Multi-Armed Bandits

Recently, it has been shown that the Stochastic Gradient Bandit (SGB) algorithm converges to a globally optimal policy with a constant learning rate. However, these guarantees rely on unrealistic assumptions about the learning process, namely that the probability of the optimal action is always bounded away from zero. We attribute this to the lack of an explicit exploration mechanism in SGB. To address these limitations, we propose to regularize the SGB objective with a log-barrier on the parametric policy, structurally enforcing a minimal amount of exploration. We prove that Log-Barrier Stochastic Gradient Bandit (LB-SGB) matches the sample complexity of SGB, but also converges (at a slower rate) without any assumptions on the learning process. We also show a connection between the log-barrier regularization and Natural Policy Gradient, as both exploit the geometry of the policy space by controlling the Fisher information. We validate our theoretical findings through numerical simulations, showing the benefits of the log-barrier regularization.

We study stochastic multi-armed bandits under simultaneous constraints on space and adaptivity: the learner interacts with the environment in $B$ batches and has only $W$ bits of persistent memory. Prior work shows that each constraint alone is surprisingly mild: near-minimax regret $\widetilde{O}(\sqrt{KT})$ is achievable with $O(\log T)$ bits of memory under fully adaptive interaction, and with a $K$-independent $O(\log\log T)$-type number of batches when memory is unrestricted. We show that this picture breaks down in the simultaneously constrained regime. We prove that any algorithm with a $W$-bit memory constraint must use at least $Ω(K/W)$ batches to achieve near-minimax regret $\widetilde{O}(\sqrt{KT})$, even under adaptive grids. In particular, logarithmic memory rules out $O(K^{1-\varepsilon})$ batch complexity. Our proof is based on an information bottleneck. We show that near-minimax regret forces the learner to acquire $Ω(K)$ bits of information about the hidden set of good arms under a suitable hard prior, whereas an algorithm with $B$ batches and $W$ bits of memory allows only $O(BW)$ bits of information. A key ingredient is a localized change-of-measure lemma that yields probability-level arm exploration guarantees, which is of independent interest. We also give an algorithm that, for any bit budget $W$ with $Ω(\log T) \le W \le O(K\log T)$, uses at most $W$ bits of memory and $\widetilde{O}(K/W)$ batches while achieving regret $\widetilde{O}(\sqrt{KT})$, nearly matching our lower bound up to polylogarithmic factors.

In this paper, we consider the problem of black-box optimization with noisy feedback revealed in batches, where the unknown function to optimize has a bounded norm in some Reproducing Kernel Hilbert Space (RKHS). We refer to this as the Batched Kernelized Bandits problem, and refine and extend existing results on regret bounds. For algorithmic upper bounds, (Li and Scarlett, 2022) shows that $B=O(\log\log T)$ batches suffice to attain near-optimal regret, where $T$ is the time horizon and $B$ is the number of batches. We further refine this by (i) finding the optimal number of batches including constant factors (to within $1+o(1)$), and (ii) removing a factor of $B$ in the regret bound. For algorithm-independent lower bounds, noticing that existing results only apply when the batch sizes are fixed in advance, we present novel lower bounds when the batch sizes are chosen adaptively, and show that adaptive batches have essentially same minimax regret scaling as fixed batches. Furthermore, we consider a robust setting where the goal is to choose points for which the function value remains high even after an adversarial perturbation. We present the robust-BPE algorithm, and show that a suitably-defined cumulative regret notion incurs the same bound as the non-robust setting, and derive a simple regret bound significantly below that of previous work.

This paper studies the optimality and complexity of Follow-the-Perturbed-Leader (FTPL) policy in $m$-set semi-bandit problems. FTPL has been studied extensively as a promising candidate of an efficient algorithm with favorable regret for adversarial combinatorial semi-bandits. Nevertheless, the optimality of FTPL has still been unknown unlike Follow-the-Regularized-Leader (FTRL) whose optimality has been proved for various tasks of online learning. In this paper, we extend the analysis of FTPL with geometric resampling (GR) to $m$-set semi-bandits, which is a special case of combinatorial semi-bandits, showing that FTPL with Fréchet and Pareto distributions with certain parameters achieves the best possible regret of $O(\sqrt{mdT})$ in adversarial setting. We also show that FTPL with Fréchet and Pareto distributions with a certain parameter achieves a logarithmic regret for stochastic setting, meaning the Best-of-Both-Worlds optimality of FTPL for $m$-set semi-bandit problems. Furthermore, we extend the conditional geometric resampling to $m$-set semi-bandits for efficient loss estimation in FTPL, reducing the computational complexity from $O(d^2)$ of the original geometric resampling to $O(md(\log(d/m)+1))$ without sacrificing the regret performance.

