Multi-Armed Sampling Framework

• Multi-Armed Sampling Framework is a model that defines sampling over discrete actions to match a target distribution instead of maximizing rewards.
• It introduces novel regret metrics and the ASE algorithm, demonstrating that minimal explicit exploration is needed for accurate sampling.
• The framework unifies principles from Bayesian inference, entropy-regularized reinforcement learning, and neural sampling to ensure high-fidelity distribution matching.

The multi-armed sampling framework formalizes the problem of sampling from a target distribution over a set of discrete actions ("arms"), contrasting with the classical multi-armed bandit (MAB) formulation that seeks to maximize cumulative reward through adaptive exploration and exploitation. The multi-armed sampling problem generalizes sampling tasks encountered in Bayesian inference, neural samplers, and entropy-regularized reinforcement learning by focusing on matching target distributions under uncertainty, rather than maximizing expected reward or identifying the optimal arm. This framework rigorously distinguishes the objectives, regret measures, and algorithmic requirements of sampling versus optimization, and establishes new theoretical results on the necessity of exploration, algorithm design, and the relation to bandit problems (2507.10797).

1. Problem Definition and Objectives

In the multi-armed sampling framework, the learner's goal is to match a predetermined target distribution $p$ on $K$ arms over $T$ rounds. Rather than maximizing reward, the algorithm samples actions such that the empirical distribution of selected arms (averaged over rounds, or at each policy step) approximates $p$ as closely as possible. This objective arises in scenarios such as

• Soft (temperature-controlled) action sampling in reinforcement learning,
• Adaptive MCMC and neural samplers striving for calibrated weighted outputs,
• Sampling in entropy-regularized or uncertainty-averse decision processes.

Performance is evaluated by statistical divergences between the empirical (or instantaneous policy) distribution and $p$, including total variation (TV) distance, Kullback–Leibler (KL) divergence (both forward and reverse), and other $f$-divergences. Two key regret notions are distinguished:

1. Simple (per-step) regret: Statistical distance between the policy distribution at a given time and $p$.
2. Cumulative regret: Aggregate divergence accrued over $T$ rounds.

Additionally, regime distinctions arise between

• Action-level regret: Assesses the aggregate mismatch of the sequence of samples.
• Policy-level regret: Evaluates the year-by-year policy’s decision distribution compared to $p$.

The relationships among these regret notions are made explicit via formal inequalities (e.g., action-level regret is upper-bounded by cumulative policy-level regret), clarifying how sampling performance is to be measured in practice.

2. Core Theoretical Insights

A central result is that, for sampling objectives, the classical need for exploration disappears: efficient sampling algorithms do not require persistent explicit exploration to achieve optimal or near-optimal regret rates.

Specifically, the paper establishes that:

• The minimal necessary exploration (number of “forced” samples per arm) scales only logarithmically with $T$ or is even unnecessary for certain divergences.
• For total variation and reverse-KL divergences, “active sampling with exploration” (ASE)—initially visiting all arms in a round-robin manner—already guarantees simple regret decays as $\widetilde{O}(T^{-1/2})$ and cumulative regret as $\widetilde{O}(T^{1/2})$ (where $\widetilde{O}$ hides logarithmic factors).
• With zero explicit exploration (setting the exploration parameter $M=1$ in ASE), only a very mild (quasi-polynomial) degradation results.

By contrast, in the MAB/optimization setting, insufficient exploration leads to linear regret. This architectural difference in regret scaling is rigorously demonstrated through lower and upper bounds.

3. Algorithmic Approach: Active Sampling with Exploration (ASE)

ASE is a two-phase algorithm designed to minimize sampling regret:

1. Exploration phase: For $M \cdot K$ initial rounds, cycle through each arm $M$ times to ensure all empirical means are initialized.
2. Exploitation phase: At time $t > M \cdot K$, use empirical means $\hat{r}_i$ to form a softmax distribution $$\tilde{p}_i = \frac{e^{\beta \hat{r}_i}}{\sum_j e^{\beta \hat{r}_j}}$$, where $\beta > 0$ is a temperature parameter (see next section). Each subsequent arm pull samples from $\tilde{p}$.

