Zhilu Zhang, Mert R. Sabuncu. Generalized Cross Entropy Loss for Training Deep Neural Networks with Noisy Labels. NeurIPS 2018.

Zhang and Sabuncu propose a generalized cross entropy loss for robust learning on noisy labels. The approach is based on the work by Gosh et al. [1] showing that the mean absolute error can be robust to label noise. Specifically, they show that a symmetric loss, under specific assumptions on the label noise, is robust. Here, symmetry corresponds to

$\sum_{j=1}^c \mathcal{L}(f(x), j) = C$ for all $x$ and $f$

where $c$ is the number of classes and $C$ some constant. The cross entropy loss is not symmetric, while the mean absolute error is. The mean absolute error however, usually results in slower learning and may reach lower accuracy. As alternative, the authors propose

$\mathcal{L}(f(x), e_j) = \frac{(1 – f_j(x)^q)}{q}$.

Here, $f$ is the classifier which is assumed to contain a softmax layer at the end. For $q \rightarrow 0$ this reduces to the cross entropy and for $q = 1$ it reduces to the mean absolute error. As shown in Figure 1, this loss (or a slightly adapted version, see paper, respectively) may obtain better performance on noisy labels. To this end, the label noise is assumed to be uniform, meaning that $p(\tilde{y} = k|y = j, x)= 1 - \eta$ where $\tilde{y}$ is the perturbed label.

Figure 1: Performance of the proposed loss for different $q$ and noise rate $\eta$ on Cifar-10. A ResNet-34 is used.

[1] Aritra Gosh, Himanshu Kumar, PS Sastry. Robust loss functions under label noise for deep neural networks. AAAI, 2017.

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