Classification algorithms in machine learning often assume a flat label space. However, most real world data have dependencies between the labels, which can often be captured by using a hierarchy. Utilizing this relation can help develop a model capable of satisfying the dependencies and improving model accuracy and interpretability. Further, as different levels in the hierarchy correspond to different granularities, penalizing each label equally can be detrimental to model learning. In this paper, we propose a loss function, hierarchical curriculum loss, with two properties: (i) satisfy hierarchical constraints present in the label space, and (ii) provide non-uniform weights to labels based on their levels in the hierarchy, learned implicitly by the training paradigm. We theoretically show that the proposed loss function is a tighter bound of 0-1 loss compared to any other loss satisfying the hierarchical constraints. We test our loss function on real world image data sets, and show that it significantly substantially outperforms multiple baselines.

Generalization is vital important for many deep network models. It becomes more challenging when high robustness is required for learning with noisy labels. The 0-1 loss has monotonic relationship between empirical adversary (reweighted) risk [8], and it is robust to outliers. However, it is also difficult to optimize. To efficiently optimize 0-1 loss while keeping its robust properties, we propose a very simple and efficient loss, i.e. curriculum loss (CL). Our CL is a tighter upper bound of the 0-1 loss compared with conventional summation based surrogate losses. Moreover, CL can adaptively select samples for training as a curriculum learning. To handle large rate of noisy label corruption, we extend our curriculum loss to a more general form that can automatically prune the estimated noisy samples during training. Experimental results on noisy MNIST, CIFAR10 and CIFAR100 dataset validate the robustness of the proposed loss.