case 3

in: [[0],5]

expected: 0

case 4

in: [[0,0],0.5]

expected: 1.3862943611198906

Starter codePython

def smooth_max(values, tau):
    # Your code here
    pass

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