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