The primary theoretical and practical outcome is that the exploration phase only needs to be mild (even $M=O(\log T)$ suffices), and the exploitation phase naturally aligns the empirical action distribution to the target $p$.

4. Unified Regret Family and the Temperature Parameter

The framework introduces a continuous, parameterized family of problems (and their corresponding regret measures) controlled by a “temperature” parameter $\beta$:

• For any finite $\beta>0$, the target distribution is $p^\beta = \text{softmax}(\beta r)$, where $r$ is the reward vector.
• As $\beta\to\infty$, $p^\beta$ concentrates on the maximizing arm(s), and the sampling problem becomes classical MAB: the objective reduces to cumulative reward maximization.
• Regret for any $\beta$ is defined as:

$$AR_T^\beta = \begin{cases} \frac{T}{\beta} D_{\rm r\text{-}KL}(q_T, p^\beta) & \text{if } \beta < \infty,\ R_T^{(\mathrm{MAB})} & \text{if } \beta = \infty, \end{cases}$$

where $q_T$ is the empirical action distribution and $R_T^{(\mathrm{MAB})}$ is the classical MAB regret.

This interpolation elucidates that the essential distinction between sampling and optimization is temperature: as the target is softened (smaller $\beta$), exploration becomes less critical for performance.

5. Implications for Exploration in Sampling Tasks

The theoretical findings decisively establish that, in the sampling regime, unlike in the MAB/optimization regime:

• Persistent exploration is not essential for minimizing regret.
• Mild, even vanishing exploration (after a brief warmup), suffices to ensure the empirical action distribution aligns with the target.
• Algorithms that focus on exploitative matching to the current empirical reward estimates (i.e., “softmax” policies) are sufficient, so brute-force exploration strategies common in bandit optimization are unnecessary for achieving optimal sampling rates.

This has significant consequences for domains such as entropy-regularized reinforcement learning, neural sampler design, adaptive importance sampling, and model calibration, where the intent is to attain high-fidelity samples from a distribution rather than simply to maximize reward.

6. Connection to Reinforcement Learning, Neural Samplers, and Beyond

The multi-armed sampling framework provides a rigorous foundation for problems where matching a soft target distribution is crucial, such as:

• Entropy-regularized reinforcement learning, where exploration bonuses and stochastic (e.g., Boltzmann) policies are commonly used,
• Fine-tuning of pretrained models and reinforcement learning from human feedback (RLHF), where actions should be sampled in proportion to reward-modulated distributions,
• Design and evaluation of neural samplers for high-dimensional generative modeling and MCMC applications,
• Bayesian adaptive importance sampling and stochastic optimization with entropy-regularized objectives.

The insights formally rationalize why, in these regimes, explicit exploration is not central—in marked contrast to classic bandit learning—and provide precise regret guarantees for algorithms that operate in this sampling-focused paradigm.

7. Experimental Results and Empirical Verification

Empirical evaluation in the referenced work confirms the theoretical predictions. ASE, even with minimal exploration, outperforms explicit-exploration designs (such as DAISEE) in terms of convergence of the empirical action distribution to the target. The analysis of regret across TV, forward-KL, and reverse-KL distances demonstrates that simple regret decays rapidly and cumulative regret grows slowly, validating the claim that exploration is not required for optimality in sampling tasks. Transition in regret scaling with the temperature parameter $\beta$ further demonstrates the unification of sampling and bandit/optimization problems and the theoretical transitions observed.

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The multi-armed sampling framework, by formalizing objectives, regret measures, and algorithmic structure for distribution learning—distinct from reward maximization—deepens the theoretical foundation for a wide variety of applications where sampling, not optimization, is the end goal. It clarifies when and why exploration is (and is not) essential, making it a foundational reference point for future work in both sampling theory and practice (2507.10797).