Multi-Armed Bandit RLHF Model

• Multi-Armed Bandit Models in RLHF are frameworks that treat candidate policies as arms, dynamically selecting based on time-varying, noisy human feedback.
• They employ exploration and exploitation epochs to balance data gathering and deployment, achieving logarithmic regret scaling for efficient learning.
• Extensions to decentralized, multi-agent settings and models for endogenous feedback drift enhance scalability and robustness in practical RLHF systems.

A Multi-Armed Bandit (MAB) model of Reinforcement Learning with Human Feedback (RLHF) captures the principle that policy selection under uncertain and possibly nonstationary reward signals—provided by human evaluators—can be formalized as a sequential decision problem in which arms correspond to candidate policies or behavioral modes and their rewards reflect potentially time-varying, noisy human feedback. The restless multi-armed bandit (RMAB) extension is the most direct abstraction for learning in RLHF when both model dynamics and human feedback evolve over time and may not be fully known a priori.

1. Reward State Evolution and Regret in Restless Bandit Models

The RMAB framework models each candidate policy $i$ in an RLHF system as an arm with a time-evolving state $s_i(t)$. When policy $i$ is enacted at time $t$, its state updates stochastically according to an unknown Markov transition matrix $P_{(i)}$: $s_i(t + 1) \sim P_{(i)}(s_i(t), \cdot)$ If policy $i$ is not selected, its state evolves via an arbitrary, unknown process $Q_{(i)}$, which may encode changes in latent human preferences or environmental context. In RLHF, this models scenarios where policy confidence, alignment, or human reward evaluations transition even without direct sampling.

Performance is universally measured by expected regret—not raw reward—relative to an oracle that always selects the optimal policy under perfect knowledge: $$r_\Phi(t) = t \cdot \mu^* - \mathbb{E}_\Phi \left[ \sum_{t'=1}^{t} r_{t'} \right]$$ $\mu^*$ denotes the expected reward (for RLHF, aggregate human feedback) under the optimal policy, and $r_{t'}$ is the true reward (human-mediated) at round $t'$. Minimizing regret operationalizes convergence to high-performing, well-aligned behavior under noisy, evolving reward signals.

2. Exploration/Exploitation Epoch Structures and Regret Bounds

The primary methodological innovation lies in structuring arm selection into deterministic epochs that alternate between exploration and exploitation stages:

• Exploration epochs: Each arm (policy) is sampled sufficiently to gather new reward (human feedback) observations. This phase ensures persistent learning about dynamic or uncertain arms, crucial in RLHF where feedback is expensive and nonstationary.
• Exploitation epochs: Sample means

$$\bar{s}_i(t) = \frac{1}{T_i(t)}\sum_{n=1}^{T_i(t)} s_i(t_n)$$

are computed for each arm and the best-performing arms are selected based on these estimates.

Epoch lengths are designed to increase geometrically, so the number of switches between policies grows only logarithmically in the horizon, minimizing regret due to transient effects: $r_\Phi(t) = O(\log t)$ This structure directly addresses RLHF demands for sample efficiency, as frequent querying for human feedback incurs significant cost. It supports alternating between data-gathering (feedback solicitation) and deployment (best-policy rollout) phases.

3. Decentralization and Multi-Agent RLHF

The RMAB model generalizes to decentralized environments: multiple learning agents operate concurrently, each with local access to feedback and without coordination. Each agent implements exploration/exploitation epochs but must resolve conflicts (collisions) when simultaneously selecting the same candidate policy.

Key principles in decentralized RLHF:

• Agents employ randomized selection or round-robin in exploitation epochs to minimize collisions.
• Epoch lengths and timings are local; global synchronization is not required.
• The decentralized algorithm preserves logarithmic regret scaling, guaranteeing that the lack of communication or pre-agreement does not degrade global learning efficiency.

This enables RLHF applications across distributed systems, crowd-sourced trainers, or modular agents learning from distinct or asynchronous human evaluators.

