Focal loss implementation

Question

In the paper introducing focal loss, they state that the loss function is formulated as such:

Where

I found an implementation of it on a Github page from another author who used it in their paper . I tried the function out on a segmentation problem dataset I have and it seems to work quite well.

Below is the implementation:

def binary_focal_loss(pred, truth, gamma=2., alpha=.25):
    eps = 1e-8
    pred = nn.Softmax(1)(pred)
    truth = F.one_hot(truth, num_classes = pred.shape[1]).permute(0,3,1,2).contiguous()

    pt_1 = torch.where(truth == 1, pred, torch.ones_like(pred))
    pt_0 = torch.where(truth == 0, pred, torch.zeros_like(pred))

    pt_1 = torch.clamp(pt_1, eps, 1. - eps)
    pt_0 = torch.clamp(pt_0, eps, 1. - eps)

    out1 = -torch.mean(alpha * torch.pow(1. - pt_1, gamma) * torch.log(pt_1)) 
    out0 = -torch.mean((1 - alpha) * torch.pow(pt_0, gamma) * torch.log(1. - pt_0))

    return out1 + out0

The part that I don't understand is the calculation of pt_0 and pt_1. I created a small example for myself to try and figure it out but it still confuses me a bit.

# one hot encoded prediction tensor
pred = torch.tensor([
                     [
                      [.2, .7, .8], # probability
                      [.3, .5, .7], # of
                      [.2, .6, .5]  # background class
                     ], 
                     [
                      [.8, .3, .2], # probability
                      [.7, .5, .3], # of
                      [.8, .4, .5]  # class 1
                     ]
                    ])

# one-hot encoded ground truth labels
truth = torch.tensor([
                      [1, 0, 0], 
                      [1, 1, 0], 
                      [1, 0, 0]
                     ])
truth = F.one_hot(truth, num_classes = 2).permute(2,0,1).contiguous()

print(truth)
# gives me:
# tensor([
#         [
#          [0, 1, 1],
#          [0, 0, 1],
#          [0, 1, 1]
#         ],
#         [
#          [1, 0, 0],
#          [1, 1, 0],
#          [1, 0, 0]
#         ]
#       ])

pt_0 = torch.where(truth == 0, pred, torch.zeros_like(pred))
pt_1 = torch.where(truth == 1, pred, torch.ones_like(pred))

print(pt_0)
# gives me:
# tensor([[
#         [0.2000, 0.0000, 0.0000],
#         [0.3000, 0.5000, 0.0000],
#         [0.2000, 0.0000, 0.0000]
#         ],
#        [
#         [0.0000, 0.3000, 0.2000],
#         [0.0000, 0.0000, 0.3000],
#         [0.0000, 0.4000, 0.5000]
#        ]
#      ])

print(pt_1)
# gives me:
# tensor([[
#          [1.0000, 0.7000, 0.8000],
#          [1.0000, 1.0000, 0.7000],
#          [1.0000, 0.6000, 0.5000]
#         ],
#         [
#          [0.8000, 1.0000, 1.0000],
#          [0.7000, 0.5000, 1.0000],
#          [0.8000, 1.0000, 1.0000]
#         ]
#       ])

What I don't understand is why in pt_0 we are placing zeros where the torch.where statement is false, and in pt_1 we place ones. From how I understood the paper, I would have thought that instead of placing zeros or ones, you would place 1-p.

Can anyone help explain this to me?

Answer 1

So the part you try to understand is a procedure people usually do when they want zero out the additional calculations that not needed.

Take another look at the formula of pt :

The following code is does exactly this by separate the two condition:

# if y=1
pt_1 = torch.where(truth == 1, pred, torch.ones_like(pred))
# otherwise
pt_0 = torch.where(truth == 0, pred, torch.zeros_like(pred)) 

Where it set to zero in pt_0 and one in pt_1 will result zero in output thus have no effect for contribute loss value, ie:

# Because pow(0., gamma) == 0. and log(1.) == 0.
# out1 == 0. if pt_1 == 1.
out1 = -torch.mean(alpha * torch.pow(1. - pt_1, gamma) * torch.log(pt_1))
# out0 == 0. if pt_0 == 0.
out0 = -torch.mean((1 - alpha) * torch.pow(pt_0, gamma) * torch.log(1. - pt_0))

And the reason for pt_0 to using value of p instead of 1-p is the same reason as your last question, ie:

1 - (1 - p) == 1 - 1 + p == p

So it can later calculate the FL(pt) by:

# -a * pow(1 - (1 - p), gamma )* log(1 - p) == -a * pow(p, gamma )* log(1 - p)
out0 = -torch.mean((1 - alpha) * torch.pow(pt_0, gamma) * torch.log(1. - pt_0))