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arxiv: 2605.24740 · v1 · pith:ECKEJ2PInew · submitted 2026-05-23 · 💻 cs.LG · cs.GT

Reinforcement Learning for Reachability: Guaranteeing Asymptotic Optimality

Amogh Palasamudram , Jakub Svoboda , Suguman Bansal , Krishnendu Chatterjee This is my paper

Pith reviewed 2026-06-30 14:12 UTC · model grok-4.3

classification 💻 cs.LG cs.GT
keywords reinforcement learningreachabilityPAC learningasymptotic optimalityMDP parametersconvergence
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The pith

Iteratively refining estimates of unknown MDP parameters allows reinforcement learning to achieve exact optimality for reachability in the limit.

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

The paper proposes an approach to reinforcement learning for reachability specifications that builds on PAC learning. While PAC methods guarantee near-optimal policies but require knowledge of internal MDP parameters such as the minimum transition probability, these parameters are typically unknown. The key insight is that these parameters can be iteratively refined and estimated with increasing accuracy. By satisfying the PAC conditions repeatedly with better estimates, the method ensures that exact optimality is achieved asymptotically. This provides deeper theoretical insights into how convergence occurs compared to prior work that only showed asymptotic convergence without detailing the dynamics.

Core claim

We argue that while these parameters are unknown in RL, they can be iteratively refined and estimated with increasing accuracy. By iteratively satisfying PAC conditions, we show that exact optimality can be achieved in the limit.

What carries the argument

Iterative refinement of MDP parameters within the PAC learning framework to satisfy guarantees with increasing accuracy until exact optimality holds in the limit.

If this is right

  • Exact optimality for reachability is achieved in the limit through the iterative process.
  • Convergence dynamics receive explicit theoretical explanation via successive PAC satisfaction.
  • Empirical evaluations on standard benchmarks confirm the predicted convergence behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same iterative estimation technique could extend to reachability variants with continuous state spaces.
  • Sample efficiency might improve if early loose estimates are used before tightening.
  • Links exist to online model learning methods where transition probabilities are updated during execution.

Load-bearing premise

Unknown MDP parameters can be iteratively refined and estimated with increasing accuracy while the PAC guarantees remain valid at each step.

What would settle it

A demonstration that parameter estimates fail to improve sufficiently over iterations or that PAC validity breaks during refinement would show the limit does not reach exact optimality.

Figures

Figures reproduced from arXiv: 2605.24740 by Amogh Palasamudram, Jakub Svoboda, Krishnendu Chatterjee, Suguman Bansal.

Figure 1
Figure 1. Figure 1: Simple MDP with EC start s0 ss T a, 0.5 a, 0.5 (s1, a), 1 (s2, a), 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance on Dining Philosophers benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. Figure 3a shows value bound convergence for Dining Philosophers (expected value [1, 1]): both U(s0) and L(s0) approach 1.0. Figure 3b compares theoretical sample size Nk versus practical implementation, showing similar con￾vergence profiles that validate our modifi… view at source ↗
Figure 4
Figure 4. Figure 4: Performance on the Consensus 2 benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance on the CSMA 2-2 benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 0.50. (a) Value Bounds vs. Stage k (b) Error vs. Stage k (c) Policy Accuracy vs. Stage k [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance on the Firewire Abst benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. (a) Value Bounds vs. Stage k (b) Error vs. Stage k (c) Policy Accuracy vs. Stage k [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Performance on the IJ 3 benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance on the IJ 10 benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. (a) Value Bounds vs. Stage k (b) Error vs. Stage k (c) Policy Accuracy vs. Stage k [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Performance on the Pacman benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 0.55. (a) Value Bounds vs. Stage k (b) Error vs. Stage k (c) Policy Accuracy vs. Stage k [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Performance on the Dining Philosophers benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance on the Rabin 3 benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 1.00. (a) Value Bounds vs. Stage k (b) Error vs. Stage k (c) Policy Accuracy vs. Stage k [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Performance on the Zeroconf benchmark. The values plotted are the median values over 10 trials. Optimal Policy Reachability Value: 0.00. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_12.png] view at source ↗
read the original abstract

Reinforcement learning (RL) for reachability specifications is fundamental in sequential decision-making, yet theoretical guarantees remain less explored. A recent work achieves asymptotic convergence to optimal policies. However, this approach provides limited insight into convergence dynamics. In this work, we present an alternative approach that provides deeper theoretical insights into convergence. Our approach builds on PAC learning with assumptions. PAC learning guarantees near-optimal policies with high confidence in finite time but requires knowing internal MDP parameters like minimum transition probability. We argue that while these parameters are unknown in RL, they can be iteratively refined and estimated with increasing accuracy. By iteratively satisfying PAC conditions, we show that exact optimality can be achieved in the limit. Empirical evaluations on standard benchmarks validate our theoretical insights into convergence dynamics.

