  STOchastic Recursive Momentum (STORM)-based algorithms have been widely
developed to solve one to $K$-level ($K \geq 3$) stochastic optimization
problems. Specifically, they use estimators to mitigate the biased gradient
issue and achieve near-optimal convergence results. However, there is
relatively little work on understanding their generalization performance,
particularly evident during the transition from one to $K$-level optimization
contexts. This paper provides a comprehensive generalization analysis of three
representative STORM-based algorithms: STORM, COVER, and SVMR, for one, two,
and $K$-level stochastic optimizations under both convex and strongly convex
settings based on algorithmic stability. Firstly, we define stability for
$K$-level optimizations and link it to generalization. Then, we detail the
stability results for three prominent STORM-based algorithms. Finally, we
derive their excess risk bounds by balancing stability results with
optimization errors. Our theoretical results provide strong evidence to
complete STORM-based algorithms: (1) Each estimator may decrease their
stability due to variance with its estimation target. (2) Every additional
level might escalate the generalization error, influenced by the stability and
the variance between its cumulative stochastic gradient and the true gradient.
(3) Increasing the batch size for the initial computation of estimators
presents a favorable trade-off, enhancing the generalization performance.
