Improve GMARE metric computation formula

Months ago I tried to improve GMARE to be compatible with DDP, but I had a very strange issue, and the distributed tests were failing, I thought I did something wrong (even I updated +10 metrics for DDP then) so I ignored it at that time, today I was trying to update it and send the PR but I had the same issue, the tests distributed tests are failing.

distributed_context_single_node_gloo = {'local_rank': 1, 'rank': 1, 'world_size': 2}

    @pytest.mark.distributed
    @pytest.mark.skipif(not idist.has_native_dist_support, reason="Skip if no native dist support")
    def test_distrib_gloo_cpu_or_gpu(distributed_context_single_node_gloo):
    
        device = idist.device()
>       _test_distrib_compute(device)


metric_device = device(type='cpu')

    def _test(metric_device):
        metric_device = torch.device(metric_device)
        m = GeometricMeanRelativeAbsoluteError(device=metric_device)
        torch.manual_seed(10 + rank)
    
        y_pred = torch.rand(size=(100,), device=device)
        y = torch.rand(size=(100,), device=device)
    
        m.update((y_pred, y))
    
        y_pred = idist.all_gather(y_pred)
        y = idist.all_gather(y)
    
        np_y = y.cpu().numpy()
        np_y_pred = y_pred.cpu().numpy()
    
        np_gmrae = np.exp(np.log(np.abs(np_y - np_y_pred) / np.abs(np_y - np_y.mean())).mean())
    
        # sum_y = 0
        # num_examples = 0
        # sum_of_errors = 0
        # np_gmrae = 0
        # sum_y += np_y.sum()
        # num_examples += np_y.shape[0]
        # y_mean = sum_y / num_examples
        # numerator = np.abs(y.view_as(y_pred) - y_pred)
        # denominator = np.abs(y.view_as(y_pred) - y_mean)
        # sum_of_errors += np.log(numerator / denominator).sum()
        # np_gmrae += np.exp((sum_of_errors / num_examples).mean())
    
>       assert m.compute() == pytest.approx(np_gmrae)
E       assert 1.3195266723632812 == 1.3320785760879517 ± 1.3e-06
E         +1.3195266723632812
E         -1.3320785760879517 ± 1.3e-06


FAILED tests/ignite/contrib/metrics/regression/test_geometric_mean_relative_absolute_error.py::test_distrib_gloo_cpu_or_gpu - assert 1.3195266723632812 == 1.3320785760879517 ± 1.3e-06
FAILED tests/ignite/contrib/metrics/regression/test_geometric_mean_relative_absolute_error.py::test_distrib_gloo_cpu_or_gpu - assert 1.3195266723632812 == 1.3320785760879517 ± 1.3e-06

And here is the code and the tests https://github.com/pytorch/ignite/compare/master...KickItLikeShika:improve-GMRAE-ddp

[sdesrozis]

Yep I cited this article in my previous post. Nothing solid, I agree.

[KickItLikeShika]

I have been looking for examples or something shows the usage of the metric, but I found nothing solid yet, but this link you sent above somehow clarifies the usage of the metric https://support.numxl.com/hc/en-us/articles/115001223403-GMRAE-Geometric-Mean-Relative-Absolute-Error