Conformal prediction provides a principled framework for uncertainty quantification with finite-sample coverage guarantees. While recent work has extended conformal prediction to online and sequential settings, existing methods typically focus on a single coverage level and do not ensure consistency across multiple confidence levels. In many real-world applications, such as weather forecasting, macroeconomic prediction, and risk management, different users operate under heterogeneous risk tolerances and require calibrated uncertainty estimates across a range of coverage levels. In such settings, it is desirable to produce prediction sets corresponding to different coverage levels that are nested and valid simultaneously. In this paper, we propose two novel online conformal prediction methods that output \emph{nested prediction sets} across a range of coverage levels, enabling simultaneous uncertainty quantification across the entire risk spectrum. Beyond interpretability, jointly estimating multiple coverage levels is known to improve statistical efficiency in classical quantile regression by enforcing non-crossing constraints and sharing information across quantiles. Our approaches leverage an online optimization perspective with small regret that translates to quantile estimation error control while enforcing nestedness of prediction sets. Empirical results on synthetic and real-world datasets, including applications in forecasting tasks with heterogeneous risk requirements, demonstrate that our method achieves stable coverage across all levels, strictly nested prediction sets, and improved efficiency compared to existing online conformal baselines.

Full (exact) conformal set vs. split or cross-validated conformal set Non-connectedness of the conformal prediction set. This was initially suggested in [18, Remark 1]. We follow the actual practice in the literature [14, Remark 5]. We did not observe violations. We will also summarize the proposed algorithm in a direct pseudo-code.

Distributional reinforcement learning (DRL) has achieved empirical success in various domains.One of the core tasks in the field of DRL is distributional policy evaluation, which involves estimating the return distribution $\eta^\pi$ for a given policy $\pi$.The distributional temporal difference learning has been accordingly proposed, whichis an extension of the temporal difference learning (TD) in the classic RL area.In the tabular case, Rowland et al. [2018] and Rowland et al. [2023] proved the asymptotic convergence of two instances of distributional TD, namely categorical temporal difference learning (CTD) and quantile temporal difference learning (QTD), respectively.In this paper, we go a step further and analyze the finite-sample performance of distributional TD.To facilitate theoretical analysis, we propose a non-parametric distributional TD learning (NTD).For a $\gamma$-discounted infinite-horizon tabular Markov decision process,we show that for NTD we need $\widetilde O\left(\frac{1}{\varepsilon^{2p}(1-\gamma)^{2p+1}}\right)$ iterations to achieve an $\varepsilon$-optimal estimator with high probability, when the estimation error is measured by the $p$-Wasserstein distance.This sample complexity bound is minimax optimal (up to logarithmic factors) in the case of the $1$-Wasserstein distance.To achieve this, we establish a novel Freedman's inequality in Hilbert spaces, which would be of independent interest.In addition, we revisit CTD, showing that the same non-asymptotic convergence bounds hold for CTD in the case of the $p$-Wasserstein distance.

We study reward-free reinforcement learning (RL) under general non-linear function approximation, and establish sample efficiency and hardness results under various standard structural assumptions. On the positive side, we propose the RFOLIVE (Reward-Free OLIVE) algorithm for sample-efficient reward-free exploration under minimal structural assumptions, which covers the previously studied settings of linear MDPs (Jin et al., 2020b), linear completeness (Zanette et al., 2020b) and low-rank MDPs with unknown representation (Modi et al., 2021). Our analyses indicate that the explorability or reachability assumptions, previously made for the latter two settings, are not necessary statistically for reward-free exploration. On the negative side, we provide a statistical hardness result for both reward-free and reward-aware exploration under linear completeness assumptions when the underlying features are unknown, showing an exponential separation between low-rank and linear completeness settings.

We thank the reviewers for their feedback. Below we respond to some of the main concerns. R1 and R3 were dissatisfied with the small scale of our experiments. B that comes with a model-free guarantee. There were two main reasons for running experiments on a small scale with a couple of data sets.

We study the generalized linear bandit (GLB) problem, a contextual multi-armed bandit framework that extends the classical linear model by incorporating a non-linear link function, thereby modeling a broad class of reward distributions such as Bernoulli and Poisson. While GLBs are widely applicable to real-world scenarios, their non-linear nature introduces significant challenges in achieving both computational and statistical efficiency. Existing methods typically trade off between two objectives, either incurring high per-round costs for optimal regret guarantees or compromising statistical efficiency to enable constant-time updates. In this paper, we propose a jointly efficient algorithm that attains a nearly optimal regret bound with $\mathcal{O}(1)$ time and space complexities per round. The core of our method is a tight confidence set for the online mirror descent (OMD) estimator, which is derived through a novel analysis that leverages the notion of mix loss from online prediction. The analysis shows that our OMD estimator, even with its one-pass updates, achieves statistical efficiency comparable to maximum likelihood estimation, thereby leading to a jointly efficient optimistic method.

Distributional reinforcement learning (DRL) has achieved empirical success in various domains.One of the core tasks in the field of DRL is distributional policy evaluation, which involves estimating the return distribution \eta \pi for a given policy \pi .The distributional temporal difference learning has been accordingly proposed, whichis an extension of the temporal difference learning (TD) in the classic RL area.In the tabular case, Rowland et al. [2018] and Rowland et al. [2023] proved the asymptotic convergence of two instances of distributional TD, namely categorical temporal difference learning (CTD) and quantile temporal difference learning (QTD), respectively.In this paper, we go a step further and analyze the finite-sample performance of distributional TD.To facilitate theoretical analysis, we propose a non-parametric distributional TD learning (NTD).For a \gamma -discounted infinite-horizon tabular Markov decision process,we show that for NTD we need \widetilde O\left(\frac{1}{\varepsilon {2p}(1-\gamma) {2p 1}}\right) iterations to achieve an \varepsilon -optimal estimator with high probability, when the estimation error is measured by the p -Wasserstein distance.This sample complexity bound is minimax optimal (up to logarithmic factors) in the case of the 1 -Wasserstein distance.To achieve this, we establish a novel Freedman's inequality in Hilbert spaces, which would be of independent interest.In addition, we revisit CTD, showing that the same non-asymptotic convergence bounds hold for CTD in the case of the p -Wasserstein distance.