We propose Confusions over Time (CoT), a novel generative framework which facilitates a multi-granular analysis of the decision making process. The CoT not only models the confusions or error properties of individual decision makers and their evolution over time, but also allows us to obtain diagnostic insights into the collective decision making process in an interpretable manner.

We used the largest batch size that could fit in memory on our limited hardware, which was 256 for an image size of 224x224. For the learning rate (Adam [2] optimizer) we searched in the range of {0.001, 0.0001, 1e04, 5e-4, 5e-5}, with weight decay {0, 5e-4. We chose a weight decay of 5e-5 and learning rate of 5e-4 until the 4:6 split and 1e-4 afterwards. We chose a prototype dimension of 256, backbone output of 512, 2 graph layers, graph hidden dimension of 512, λh of 10, Clst and Sep of 0.01. UT-Zappos we again used the Adam optimizer, with learning rate in the ranges {5e-5, 5e-4, 5e-3}, and weight decay {0, 5e-4.

We investigate the degree to which our conditional independence assumption is satisfied empirically in the datasets used in the paper. Specifically, of interest is the assumption of conditional independence of m(x) and h(x), given y. Assessing conditional independence is not straightforward given that m(x) is a K-dimensional real-valued vector and h(x) and yeach take one of K categorical values, with K = 10 for CIFAR-10H and K = 16 for ImageNet-16H. While there exist statistical tests for assessing conditional independence for categorical random variables, with real-valued variables the situation is less straightforward and there are multiple options such as different non-parametric tests involving different tradeoffs [Runge, 2018, Marx and Vreeken, 2019, Mukherjee et al., 2020, Berrett et al., 2020]. Given these issues we investigate the degree of conditional dependence using two relatively simple approaches. The first approach looks at the conditional mutual information (CMI) between the predicted label from the model and the predicted label from the human, conditioned on the true label.

The generation of label noise is often modeled as a process involving a probability transition matrix (also interpreted as the) imposed onto the label distribution. Under this model, learning the ``ground-truth classifier''---i.e., the classifier that can be learned if no noise was present---and the confusion matrix boils down to a model identification problem. Prior works along this line demonstrated appealing empirical performance, yet identifiability of the model was mostly established by assuming an instance-invariant confusion matrix. Having an (occasionally) instance-dependent confusion matrix across data samples is apparently more realistic, but inevitably introduces outliers to the model. Our interest lies in confusion matrix-based noisy label learning with such outliers taken into consideration. We begin with pointing out that under the model of interest, using labels produced by only one annotator is fundamentally insufficient to detect the outliers or identify the ground-truth classifier. Then, we prove that by employing a crowdsourcing strategy involving multiple annotators, a carefully designed loss function can establish the desired model identifiability under reasonable conditions. Our development builds upon a link between the noisy label model and a column-corrupted matrix factorization mode---based on which we show that crowdsourced annotations distinguish nominal data and instance-dependent outliers using a low-dimensional subspace. Experiments show that our learning scheme substantially improves outlier detection and the classifier's testing accuracy.

Among them, over-sampling and data augmentation aim to balance the data distribution by oversampling or generating closely related minority classes.