  We propose a new family of inexact sequential quadratic approximation (SQA)
methods, which we call the inexact regularized proximal Newton
($\textsf{IRPN}$) method, for minimizing the sum of two closed proper convex
functions, one of which is smooth and the other is possibly non-smooth. Our
proposed method features strong convergence guarantees even when applied to
problems with degenerate solutions while allowing the inner minimization to be
solved inexactly. Specifically, we prove that when the problem possesses the
so-called Luo-Tseng error bound (EB) property, $\textsf{IRPN}$ converges
globally to an optimal solution, and the local convergence rate of the sequence
of iterates generated by $\textsf{IRPN}$ is linear, superlinear, or even
quadratic, depending on the choice of parameters of the algorithm. Prior to
this work, such EB property has been extensively used to establish the linear
convergence of various first-order methods. However, to the best of our
knowledge, this work is the first to use the Luo-Tseng EB property to establish
the superlinear convergence of SQA-type methods for non-smooth convex
minimization. As a consequence of our result, $\textsf{IRPN}$ is capable of
solving regularized regression or classification problems under the
high-dimensional setting with provable convergence guarantees. We compare our
proposed $\textsf{IRPN}$ with several empirically efficient algorithms by
applying them to the $\ell_1$-regularized logistic regression problem.
Experiment results show the competitiveness of our proposed method.
