  Higher-Order Influence Functions (HOIFs) provide a unified theory for
constructing rate-optimal estimators for a large class of low-dimensional
(smooth) statistical functionals/parameters (and sometimes even
infinite-dimensional functions) that arise in substantive fields including
epidemiology, economics, and the social sciences. Since the introduction of
HOIFs by Robins et al. (2008), they have been viewed mostly as a theoretical
benchmark rather than a useful tool for statistical practice. Works aimed to
flip the script are scant, but a few recent papers Liu et al. (2017, 2021b)
make some partial progress. In this paper, we take a fresh attempt at achieving
this goal by constructing new, numerically stable HOIF estimators (or sHOIF
estimators for short with ``s'' standing for ``stable'') with provable
statistical, numerical, and computational guarantees. This new class of sHOIF
estimators (up to the 2nd order) was foreshadowed in synthetic experiments
conducted by Liu et al. (2020a).
