We propose a joint sparse sample weighting and parameter estimation (SWAPE) framework that jointly optimizes model parameters and sample weights to train a model while simultaneously assigning an individual weight to each training data point. Our framework accommodates several widely used loss functions for the model parameters, including convex, weakly convex, and difference-of-convex loss, and weakly convex sparsity functions to control the complexity of the sample weights. To solve the resulting bivariate problem, we propose an alternating minimization method in which the subproblems for updating the model parameters and weights are each approximated using majorize-minimize techniques. We provide several approximation methods for the loss functions considered in this paper, and show the overall algorithm's convergence. Numerical experiments on various learning tasks and data anomaly settings demonstrate the effectiveness of the proposed framework in minimizing the impact of undesirable data points compared to benchmark methods.

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