arXiv:2610.29381v1 Announce Type: cross Abstract: We propose a novel stochastic cascading bandit setup where probing an ordered shortlist of $K$ out of $L$ arms incurs position-dependent costs, and the learner observes only noisy feedback regarding each arm's true output. We identify a sub-class of noise models for which the optimal shortlist can be found by sorting arms according to a per-arm score: such a score exists for both selection and ordering when the channel has no false positives, and for ordering alone (but not selection) when measurements are costless; however in general neither selection nor ordering is separable once noise and cost are both present. For the case where both selection and ordering scores exist, we provide a UCB-based algorithm with $O((L-K)\log T)$ instance-dependent regret over a time horizon $T$, which matches an algorithm-agnostic lower bound (up to constant factors). For the more challenging scenario where scores don't exist, we develop a new algorithmic template, DP-UCB, that embeds optimism directly inside an exact dynamic program, yielding $O\big(LK\log T\big)$ regret. Finally, we validate our theoretical results through extensive numerical evaluations and simulations over both synthetic and real-world datasets.