Personalized AI-based services involve a population of individual reinforcement learning agents. However, most reinforcement learning algorithms focus on harnessing individual learning and fail to leverage the social learning capabilities commonly exhibited by humans and animals. Social learning integrates individual experience with observing others' behavior, presenting opportunities for improved learning outcomes. In this study, we focus on a social bandit learning scenario where a social agent observes other agents' actions without knowledge of their rewards. The agents independently pursue their own policy without explicit motivation to teach each other. We propose a free energy-based social bandit learning algorithm over the policy space, where the social agent evaluates others' expertise levels without resorting to any oracle or social norms. Accordingly, the social agent integrates its direct experiences in the environment and others' estimated policies. The theoretical convergence of our algorithm to the optimal policy is proven. Empirical evaluations validate the superiority of our social learning method over alternative approaches in various scenarios. Our algorithm strategically identifies the relevant agents, even in the presence of random or suboptimal agents, and skillfully exploits their behavioral information. In addition to societies including expert agents, in the presence of relevant but non-expert agents, our algorithm significantly enhances individual learning performance, where most related methods fail. Importantly, it also maintains logarithmic regret.

We study the fixed-budget best-arm identification (BAI) problem in non-stationary linear bandits. Concretely, given a fixed time budget $T\in \mathbb{N}$, finite arm set $\mathcal{X} \subset \mathbb{R}^d$, and a potentially adversarial sequence of unknown parameters $\lbrace θ_t\rbrace_{t=1}^{T}$ (hence non-stationary), a learner aims to identify the arm with the largest cumulative reward $x_* = \arg\max_{x \in \mathcal{X}} x^\top\sum_{t=1}^T θ_t$ with high probability. In this setting, it is well-known that uniformly sampling arms from the G-optimal design yields a minimax-optimal error probability of $\exp\left(-Θ\left(T / H_{G}\right)\right)$, where $H_{G}$ scales proportionally with the dimension $d$. However, this notion of complexity is overly pessimistic, as it is derived from a lower bound in which the arm set consists only of the standard basis vectors, thus masking any potential advantages arising from arm sets with richer geometric structure. To address this, we establish an arm-set-dependent lower bound that, in contrast, holds for any arm set. Motivated by the ideas underlying our lower bound, we propose the Adjacent-optimal design, a specialization of the well-known $\mathcal{X}\mathcal{Y}$-optimal design, and develop the $\textsf{Adjacent-BAI}$ algorithm. We prove that the error probability of $\textsf{Adjacent-BAI}$ matches our lower bound up to constants, verifying the tightness of our lower bound, and establishing the arm-set-dependent complexity of this setting.

We study a continuous-time diffusion approximation of policy gradient for $k$-armed stochastic bandits. We prove that with a learning rate $η= O(Δ^2/\log(n))$ the regret is $O(k \log(k) \log(n) / η)$ where $n$ is the horizon and $Δ$ the minimum gap. Moreover, we construct an instance with only logarithmically many arms for which the regret is linear unless $η= O(Δ^2)$.

Statistical inference with bandit data presents fundamental challenges owing to adaptive sampling, which violates the independence assumptions underlying classical asymptotic theory. Recent work has identified stability~\citep{laiwei82} as a sufficient condition for valid inference under adaptivity. This paper first provides a refined stability condition, stated in terms of the iterates of an online algorithm, and shows that a large class of regularized stochastic-mirror-descent-style algorithms satisfy it. This refined condition allows us to strengthen the asymptotic results of~\citet{laiwei82} in several ways. First, we derive a non-asymptotic Berry--Esseen bound for the empirical reward estimates under adaptive sampling. Second, we derive matching non-asymptotic upper and lower bounds on the regret of the proposed algorithm, yielding a precise characterization of its regret. Third, we show that these regularized algorithms preserve asymptotic normality and valid inference under a prescribed level of adversarial corruption. Finally, we show that regularization is necessary rather than incidental: Lai--Wei stability is incompatible with the optimal $O(\sqrt{T})$ regret rate -- the rate attained by unregularized algorithms such as EXP3 -- so that a controlled, polylogarithmic inflation in regret is the price of valid inference.

