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There has been a surge of interest in developing robust estimators for modelswith heavy-tailed and bounded variance data in statistics and machine learning,while few works impose unbounded variance. This paper proposes two type ofrobust estimators, the ridge log-truncated M-estimator and the elastic netlog-truncated M-estimator. The first estimator is applied to convex regressionssuch as quantile regression and generalized linear models, while the other oneis applied to high dimensional non-convex learning problems such as regressionsvia  deep neural networks. Simulations and real data analysis demonstrate the{robustness} of log-truncated estimations over standard estimations.

Non-Asymptotic Guarantees for Robust Statistical Learning under Infinite Variance Assumption