Cascading Bandits with Noisy Observations: Algorithms and Regret Bounds

The classic multi-armed bandit (MAB) problem tackles the challenge of accruing maximum reward while making decisions under uncertainty. However, in applications, often the goal is to minimize cost subject to a constraint on the minimum permissible reward, an objective captured by multi-armed bandits with cost-subsidy (MAB-CS). Of interest to this paper is the setting where the quality (reward) constraint is specified relative to the unknown best reward and the cost of each arm is known. We characterize the expected sub-optimal samples required by any policy by proving instance-dependent lower bounds that offer new insight into the problem and are a strict generalization of prior bounds. Then, we propose an algorithm called Cost-Ordered Feasibility (COF) that leverages our insight and intelligently combine samples from all arms to gauge the feasibility of a cheap arm. Thereafter, we analyze COF to establish instance-dependent upper bounds on its expected cumulative cost and quality regret, i.e., relative to the cheapest feasible arm. Finally, we empirically validate the merits of COF, comparing it to baselines from the literature through extensive simulation experiments on the MovieLens and Goodreads datasets as well as representative synthetic instances. Not only does our paper develop qualitatively better theoretical regret upper bounds, but COF also convincingly demonstrates improved empirical performance.

This paper studies the problem of regret minimization in Markovian bandits with non-observable states and possibly constrained decision epochs. The focus is restricted to a ``pure'' regret benchmark, that compares the performance of the learning algorithm to the best pure policy which -- akin to optimal policies of stochastic bandits -- picks the optimal arm from start to finish without ever switching. We introduce a generalization of rested Markovian bandits, self-degrading Markovian bandits, for which pure policies are always asymptotically optimal.We show that without prior knowledge on the underlying bandit, the regret of algorithms that switch arms rarely necessarily scales super-logarithmically for every bandit, i.e., as $ω(\log(T))$, where $T$ is the learning horizon. Despite the unreachability of the logarithmic regime, we design UCB-NOM, an optimistic algorithm inspired by UCB, of which the regret is nearly logarithmic. Lastly, we show that given prior knowledge on the Markovian bandit in the form of a bound on the bias functions of its arm, a proper instantiation of UCB-NOM achieves $O(\log(T))$ regret. We further show that this prior knowledge allows for a $O(\sqrt{T \log(T)})$ worst-case regret bound for UCB-NOM. Notably, our regret bounds do not depend on the number of states of the underlying Markov chains. Our findings suggest that the non-observability of states is a mild inconvenience in self-degrading Markovian bandits.

In the improving multi-armed bandits problem, each of $k$ arms has an unknown nondecreasing, discretely concave reward curve $f_i$, and pulling arm $i$ for the $t$-th time yields $f_i(t)$. For sufficiently long horizons, Blum and Ravichandran (ALT 2025) proved that randomized algorithms achieve an $O(\sqrt k)$ approximation to the best single arm when the scale $m=f^*(T)$ of the optimal arm is known ($T\ge2k$), and $O(\sqrt k\log k)$ when it is not ($T>4k$), against an $Ω(\sqrt k)$ lower bound. The logarithmic factor is unnecessary: a one-page probe-and-commit algorithm achieves competitive ratio $4\sqrt3\,\sqrt k$ for $T\ge2\lfloor\sqrt k\rfloor$, without any knowledge of the scale, and we determine the optimal ratio for every horizon, $Θ(\sqrt k+k/T)$, also for unknown horizons. Without noise, no prior is needed at all: a random-marginal probing algorithm reading neither the scale $m$, nor the concavity-envelope exponent $β$ of Blum, Garicano, Ravichandran and Sharma (UAI 2026), nor the horizon $T$, achieves the optimal $Θ(k^{β/(1+β)}+k/T)$ simultaneously for every $β$ and every horizon. Under the multiplicative noise model of Blum and Ravichandran, probe-and-commit keeps the same all-horizon order $Θ(\sqrt k+k/T)$ without knowing the noise level (and $Θ(\sqrt k)$ on the same range), but the price of priors jumps: for any fixed noise level $\varepsilon\in(0,1/2]$, the uniform price of adaptation $φ_\varepsilon(k)$ --- the worst case over horizons $T\ge16k$ of the loss relative to $k^{β/(1+β)}$ for algorithms knowing neither $m$ nor $β$ --- is $Θ_\varepsilon(\sqrt{\log k/\log\log k})$, the lower bound asymptotic in $k$ at fixed positive $\varepsilon$ and matched by a nested random-permutation probing algorithm, whereas knowing either $m$ or $β$ alone restores a constant price.