Hinge loss function gradient w.r.t. input prediction

Question

For an assignment I have to implement both the Hinge loss and its partial derivative calculation functions. I got the Hinge loss function itself but I'm having hard time understanding how to calculate its partial derivative wrt prediction input. I tried different approaches but none worked.

Any help, hints, suggestions will be much appreciated!

Here is the analytical expression for Hinge loss function itself:

And here is my Hinge loss function implementation:

def hinge_forward(target_pred, target_true):
    """Compute the value of Hinge loss 
        for a given prediction and the ground truth
    # Arguments
        target_pred: predictions - np.array of size `(n_objects,)`
        target_true: ground truth - np.array of size `(n_objects,)`
    # Output
        the value of Hinge loss 
        for a given prediction and the ground truth
        scalar
    """
    output = np.sum((np.maximum(0, 1 - target_pred * target_true)) / target_pred.size)

    return output

Now I need to calculate this gradient:

This is what I tried for the Hinge loss gradient calculation:

def hinge_grad_input(target_pred, target_true):
    """Compute the partial derivative 
        of Hinge loss with respect to its input
    # Arguments
        target_pred: predictions - np.array of size `(n_objects,)`
        target_true: ground truth - np.array of size `(n_objects,)`
    # Output
        the partial derivative 
        of Hinge loss with respect to its input
        np.array of size `(n_objects,)`
    """
# ----------------
#     try 1
# ----------------
#     hinge_result = hinge_forward(target_pred, target_true)

#     if hinge_result == 0:
#         grad_input = 0
#     else:
#         hinge = np.maximum(0, 1 - target_pred * target_true)
#         grad_input = np.zeros_like(hinge)
#         grad_input[hinge > 0] = 1
#         grad_input = np.sum(np.where(hinge > 0))
# ----------------
#     try 2
# ----------------
#     hinge = np.maximum(0, 1 - target_pred * target_true)
#     grad_input = np.zeros_like(hinge)

#     grad_input[hinge > 0] = 1
# ----------------
#     try 3
# ----------------
    hinge_result = hinge_forward(target_pred, target_true)

    if hinge_result == 0:
        grad_input = 0
    else:
        loss = np.maximum(0, 1 - target_pred * target_true)
        grad_input = np.zeros_like(loss)
        grad_input[loss > 0] = 1
        grad_input = np.sum(grad_input) * target_pred

    return grad_input

Answer 1

I've managed to solve this by using np.where() function. Here is the code:

def hinge_grad_input(target_pred, target_true):
    """Compute the partial derivative 
        of Hinge loss with respect to its input
    # Arguments
        target_pred: predictions - np.array of size `(n_objects,)`
        target_true: ground truth - np.array of size `(n_objects,)`
    # Output
        the partial derivative 
        of Hinge loss with respect to its input
        np.array of size `(n_objects,)`
    """
    grad_input = np.where(target_pred * target_true < 1, -target_true / target_pred.size, 0)

    return grad_input

Basically the gradient equals -y/N for all the cases where y*y < 1, otherwise 0.