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.

Referee Report

2 major / 0 minor

Summary. The paper claims that for RL with reachability specifications, an iterative procedure can achieve asymptotic optimality by starting from PAC-MDP bounds (which require known parameters such as the minimum transition probability p_min) and successively refining estimates of those parameters from data collected under the current policy; by satisfying the PAC conditions at each iteration with increasingly accurate estimates, the sequence of policies converges to the exact optimum in the limit, yielding deeper insight into convergence dynamics than prior asymptotic results.

Significance. If the derivation is free of circularity, the result would supply a concrete mechanism linking finite-time PAC guarantees to asymptotic optimality for reachability objectives, which is a non-trivial theoretical contribution. The manuscript supplies no machine-checked proofs, reproducible code, or explicit parameter-free derivations, so its primary value would rest on whether the iterative estimation argument is rigorously closed.

major comments (2)
  1. [Abstract] Abstract (and the theoretical argument summarized therein): the claim that PAC conditions can be iteratively satisfied while p_min is estimated on the fly is load-bearing for the asymptotic-optimality result, yet the abstract supplies no derivation showing how sample sizes are chosen to dominate the unknown true p_min at every finite iteration. An overestimate voids the high-probability guarantee; an underestimate may starve exploration needed to refine the estimate further. Without an explicit dominating schedule or a proof that the sequence of PAC policies still converges w.p. 1, the central claim remains unverified.
  2. [Abstract] The weakest assumption identified in the reader's report (unknown MDP parameters can be iteratively refined while PAC guarantees remain valid) is not discharged by any cited lemma or theorem in the provided text; the circularity risk noted by the skeptic therefore directly threatens the limit claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for pinpointing the need for a more explicit treatment of the iterative estimation procedure. The comments correctly identify that the abstract summarizes a claim whose supporting derivation requires additional formalization to rule out circularity. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the theoretical argument summarized therein): the claim that PAC conditions can be iteratively satisfied while p_min is estimated on the fly is load-bearing for the asymptotic-optimality result, yet the abstract supplies no derivation showing how sample sizes are chosen to dominate the unknown true p_min at every finite iteration. An overestimate voids the high-probability guarantee; an underestimate may starve exploration needed to refine the estimate further. Without an explicit dominating schedule or a proof that the sequence of PAC policies still converges w.p. 1, the central claim remains unverified.

    Authors: We agree that the abstract is too terse on this point. The current manuscript develops the iterative argument at a high level but does not supply an explicit schedule that selects sample sizes to dominate the unknown p_min at each finite step, nor a self-contained proof of almost-sure convergence of the policy sequence. We will add a dedicated lemma and subsection that (i) constructs a conservative, data-driven lower bound sequence for p_min, (ii) gives the corresponding sample-complexity schedule, and (iii) proves that the resulting policies converge to the true optimum with probability 1. revision: yes

  2. Referee: [Abstract] The weakest assumption identified in the reader's report (unknown MDP parameters can be iteratively refined while PAC guarantees remain valid) is not discharged by any cited lemma or theorem in the provided text; the circularity risk noted by the skeptic therefore directly threatens the limit claim.

    Authors: The manuscript presents the iterative-refinement argument narratively rather than via a cited lemma that explicitly rules out circularity. We accept that this leaves the central limit claim insufficiently grounded. In the revision we will insert a new lemma that shows the process is well-defined: at each iteration the data collected under the current PAC policy is sufficient to produce a strictly tighter estimate of p_min, allowing the next PAC condition to be satisfied with the updated bound, thereby closing the argument without circularity. revision: yes

Circularity Check

0 steps flagged

No circularity: iterative PAC refinement presented as independent argument

full rationale

The abstract describes building on PAC learning (which requires known parameters such as p_min) by arguing that unknown parameters can be iteratively refined with increasing accuracy, allowing exact optimality in the limit via repeated PAC satisfaction. No equations, self-citations, or explicit reductions are quoted that make the claimed optimality equivalent to its inputs by construction, nor is any fitted parameter renamed as a prediction. The derivation is presented as an extension of standard PAC guarantees with an additional iterative estimation step whose validity is asserted separately; this does not trigger any of the enumerated circularity patterns on the basis of the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard PAC-MDP assumption that finite-time near-optimality holds when parameters are known, plus the additional domain assumption that those parameters can be estimated iteratively without breaking the guarantees.

axioms (1)
  • domain assumption PAC learning guarantees near-optimal policies with high confidence in finite time but requires knowing internal MDP parameters like minimum transition probability.
    Explicitly stated in the abstract as the foundation that must be relaxed via iterative estimation.

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Reference graph

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