We test whether LLMs show robust decision biases. Treating models as participants in two-arm bandits, we ran 20000 trials per condition across four decoding configurations. Under symmetric rewards, models amplified positional order into stubborn one-arm policies. Under asymmetric rewards, they exploited rigidly yet underperformed an oracle and rarely re-checked. The observed patterns were consistent across manipulations of temperature and top-p, with top-k held at the provider default, indicating that the qualitative behaviours are robust to the two decoding knobs typically available to practitioners. Crucially, moving beyond descriptive metrics to computational modelling, a hierarchical Rescorla-Wagner-softmax fit revealed the underlying strategies: low learning rates and very high inverse temperatures, which together explain both noise-to-bias amplification and rigid exploitation. These results position minimal bandits as a tractable probe of LLM decision tendencies and motivate hypotheses about how such biases could shape human-AI interaction.

Multi-modal large language model (MLLM) inference scheduling enables strong response quality under practical and heterogeneous budgets, beyond what a homogeneous single-backend setting can offer. Yet online MLLM task scheduling is nontrivial, as requests vary sharply in modality composition and latent reasoning difficulty, while execution backends incur distinct, time-varying costs due to system jitter and network variation. These coupled uncertainties pose two core challenges: deriving semantically faithful yet scheduling-relevant multi-modal task representations, and making low-overhead online decisions over irreversible multi-dimensional budgets. Accordingly, we propose M-CMAB (Multi-modal Multi-constraint Contextual Multi-Armed Bandit), a multi-adapter-enhanced MLLM inference scheduling framework with three components: (i) a CLS-attentive, frozen-backbone Predictor that extracts compact task representations and updates only lightweight adapters for action-specific estimation; (ii) a primal-dual Constrainer that maintains online Lagrange multipliers to enforce long-horizon constraints via per-round objectives; and (iii) a two-phase Scheduler that balances exploration and exploitation under irreversible budgets. We establish a regret guarantee under multi-dimensional knapsack constraints. On a composite multimodal benchmark with heterogeneous backends, M-CMAB consistently outperforms state-of-the-art baselines across budget regimes, achieving up to 14.18% higher reward and closely tracking an oracle-aided upper bound. Codes are available at https://anonymous.4open.science/r/M2CMAB/.

Online platforms routinely compare multi-armed bandit algorithms, such as UCB and Thompson Sampling, to select the best-performing policy. Unlike standard A/B tests for static treatments, each run of a bandit algorithm over $T$ users produces only one trajectory, because the algorithm's decisions depend on all past interactions. Reliable inference therefore demands many independent restarts of the algorithm, making experimentation costly and delaying deployment decisions. We propose Artificial Replay (AR) as a new experimental design for this problem. AR first runs one policy and records its trajectory. When the second policy is executed, it reuses a recorded reward whenever it selects an action the first policy already took, and queries the real environment only otherwise. We develop a new analytical framework for this design and prove three key properties of the resulting estimator: it is unbiased; it requires only $T + o(T)$ user interactions instead of $2T$ for a run of the treatment and control policies, nearly halving the experimental cost when both policies have sub-linear regret; and its variance grows sub-linearly in $T$, whereas the estimator from a naïve design has a linearly-growing variance. Numerical experiments with UCB, Thompson Sampling, and $ε$-greedy policies confirm these theoretical gains.

Decision makers exploiting photonic chaotic dynamics obtained by semiconductor lasers provide an ultrafast approach to solving multi-armed bandit problems by using a temporal optical signal as the driving source for sequential decisions. In such systems, the sampling interval of the chaotic waveform shapes the temporal correlation of the resulting time series, and experiments have reported that decision accuracy depends strongly on this autocorrelation property. However, it remains unclear whether the benefit of autocorrelation can be explained by a minimal mathematical model. Here, we analyze a stochastic-process model for solving the two-armed bandit problem based on time series, where the threshold and a two-valued Markov signal evolve jointly. Numerical results reveal an environment-dependent structure: negative (positive) autocorrelation is optimal in reward-rich (reward-poor) environments. These findings show that negative autocorrelation of the time series is advantageous when the sum of the winning probabilities is more than one, whereas positive autocorrelation is useful when the sum of the winning probabilities is less than one. Moreover, the performance is independent of autocorrelation if the sum of the winning probabilities equals one, which is mathematically clarified. This study paves the way for solving the two-armed bandit problems for reinforcement learning applications in wireless communications and robotics.

