  Regret minimization in streaming multi-armed bandits (MABs) has been studied
extensively in recent years. In the single-pass setting with $K$ arms and $T$
trials, a regret lower bound of $\Omega(T^{2/3})$ has been proved for any
algorithm with $o(K)$ memory (Maiti et al. [NeurIPS'21]; Agarwal at al.
[COLT'22]). On the other hand, however, the previous best regret upper bound is
still $O(K^{1/3} T^{2/3}\log^{1/3}(T))$, which is achieved by the streaming
implementation of the simple uniform exploration. The $O(K^{1/3}\log^{1/3}(T))$
gap leaves the open question of the tight regret bound in the single-pass MABs
with sublinear arm memory.
  In this paper, we answer this open problem and complete the picture of regret
minimization in single-pass streaming MABs. We first improve the regret lower
bound to $\Omega(K^{1/3}T^{2/3})$ for algorithms with $o(K)$ memory, which
matches the uniform exploration regret up to a logarithm factor in $T$. We then
show that the $\log^{1/3}(T)$ factor is not necessary, and we can achieve
$O(K^{1/3}T^{2/3})$ regret by finding an $\varepsilon$-best arm and committing
to it in the rest of the trials. For regret minimization with high constant
probability, we can apply the single-memory $\varepsilon$-best arm algorithms
in Jin et al. [ICML'21] to obtain the optimal bound. Furthermore, for the
expected regret minimization, we design an algorithm with a single-arm memory
that achieves $O(K^{1/3} T^{2/3}\log(K))$ regret, and an algorithm with
$O(\log^{*}(n))$-memory with the optimal $O(K^{1/3} T^{2/3})$ regret following
the $\varepsilon$-best arm algorithm in Assadi and Wang [STOC'20].
  We further tested the empirical performances of our algorithms. The
simulation results show that the proposed algorithms consistently outperform
the benchmark uniform exploration algorithm by a large margin, and on occasion,
reduce the regret by up to 70%.
