  The generalized linear model (GLM) plays a key role in regression analyses.
In high-dimensional data, the sparse GLM has been used but it is not robust
against outliers. Recently, the robust methods have been proposed for the
specific example of the sparse GLM. Among them, we focus on the robust and
sparse linear regression based on the $\gamma$-divergence. The estimator of the
$\gamma$-divergence has strong robustness under heavy contamination. In this
paper, we extend the robust and sparse linear regression based on the
$\gamma$-divergence to the robust and sparse GLM based on the
$\gamma$-divergence with a stochastic optimization approach in order to obtain
the estimate. We adopt the randomized stochastic projected gradient descent as
a stochastic optimization approach and extend the established convergence
property to the classical first-order necessary condition. By virtue of the
stochastic optimization approach, we can efficiently estimate parameters for
very large problems. Particularly, we show the linear regression, logistic
regression and Poisson regression with $L_1$ regularization in detail as
specific examples of robust and sparse GLM. In numerical experiments and real
data analysis, the proposed method outperformed comparative methods.
