Weighted Log-Sum-Exp Hessian Diagonal

Problem

For f(w) = log(sum_i exp(a_i w_i)), the diagonal of the Hessian with respect to w is p_i (1 - p_i) a_i^2, where p is the softmax of a_i w_i.

Given a and w, return the diagonal list.

Hint

Differentiate each softmax entry once more and apply the chain rule factor a_i^2.

case 1

in: [[1,1],[0,0]]

expected: [0.25,0.25]

case 2

in: [[2,0.5],[1,2]]

expected: [0.7864477329659274,0.04915298331037046]

case 3

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

expected: [0.08192506906499324,0.1848364465099787,0.2226954265346234]

Starter codePython

def weighted_logsumexp_hessian_diag(a, w):
    # Your code here
    pass

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