We study fixed budget constrained best-arm identification in grouped bandits, where each arm consists of multiple independent attributes with stochastic rewards. An arm is considered feasible only if all its attributes' means are above a given threshold. The aim is to find the feasible arm with the largest overall mean. We first derive a lower bound on the error probability for any algorithm on this setting. We then propose Feasibility Constrained Successive Rejects (FCSR), a novel algorithm that identifies the best arm while ensuring feasibility. We show it attains optimal dependence on problem parameters up to constant factors in the exponent. Empirically, FCSR outperforms natural baselines while preserving feasibility guarantees.

We consider stochastic non-stationary linear bandits where the linear parameter connecting contexts to the reward changes over time. Existing algorithms in this setting localize the policy by gradually discarding or down-weighting past data, effectively shrinking the time horizon over which learning can occur. However, in many settings historical data may still carry partial information about the reward model. We propose to leverage such data while adapting to changes, by assuming the reward model decomposes into stationary and non-stationary components. Based on this assumption, we introduce ISD-linUCB, an algorithm that uses past data to learn invariances in the reward model and subsequently exploits them to improve online performance. We show both theoretically and empirically that leveraging invariance reduces the problem dimensionality, yielding significant regret improvements in fast-changing environments when sufficient historical data is available.

Recent studies have shown that reinforcement learning with KL-regularized objectives can enjoy faster rates of convergence or logarithmic regret, in contrast to the classical $\sqrt{T}$-type regret in the unregularized setting. However, the statistical efficiency of online learning with respect to KL-regularized objectives remains far from completely characterized, even when specialized to multi-armed bandits (MABs). We address this problem for MABs via a sharp analysis of KL-UCB using a novel peeling argument, which yields a $\tilde{O}(ηK\log^2T)$ upper bound: the first high-probability regret bound with linear dependence on $K$. Here, $T$ is the time horizon, $K$ is the number of arms, $η^{-1}$ is the regularization intensity, and $\tilde{O}$ hides all logarithmic factors except those involving $\log T$. The near-tightness of our analysis is certified by the first non-constant lower bound $Ω(ηK \log T)$, which follows from subtle hard-instance constructions and a tailored decomposition of the Bayes prior. Moreover, in the low-regularization regime (i.e., large $η$), we show that the KL-regularized regret for MABs is $η$-independent and scales as $\tildeΘ(\sqrt{KT})$. Overall, our results provide a thorough understanding of KL-regularized MABs across all regimes of $η$ and yield nearly optimal bounds in terms of $K$, $η$, and $T$.

Motivated by wireless networks where interference or channel state estimates provide partial insight into throughput, we study a variant of the classical stochastic multi-armed bandit problem in which the learner has limited access to auxiliary information. Recent work has shown that such auxiliary information, when available as control variates, can be used to get tighter confidence bounds, leading to lower regret. However, existing works assume that control variates are available in every round, which may not be realistic in several real-life scenarios. To address this, we propose UCB-LCV, an upper confidence bound (UCB) based algorithm that effectively combines the estimators obtained from rewards and control variates. When there is no control variate, UCB-LCV leads to a novel algorithm that we call UCB-NORMAL, outperforming its existing algorithms for the standard MAB setting with normally distributed rewards. Finally, we discuss variants of the proposed UCB-LCV that apply to general distributions and experimentally demonstrate that UCB-LCV outperforms existing bandit algorithms.

Determining a winner among a set of items using active pairwise comparisons under a limited budget is a challenging problem in preference-based learning. The goal of this study is to implement and evaluate the PARWiS algorithm, which shows spectral ranking and disruptive pair selection to identify the best item under shoestring budgets. This work have extended the PARWiS with a contextual variant (Contextual PARWiS) and a reinforcement learning-based variant (RL PARWiS), comparing them against baselines, including Double Thompson Sampling and a random selection strategy. This evaluation spans synthetic and real-world datasets (Jester and MovieLens), using budgets of 40, 60, and 80 comparisons for 20 items. The performance is measured through recovery fraction, true rank of reported winner, reported rank of true winner, and cumulative regret, alongside the separation metric $Δ_{1,2}$. Results show that PARWiS and RL PARWiS outperform baselines across all datasets, particularly in the Jester dataset with a higher $Δ_{1,2}$, while performance gaps narrow in the more challenging MovieLens dataset with a smaller $Δ_{1,2}$. Contextual PARWiS shows comparable performance to PARWiS, indicating that contextual features may require further tuning to provide significant benefits.

