I have set the two loss functions as:
criterion_label = nn.CrossEntropyLoss()
criterion_bbox = nn.MSELoss()

The training epoch is set as follows:
for epoch in range(1):

    running_loss = 0.0
    for i, data in enumerate(trainloader, 0):
        images, labels, bbox = data
        images = Variable(images)
        labels = Variable(labels)
        bbox = Variable(bbox).float()
        optimizer.zero_grad()
        outputs = net(images)
        loss_label= criterion_label(outputs, labels)
        loss_bbox = criterion_bbox(outputs, bbox)
        loss = loss_label + loss_bbox
        loss.backward()
        optimizer.step()
        
        running_loss += loss.item()
        print('[%d, %2d] loss: %.3f' % (epoch + 1, i + 1, running_loss))
        running_loss = 0.0

However, this is the output that I get:
[1,  1] loss: nan
[1,  2] loss: nan
[1,  3] loss: nan
[1,  4] loss: nan
[1,  5] loss: nan
[1,  6] loss: nan
[1,  7] loss: nan
[1,  8] loss: nan
[1,  9] loss: nan
[1, 10] loss: nan
....
...
..

I’m not sure if this is what you meant.