Apologies, I didn’t include the code – here it is:
# run n_iter iterations of training
    for t in range(n_iterations):

        # get a single batch
        for (it, batch) in enumerate(data_loader):
            # get the X and the targets values
            X = batch[0]
            ts = batch[1]
            if torch.cuda.is_available():
                X = X.cuda()
                ts = ts.cuda()

            # evaluate each task loss L_i(t)
            task_loss = model(X, ts) # this will do a forward pass in the model and will also evaluate the loss
            # compute the weighted loss w_i(t) * L_i(t)
            weighted_task_loss = torch.mul(model.weights, task_loss)
            # initialize the initial loss L(0) if t=0
            if t == 0:
                # set L(0)
                if torch.cuda.is_available():
                    initial_task_loss = task_loss.data.cpu()
                else:
                    initial_task_loss = task_loss.data
                initial_task_loss = initial_task_loss.numpy()

            # get the total loss
            loss = torch.sum(weighted_task_loss)
            # clear the gradients
            optimizer.zero_grad()
            # do the backward pass to compute the gradients for the whole set of weights
            # This is equivalent to compute each \nabla_W L_i(t)
            loss.backward(retain_graph=True)

            # set the gradients of w_i(t) to zero because these gradients have to be updated using the GradNorm loss
            #print('Before turning to 0: {}'.format(model.weights.grad))
            model.weights.grad.data = model.weights.grad.data * 0.0
            #print('Turning to 0: {}'.format(model.weights.grad))


            # switch for each weighting algorithm:
            # --> grad norm
            if args.mode == 'grad_norm':
                
                # get layer of shared weights
                W = model.get_last_shared_layer()

                # get the gradient norms for each of the tasks
                # G^{(i)}_w(t) 
                norms = []
                for i in range(len(task_loss)):
                    # get the gradient of this task loss with respect to the shared parameters
                    gygw = torch.autograd.grad(task_loss[i], W.parameters(), retain_graph=True)
                    # compute the norm
                    norms.append(torch.norm(torch.mul(model.weights[i], gygw[0])))
                norms = torch.stack(norms)
                #print('G_w(t): {}'.format(norms))


                # compute the inverse training rate r_i(t) 
                # \curl{L}_i 
                if torch.cuda.is_available():
                    loss_ratio = task_loss.data.cpu().numpy() / initial_task_loss
                else:
                    loss_ratio = task_loss.data.numpy() / initial_task_loss
                # r_i(t)
                inverse_train_rate = loss_ratio / np.mean(loss_ratio)
                #print('r_i(t): {}'.format(inverse_train_rate))


                # compute the mean norm \tilde{G}_w(t) 
                if torch.cuda.is_available():
                    mean_norm = np.mean(norms.data.cpu().numpy())
                else:
                    mean_norm = np.mean(norms.data.numpy())
                #print('tilde G_w(t): {}'.format(mean_norm))


                # compute the GradNorm loss 
                # this term has to remain constant
                constant_term = torch.tensor(mean_norm * (inverse_train_rate ** args.alpha), requires_grad=False)
                if torch.cuda.is_available():
                    constant_term = constant_term.cuda()
                #print('Constant term: {}'.format(constant_term))
                # this is the GradNorm loss itself
                grad_norm_loss = torch.tensor(torch.sum(torch.abs(norms - constant_term)))
                #print('GradNorm loss {}'.format(grad_norm_loss))

                # compute the gradient for the weights
                model.weights.grad = torch.autograd.grad(grad_norm_loss, model.weights)[0]

            # do a step with the optimizer
            optimizer.step()