We study an identification problem in multi-armed bandits. In each round a learner selects one of $K$ arms and observes its reward, with the goal of eventually identifying an arm that will perform best at a {\it future} time. In adversarial environments, however, past performance may offer little information about the future, raising the question of whether meaningful identification is possible at all. In this work, we introduce lookahead identification, a task in which the goal of the learner is to select a future prediction window and commit in advance to an arm whose average reward over that window is within $\varepsilon$ of optimal. Our analysis characterizes both the achievable accuracy of lookahead identification and the memory resources required to obtain it. From an accuracy standpoint, for any horizon $T$ we give an algorithm achieving $\varepsilon = O\bigl(1/\sqrt{\log T}\bigr)$ over $Ω(\sqrt{T})$ prediction windows. This demonstrates that, perhaps surprisingly, identification is possible in adversarial settings, despite significant lack of information. We also prove a near-matching lower bound showing that $\varepsilon = Ω\bigl(1/\log T\bigr)$ is unavoidable. We then turn to investigate the role of memory in our problem, first proving that any algorithm achieving nontrivial accuracy requires $Ω(K)$ bits of memory. Under a natural local sparsity condition, we show that the same accuracy guarantees can be achieved using only poly-logarithmic memory.

In many applications, evaluating the effectiveness of different alternatives comes with varying costs or resource usage. Motivated by such heterogeneity, we study the Best Arm Identification with Resource Constraints (BAIwRC) problem, where an agent seeks to identify the best alternative (aka arm) in the presence of resource constraints. Each arm pull consumes one or more types of limited resources. We make two key contributions. First, we propose the Successive Halving with Resource Rationing (SH-RR) algorithm, which integrates resource-aware allocation into the classical successive halving framework on best arm identification. The SH-RR algorithm unifies the theoretical analysis for both the stochastic and deterministic consumption settings, with a new \textit{effective consumption measure

Crowdsourcing platforms face a four-way tension between allocation quality, workforce sustainability, operational feasibility, and strategic contractor behaviour--a dilemma we formalise as the Cold-Start, Burnout, Utilisation, and Strategic Agency Dilemma. Existing methods resolve at most two of these tensions simultaneously: greedy heuristics and multi-criteria decision making (MCDM) methods achieve Day-1 quality but cause catastrophic burnout, while bandit algorithms eliminate burnout only through operationally infeasible 100% workforce utilisation.To address this, we introduce FORGE, a physics-grounded $K+1$ multi-agent simulator in which each contractor is a rational agent that declares its own load-acceptance threshold based on its fatigue state, converting the standard passive Restless Multi-Armed Bandit (RMAB) into a genuine Stackelberg game. Operating within FORGE, we propose a Neural-Linear UCB allocator that fuses a Two-Tower embedding network with a Physics-Informed Covariance Prior derived from offline simulator interactions. The prior simultaneously warm-starts skill-cluster geometry and UCB exploration landscape, providing a geometry-aware belief state from episode 1 that measurably reduces cold-start regret.Over $T = 200$ cold-start episodes, the proposed method achieves the highest reward of all non-oracle methods ($\text{LRew} = 0.555 \pm 0.041$) at only 7.6% workforce utilisation--a combination no conventional baseline achieves--while maintaining robustness to workforce turnover up to 50% and observation noise up to $σ= 0.20$.

In bandit settings, optimizing long-term regret metrics requires exploration, which corresponds to sometimes taking myopically sub-optimal actions. When a long-lived principal merely recommends actions to be executed by a sequence of different agents (as in an online recommendation platform) this provides an incentive misalignment: exploration is "worth it" for the principal but not for the agents. Prior work studies regret minimization under the constraint of Bayesian Incentive-Compatibility in a static stochastic setting with a fixed and common prior shared amongst the agents and the algorithm designer. We show that (weighted) swap regret bounds on their own suffice to cause agents to faithfully follow forecasts in an approximate Bayes Nash equilibrium, even in dynamic environments in which agents have conflicting prior beliefs and the mechanism designer has no knowledge of any agents beliefs. To obtain these bounds, it is necessary to assume that the agents have some degree of uncertainty not just about the rewards, but about their arrival time -- i.e. their relative position in the sequence of agents served by the algorithm. We instantiate our abstract bounds with concrete algorithms for guaranteeing adaptive and weighted regret in bandit settings.

