Ndtyjky

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 w.r.t. 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

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.