4. Exogenous and Endogenous Restlessness: Modeling Feedback Change

Restless bandits model both exogenous and endogenous evolution:

• Exogenous model: State changes occur only if the policy is actively sampled. In RLHF, this implies human feedback and behavioral updates only on direct interaction.
• Endogenous model: States continue to evolve without active sampling, capturing shifting human preferences or context-sensitive feedback even for passive policies.

Mathematically: $$s_i(t+1) = \begin{cases} P_i(s_i(t), s') & \text{if } i \text{ played (active)}, \ Q_i(s_i(t), s') & \text{if } i \text{ passive (not played)}. \end{cases}$$ Endogenous models recognize that policy ratings, anthropic assessments, or contextual relevance may change with time and external events; thus, RLHF systems must adapt exploration frequency, epoch lengths, or feedback solicitation policies to avoid staleness or drift.

5. Connections to Broader Bandit Algorithms and RLHF Extensions

The RMAB model for RLHF sits within a broader taxonomy of bandit-based RL methods:

• Classical stochastic and adversarial bandits (Bubeck et al., 2012, Agrawal et al., 2011) provide foundational regret bounds and exploration/exploitation strategies, but fail to capture nonstationarity and passive-state evolution.
• Contextual and dueling bandits (Chen et al., 18 May 2024, Gornet et al., 15 May 2024, Scheid et al., 22 Oct 2024) generalize to model human pairwise preferences or non-numeric reward signals, aligning closely with practical RLHF pipelines for LLM alignment.
• Multi-fidelity bandits (Kandasamy et al., 2016) model evaluation cost heterogeneity, relevant for stages where policy evaluation can alternate between automated and costly human-in-the-loop feedback.
• Risk-aware extensions (Alami et al., 2023) introduce risk measures (e.g., CVaR, mean-variance) to encode safety in high-volatility domains, reflecting RLHF requirements for robustness against adverse reward outcomes and preference shifts.
• Online exploration optimization (Li et al., 26 Sep 2025) addresses sample efficiency and uncertainty minimization in reward differences, targeting RLHF-specific regret bounds in adaptive preference querying.

Tabulated summary of core RMAB modeling features in RLHF context:

Feature	Mathematical Encoding	RLHF Significance
State transitions	$P_i$, $Q_i$	Human feedback evolution, alignment drift
Regret formulation	$r_\Phi(t) = t\mu^* - \mathbb{E}_\Phi[R(t)]$	Guarantees on long-term human alignment
Epoch structure	Growing deterministic epochs	Sample efficiency, reduced querying cost
Decentralization	Randomized local exploration, collisions	Multi-agent, crowdsourced RLHF settings
Restlessness model	Exogenous vs. endogenous $P_i$, $Q_i$	Modeling passive preference drift

6. Summary of Theoretical and Practical Implications

Applying RMAB principles to RLHF enables:

• Systematic selection and improvement of candidate policies (arms) under unknown, time-evolving human feedback.
• Rigorous performance guarantees: logarithmic regret scaling in both centralized and decentralized learning, sample-efficient exploration tailored to expensive human input.
• Modeling of both exogenous and endogenous feedback dynamics, supporting robust adaptation to preference drift and nonstationary environments.
• Scalability across distributed learners, modules, or human-agent collectives without synchronization, collision penalties, or information sharing requirements.
• Direct mapping between RMAB theoretical constructs (state evolution rules, exploration/exploitation, epoch design, regret minimization) and operational RLHF pipelines for aligned, efficient LLM fine-tuning.

Formally: $$\left. \begin{aligned} & s_i(t+1) = \begin{cases} P_i(s_i(t), s') & \text{if policy } i \text{ active}, \ Q_i(s_i(t), s') & \text{if } i \text{ passive} \end{cases} \ & r_\Phi(t) = t \cdot \mu^* - \mathbb{E}_\Phi \left[ \sum_{t'=1}^t r_{t'} \right] = O(\log t) \end{aligned} \right.$$ The RMAB model thereby abstracts and informs the design of scalable, adaptive RLHF procedures with guarantees on the cost-performance tradeoff, resilience to feedback nonstationarity, and robust multi-agent policy selection (Liu et al., 2010).