We introduce Box Thirding (B3), a flexible and efficient algorithm for Best Arm Identification (BAI) under fixed-budget constraints. It is designed for both anytime BAI and scenarios with large N, where the number of arms is too large for exhaustive evaluation within a limited budget T. The algorithm employs an iterative ternary comparison: in each iteration, three arms are compared--the best-performing arm is explored further, the median is deferred for future comparisons, and the weakest is discarded. Even without prior knowledge of T, B3 achieves an epsilon-best arm misidentification probability comparable to Successive Halving (SH), which requires T as a predefined parameter, applied to a randomly selected subset of c0 arms that fit within the budget. Empirical results show that B3 outperforms existing methods under limited-budget constraints in terms of simple regret, as demonstrated on the New Yorker Cartoon Caption Contest dataset.

Online model selection in Bayesian bandits raises a fundamental exploration challenge: When an environment instance is sampled from a prior distribution, how can we design an adaptive strategy that explores multiple bandit learners and competes with the best one in hindsight? We address this problem by introducing a new Bayesian algorithm for online model selection in stochastic bandits. We prove an oracle-style guarantee of $O\left( d^* M \sqrt{T} + \sqrt{(MT)} \right)$ on the Bayesian regret, where $M$ is the number of base learners, $d^*$ is the regret coefficient of the optimal base learner, and $T$ is the time horizon. We also validate our method empirically across a range of stochastic bandit settings, demonstrating performance that is competitive with the best base learner. Additionally, we study the effect of sharing data among base learners and its role in mitigating prior mis-specification.

We study an algorithm-independent, worst-case lower bound for the Gaussian process (GP) bandit problem in the frequentist setting, where the reward function is fixed and has a bounded norm in the known reproducing kernel Hilbert space (RKHS). Specifically, we focus on the squared exponential (SE) kernel, one of the most widely used kernel functions in GP bandits. One of the remaining open questions for this problem is the gap in the dimension-dependent logarithmic factors between upper and lower bounds. This paper partially resolves this open question under a hyperspherical input domain. We show that any algorithm suffers $Ω(\sqrt{T (\ln T)^{d} (\ln \ln T)^{-d}})$ cumulative regret, where $T$ and $d$ represent the total number of steps and the dimension of the hyperspherical domain, respectively. Regarding the simple regret, we show that any algorithm requires $Ω(ε^{-2}(\ln \frac{1}ε)^d (\ln \ln \frac{1}ε)^{-d})$ time steps to find an $ε$-optimal point. We also provide the improved $O((\ln T)^{d+1}(\ln \ln T)^{-d})$ upper bound on the maximum information gain for the SE kernel. Our results guarantee the optimality of the existing best algorithm up to dimension-independent logarithmic factors under a hyperspherical input domain.

We introduce Flickering Multi-Armed Bandits (FMAB) to model sequential decision-making in environments with changing action availability, where accessibility of the next action is restricted to a subset dependent on the agent's current choice. We formalize these constraints through stochastically evolving graphs where actions are limited to local neighborhoods. This mobility-constrained structure imposes a dual challenge: the statistical requirement of information acquisition and the physical overhead of navigation. We analyze FMAB under i.i.d. Erdős--R'enyi and Edge-Markovian process, proposing a two-phase lazy random walk algorithm for robust exploration. We establish high-probability sublinear regret bounds and prove near-optimality via a matching information-theoretic lower bound. Our results characterize the intrinsic cost of learning under local-move constraints, complemented by a robotic disaster-response simulation.

