arxiv: 2605.03921 · v1 · submitted 2026-05-05 · 💻 cs.LG · stat.ML

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Optimal Posterior Sampling for Policy Identification in Tabular Markov Decision Processes

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Pith reviewed 2026-05-07 16:17 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords policy identificationMarkov decision processesposterior samplingPAC identificationsample complexitytabular MDPsasymptotic optimalityreinforcement learning

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The pith

Posterior sampling combined with online learning achieves asymptotically optimal sample complexity for best-policy identification in tabular MDPs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a randomized algorithm for identifying the best policy in finite-horizon episodic Markov Decision Processes that combines posterior sampling with an online learning method to direct exploration. This method reaches asymptotic optimality in both sample complexity and the rate at which the posterior concentrates on the true model. A sympathetic reader would care because it provides guarantees that hold even as the allowed failure probability shrinks, while keeping computation per episode linear in the number of states, actions, and horizon length, comparable to standard planning methods.

Core claim

By integrating posterior sampling for model estimation with an online learning subroutine that guides which episodes to sample, the algorithm attains the optimal sample complexity for (ε, δ)-PAC policy identification and matches the posterior contraction rates of ideal estimators, all while executing in O(S² A H) time per episode.

What carries the argument

Posterior sampling guided by an online learning algorithm for exploration in the MDP, which allows efficient and optimal identification.

Load-bearing premise

The analysis requires that posterior samples can be drawn exactly and that the online learning component delivers regret bounds free of additional factors depending on the failure probability.

What would settle it

Running the algorithm on a simple MDP and measuring whether the total samples required scales without a polynomial dependence on log(1/δ) as δ decreases would test the claim; deviation from the predicted asymptotic rate would falsify it.

Figures

Figures reproduced from arXiv: 2605.03921 by Cyrille Kone, Kevin Jamieson.

**Figure 2.** Figure 2: Custom MOCA environment where reaching the optimal policy requires exploratory steps involving view at source ↗

**Figure 3.** Figure 3: The H-state CombinationLock environment. Each nonterminal state offers A actions. The unique optimal action at state sl advances the agent to sl+1 with probability 1 − ε (solid black arrow) and returns it to the initial state sinit with probability ε (dashed arrow); any suboptimal action (red arrow) sends the agent back to sinit with probability 1. The terminal state sH is absorbing and yields a reward 1 (… view at source ↗

**Figure 4.** Figure 4: Performance factor averaged on 64 runs with 1 view at source ↗

**Figure 5.** Figure 5: Number of samples for 1 finding an alternative model averaged on 100 runs. 0 2000 4000 6000 8000 10000 episode 100 101 102 nbr. of samples PIPS view at source ↗

**Figure 8.** Figure 8: The L-state RiverSwim environment. Each state allows two actions: left (red arrows) and right (black arrows). Rewards are placed at the two ends of the chain. Transition probabilities are annotated above the corresponding arrows view at source ↗

read the original abstract

We study the $(\varepsilon, \delta)$-PAC policy identification problem in finite-horizon episodic Markov Decision Processes. Existing approaches provide finite-time guarantees for approximate settings ($\varepsilon>0$) but suffer from high computational cost, rendering them hard to implement, and also suffer from suboptimal dependence on $\log(1/\delta)$. We propose a randomized and computationally efficient algorithm for best policy identification that combines posterior sampling with an online learning algorithm to guide exploration in the MDP. Our method achieves asymptotic optimality in sample complexity, also in terms of posterior contraction rate, and runs in $O(S^2AH)$ per episode, matching standard model-based approaches. Unlike prior algorithms such as MOCA and PEDEL, our guarantees remain meaningful in the asymptotic regime and avoid sub-optimal polynomial dependence on $\log(1/\delta)$. Our results provide both theoretical insights and practical tools for efficient policy identification in tabular MDPs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims an asymptotically optimal posterior sampling algorithm for best policy identification in tabular MDPs that dodges the usual polynomial log(1/δ) costs while matching standard O(S²AH) runtime.

read the letter

The core contribution is a randomized method that pairs posterior sampling over the MDP transitions with an online learning subroutine to steer exploration. It targets the (ε, δ)-PAC setting in finite-horizon episodic tabular MDPs and asserts asymptotic optimality in sample complexity plus posterior contraction rate, plus per-episode time that lines up with ordinary model-based planning. The authors contrast this with MOCA and PEDEL, noting that their guarantees stay useful as δ shrinks instead of picking up extra polynomial factors in log(1/δ).

