The initialisation to compute the gradient descent weight updates for the output units should be wrong:
In the comment: "dy / dw is just w since y = x' * w + b."
This is wrong. dy/dw is x (ignoring the indices). The same initialisation is done in the code.
Check by using neural network terminology:
The gradient machine is a specialized version of a multi layer perceptron (MLP).
In a MLP the gradient for computing the "weight change" for the output units is:
dE / dw_ij = dE / dz_i * dz_i / d_ij with z_i = sum_j (w_ij * a_j)
here: i index of the output layer; j index of the hidden layer
(d stands for the partial derivatives)
here: z_i = a_i (no squashing in the output layer)
with the special loss (cost function) is E = 1 - a_g + a_b = 1 - z_g + z_b
g index of output unit with target value: +1 (positive class)
b: random output unit with target value: 0
dE / dw_gj = -dE/dz_g * dz_g/dw_gj = -1 * a_j (a_j: activity of the hidden unit
dE / dw_bj = -dE/dz_b * dz_b/dw_bj = +1 * a_j (a_j: activity of the hidden unit
That's the same if the comment would be correct:
dy /dw = x (x is here the activation of the hidden unit) * (-1) for weights to
the output unit with target value +1.
In neural network implementations it's common to compute the gradient
numerically for a test of the implementation. This can be done by:
dE/dw_ij = (E(w_ij + epsilon) -E(w_ij - epsilon) ) / (2* (epsilon))