Conformal-Style Quantile Analyses for Stochastic Bandits

We study the tail behavior of regret in stochastic multi-armed bandits for algorithms that are asymptotically optimal in expectation. While minimizing expected regret is the classical objective, recent work shows that even such algorithms can exhibit heavy regret tails, incurring large regret with non-negligible probability. Existing sharp characterizations of regret tails are largely restricted to parametric settings, such as single-parameter exponential families. In this work, we extend the $\KLinf$-UCB algorithm of to a broad nonparametric class of reward distributions satisfying mild assumptions, and establish its asymptotic optimality in expectation. We then analyze the tail behavior of its regret and derive a novel upper bound on the regret tail probability. As special cases, our results recover regret-tail guarantees for both bounded-support and heavy-tailed (moment-bounded) bandit models. Moreover, for the special case of finitely-supported reward distributions, our upper bound matches the known lower bound exactly. Our results thus provide a unified and tight characterization of regret tails for asymptotically optimal KL-based UCB algorithms, going beyond parametric models.

Uncertainty quantification is crucial in safety-critical systems, where decisions must be made under uncertainty. In particular, we consider the problem of online uncertainty quantification, where data points arrive sequentially. Online conformal prediction is a principled online uncertainty quantification method that dynamically constructs a prediction set at each time step. While existing methods for online conformal prediction provide long-run coverage guarantees without any distributional assumptions, they typically assume a full feedback setting in which the true label is always observed. In this paper, we propose a novel learning method for online conformal prediction with partial feedback from an adaptive adversary-a more challenging setup where the true label is revealed only when it lies inside the constructed prediction set. Specifically, we formulate online conformal prediction as an adversarial bandit problem by treating each candidate prediction set as an arm. Building on an existing algorithm for adversarial bandits, our method achieves a long-run coverage guarantee by explicitly establishing its connection to the regret of the learner. Finally, we empirically demonstrate the effectiveness of our method in both independent and identically distributed (i.i.d.) and non-i.i.d. settings, showing that it successfully controls the miscoverage rate while maintaining a reasonable size of the prediction set.

We study replicable algorithms for stochastic multi-armed bandits (MAB) and linear bandits with UCB (Upper Confidence Bound) based exploration. A bandit algorithm is $ρ$-replicable if two executions using shared internal randomness but independent reward realizations produce the same action sequence with probability at least $1-ρ$. Prior approaches to this problem are elimination-based and, in linear bandits with infinitely many actions, rely on discretization, leading to suboptimal dependence on the dimension $d$ and $ρ$. We develop optimistic alternatives for both settings. For stochastic multi-armed bandits, we propose RepUCB, a replicable batched UCB algorithm and show that it attains a regret $O\!\left(\frac{K^2\log^2 T}{ρ^2}\sum_{a:Δ_a>0}\left(Δ_a+\frac{\log(KT\log T)}{Δ_a}\right)\right)$. For stochastic linear bandits, we first introduce RepRidge, a replicable ridge regression estimator that satisfies both a confidence guarantee and a $ρ$-replicability guarantee. Beyond its role in our bandit algorithm, this may also be of independent interest in other statistical estimation settings. We then use RepRidge to design RepLinUCB, a replicable optimistic algorithm for stochastic linear bandits, and show that its regret is bounded by $\widetilde{O}\!\big(\big(d+\frac{d^3}ρ\big)\sqrt{T}\big)$. This improves the best prior regret guarantee by a factor of $O(d/ρ)$, showing that our optimistic algorithm can substantially reduce the price of replicability. This is the first linear-bandit algorithm with an optimal dependence on $ρ$ for large number of arms. Finally, we extend our framework to stochastic generalized linear bandits by developing RepGLM, a replicable penalized GLM estimator, and RepGLMUCB, a replicable optimistic algorithm for this setting.