  Stochastic second-order methods achieve fast local convergence in strongly
convex optimization by using noisy Hessian estimates to precondition the
gradient. However, these methods typically reach superlinear convergence only
when the stochastic Hessian noise diminishes, increasing per-iteration costs
over time. Recent work in [arXiv:2204.09266] addressed this with a Hessian
averaging scheme that achieves superlinear convergence without higher
per-iteration costs. Nonetheless, the method has slow global convergence,
requiring up to $\tilde{O}(\kappa^2)$ iterations to reach the superlinear rate
of $\tilde{O}((1/t)^{t/2})$, where $\kappa$ is the problem's condition number.
In this paper, we propose a novel stochastic Newton proximal extragradient
method that improves these bounds, achieving a faster global linear rate and
reaching the same fast superlinear rate in $\tilde{O}(\kappa)$ iterations. We
accomplish this by extending the Hybrid Proximal Extragradient (HPE) framework,
achieving fast global and local convergence rates for strongly convex functions
with access to a noisy Hessian oracle.
