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 provide theoretical guarantees for label consistency in generalized k-means problems, with an emphasis on the overfitted case where the number of clusters used by the algorithm is more than the ground truth. We provide conditions under which the estimated labels are close to a refinement of the true cluster labels. We consider both exact and approximate recovery of the labels. Our results hold for any constant-factor approximation to the k-means problem. The results are also model-free and only based on bounds on the maximum or average distance of the data points to the true cluster centers. These centers themselves are loosely defined and can be taken to be any set of points for which the aforementioned distances can be controlled. We show the usefulness of the results with applications to some manifold clustering problems.

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.

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Learning from noisy data is a fundamental problem in many machine learning applications.

SGD until 100% accuracy is achieved, in four different settings: 1. Random initialization + Training with true labels.

We consider both exact andapproximate recoveryofthelabels. Our results hold for anyconstant-factor approximation tothek-means problem.