  The generalized smooth condition, $(L_{0},L_{1})$-smoothness, has triggered
people's interest since it is more realistic in many optimization problems
shown by both empirical and theoretical evidence. Two recent works established
the $O(\epsilon^{-3})$ sample complexity to obtain an $O(\epsilon)$-stationary
point. However, both require a large batch size on the order of
$\mathrm{ploy}(\epsilon^{-1})$, which is not only computationally burdensome
but also unsuitable for streaming applications. Additionally, these existing
convergence bounds are established only for the expected rate, which is
inadequate as they do not supply a useful performance guarantee on a single
run. In this work, we solve the prior two problems simultaneously by revisiting
a simple variant of the STORM algorithm. Specifically, under the
$(L_{0},L_{1})$-smoothness and affine-type noises, we establish the first
near-optimal $O(\log(1/(\delta\epsilon))\epsilon^{-3})$ high-probability sample
complexity where $\delta\in(0,1)$ is the failure probability. Besides, for the
same algorithm, we also recover the optimal $O(\epsilon^{-3})$ sample
complexity for the expected convergence with improved dependence on the
problem-dependent parameter. More importantly, our convergence results only
require a constant batch size in contrast to the previous works.