Dynamic decision-making under model uncertainty is central to many economic environments, yet existing bandit and reinforcement learning algorithms rely on the assumption of correct model specification. This paper studies the behavior and performance of one of the most commonly used Bayesian reinforcement learning algorithms, Thompson Sampling (TS), when the model class is misspecified. We first provide a complete dynamic classification of posterior evolution in a misspecified two-armed Gaussian bandit, identifying distinct regimes: correct model concentration, incorrect model concentration, and persistent belief mixing, characterized by the direction of statistical evidence and the model-action mapping. These regimes yield sharp predictions for limiting beliefs, action frequencies, and asymptotic regret. We then extend the analysis to a general finite model class and develop a unified stochastic stability framework that represents posterior evolution as a Markov process on the belief simplex. This approach characterizes two sufficient conditions to classify the ergodic and transient behaviors and provides inductive dimensional reductions of the posterior dynamics. Our results offer the first qualitative and geometric classification of TS under misspecification, bridging Bayesian learning with evolutionary dynamics, and also build the foundations of robust decision-making in structured bandits.

We study the decentralized multi-player stochastic bandit problem over a continuous, Lipschitz-structured action space where hard collisions yield zero reward. Our objective is to design a communication-free policy that maximizes collective reward, while separating coordination costs from learning costs. We propose a modular protocol that first solves the multi-agent coordination problem by identifying and seating players on distinct, high-value regions via a novel maxima-directed search and then decouples the problem into $N$ independent single-player Lipschitz bandits. In the consensus regime, we obtain an end-to-end regret bound whose dominant learning term is $\tilde{O}(T^{(d+1)/(d+2)})$, matching the single-player Lipschitz rate; the upfront coordination cost is horizon-independent at fixed confidence and only polylogarithmic in $T$ in the expected-regret form. Under an additional public coverage/scheduling assumption for the epochic extension, we also obtain a gap-free $\tilde{O}(T^{(d+1)/(d+2)})$ guarantee. We further derive a matching lower bound for the dominant learning term and extend the framework to general distance-threshold collision models.

We study dynamic joint assortment and pricing where a seller updates decisions at regular accounting/operating intervals to maximize the cumulative per-period revenue over a horizon $T$. In many settings, assortment and prices affect not only what an arriving customer buys but also how many customers arrive within the period, whereas classical multinomial logit (MNL) models assume arrivals as fixed, potentially leading to suboptimal decisions. We propose a Poisson-MNL model that couples a contextual MNL choice model with a Poisson arrival model whose rate depends on the offered assortment and prices. Building on this model, we develop an efficient algorithm PMNL based on the idea of upper confidence bound (UCB). We establish its (near) optimality by proving a non-asymptotic regret bound of order $\sqrt{T\log{T}}$ and a matching lower bound (up to $\log T$). Simulation studies underscore the importance of accounting for the dependency of arrival rates on assortment and pricing: PMNL effectively learns customer choice and arrival models and provides joint assortment-pricing decisions that outperform others that assume fixed arrival rates.

We study the Submodular Welfare Problem (SWP), where items are partitioned among agents with monotone submodular utilities to maximize the total welfare under bandit feedback. Classical SWP assumes full value-oracle access, achieving $(1-1/e)$ approximations via continuous-greedy algorithms. We extend this to a multi-agent combinatorial bandit framework (MA-CMAB), where actions are partitions under full-bandit feedback with non-communicating agents. Unlike prior single-agent or separable multi-agent CMAB models, our setting couples agents through shared allocation constraints. We propose an explore-then-commit strategy with randomized assignments, achieving $\tilde{\mathcal{O}}(T^{2/3})$ regret against a $(1-1/e)$ benchmark, the first such guarantee for partition-based submodular welfare problem under bandit feedback.

We study a type of Multi-Armed Bandit (MAB) problems in which arms with a Gaussian reward feedback are clustered. Such an arm setting finds applications in many real-world problems, for example, mmWave communications and portfolio management with risky assets, as a result of the universality of the Gaussian distribution. Based on the Thompson Sampling algorithm with Gaussian prior (TSG) algorithm for the selection of the optimal arm, we propose our Thompson Sampling with Clustered arms under Gaussian prior (TSCG) specific to the 2-level hierarchical structure. We prove that by utilizing the 2-level structure, we can achieve a lower regret bound than we do with ordinary TSG. In addition, when the reward is Unimodal, we can reach an even lower bound on the regret by our Unimodal Thompson Sampling algorithm with Clustered Arms under Gaussian prior (UTSCG). Each of our proposed algorithms are accompanied by theoretical evaluation of the upper regret bound, and our numerical experiments confirm the advantage of our proposed algorithms.