Referee Report

2 major / 1 minor

Summary. The paper proposes a randomized algorithm for (ε, δ)-PAC policy identification in finite-horizon episodic tabular MDPs. It combines posterior sampling with an online learning subroutine to guide exploration, claiming asymptotic optimality in sample complexity (including posterior contraction rate), per-episode runtime O(S²AH), and avoidance of the suboptimal polynomial dependence on log(1/δ) that affects prior methods such as MOCA and PEDEL.

Significance. If the guarantees hold, the work would advance the field by delivering both asymptotically optimal sample complexity and practical computational efficiency for policy identification, while providing theoretical insights into the integration of posterior sampling with online learning in MDPs.

major comments (2)

[Main theorem and regret invocation] The central claim of asymptotic optimality and avoidance of polynomial log(1/δ) dependence rests on the online learning subroutine delivering regret bounds free of extra logarithmic factors in 1/δ. The manuscript must explicitly state the regret bound used (e.g., in the main theorem) and show how it integrates with the stopping-time analysis without reintroducing δ-dependent terms, as standard UCB/EXP3-style bounds typically contain such factors.
[Posterior sampling analysis] The optimality claim with respect to posterior contraction rate relies on standard contraction properties of the MDP posterior without a self-contained derivation or explicit handling of approximation errors in the sampling process. This needs to be derived or verified in the analysis section to confirm the rate remains meaningful in the asymptotic regime.

minor comments (1)

[Abstract] The abstract states the claims clearly but could briefly note the key technical step (the δ-independent regret integration) that distinguishes the approach from MOCA/PEDEL.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, outlining the revisions we will implement.

read point-by-point responses

Referee: [Main theorem and regret invocation] The central claim of asymptotic optimality and avoidance of polynomial log(1/δ) dependence rests on the online learning subroutine delivering regret bounds free of extra logarithmic factors in 1/δ. The manuscript must explicitly state the regret bound used (e.g., in the main theorem) and show how it integrates with the stopping-time analysis without reintroducing δ-dependent terms, as standard UCB/EXP3-style bounds typically contain such factors.

Authors: We agree that greater explicitness is warranted. In the revised manuscript we will state the precise regret bound of the online learning subroutine directly in the statement of the main theorem. We will also augment the stopping-time analysis to show, step by step, that the chosen bound combines with the posterior-sampling stopping rule in a manner that keeps all δ-dependent factors from becoming polynomial; the argument relies on the fact that the online-learning regret is invoked only on an asymptotically vanishing fraction of episodes, which is already implicit in our current proof but will now be spelled out. revision: yes
Referee: [Posterior sampling analysis] The optimality claim with respect to posterior contraction rate relies on standard contraction properties of the MDP posterior without a self-contained derivation or explicit handling of approximation errors in the sampling process. This needs to be derived or verified in the analysis section to confirm the rate remains meaningful in the asymptotic regime.

Authors: We will add a self-contained derivation of the posterior contraction rate (including the handling of sampling approximation errors) to the analysis section. The derivation will follow the standard martingale arguments for Dirichlet posteriors but will be written out in full so that the asymptotic validity of the contraction rate is verified directly within the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained against external benchmarks

full rationale

The paper's central claim is that combining posterior sampling with an online learning subroutine yields asymptotic optimality in sample complexity and posterior contraction rate for (ε,δ)-PAC policy identification, with O(S²AH) per-episode runtime and no polynomial log(1/δ) blowup. This rests on two external standard results: (1) known posterior contraction rates for tabular MDPs under exact sampling, and (2) regret bounds from a generic online learning subroutine that are assumed to be free of extra δ-dependent logarithmic factors. Neither is derived inside the paper via self-definition, parameter fitting, or a load-bearing self-citation chain; both are invoked as black-box inputs whose validity is independent of the present work. No equations reduce a claimed prediction to a fitted input by construction, no uniqueness theorem is imported from the authors' prior papers, and no ansatz is smuggled via citation. The derivation chain therefore remains non-circular even though it depends on external subroutine properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard finite MDP assumptions (finite states, actions, horizon) and on the existence of an online learning algorithm with suitable regret bounds; no new entities are introduced and no parameters appear to be fitted to data.

axioms (2)

domain assumption The MDP is finite-horizon episodic with known horizon H and finite state-action space.
Stated in the problem setup for tabular MDPs.
domain assumption Posterior sampling over transition and reward models can be performed exactly.
Implicit in the algorithm description and required for the contraction-rate claim.

pith-pipeline@v0.9.0 · 5448 in / 1480 out tokens · 46104 ms · 2026-05-07T16:17:34.109066+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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