MediumCalculusca-233

Logistic Hessian-Vector Product

Problem

For the logistic loss L(w) = log(1 + exp(-y * w dot x)) with label y in {+1, -1}, let p = 1 / (1 + exp(y * w dot x)) and c = p * (1 - p).

The Hessian is c * x x^T, so the Hessian-vector product with a vector v is c * (x dot v) * x. Given w, x, y, and v, return that product.

Hint

The Hessian is a rank-one outer product scaled by the sigmoid variance.

case 1

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

expected: [0.25,0.25]

case 2

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

expected: [0.419974341614026,0]

case 3

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

expected: [0,0]

case 4

in: [[0.5,-0.5],[1,1],1,[1,-1]]

expected: [0,0]

Starter codePython

def logistic_hessian_vector_product(w, x, y, v):
    # Return H @ v
    # Your code here
    pass

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