arxiv: 2605.11975 · v1 · submitted 2026-05-12 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Stochastic Minimum-Cost Reach-Avoid Reinforcement Learning

Jingduo Pan , Taoran Wu , Yiling Xue , Bai Xue

Authors on Pith no claims yet

Pith reviewed 2026-05-13 07:23 UTC · model grok-4.3

classification 💻 cs.LG

keywords stochastic reinforcement learningreach-avoid constraintssafe reinforcement learningprobability certificatesBellman operatorconstrained optimizationconvergence analysis

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The pith

Reach-avoid probability certificates turn stochastic safety constraints into a surrogate objective that reinforcement learners can optimize for minimum cost.

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

The paper develops a way for agents to minimize expected cumulative costs while still reaching a target set and avoiding an unsafe set with probability at least p in fully stochastic environments. It does so by defining reach-avoid probability certificates that label states from which the probabilistic specification remains satisfiable, then embedding those labels into a contraction mapping on the Bellman operator. The resulting formulation serves as a drop-in surrogate that lets standard value-based or policy-based algorithms handle the constraint without separate penalty or shielding terms. If the method succeeds, it produces policies whose cost performance improves while the probability of satisfying the reach-avoid spec stays above the required threshold. This addresses the common failure of existing safe RL techniques to enforce hard probabilistic specifications jointly with cost minimization during learning.

Core claim

The authors introduce reach-avoid probability certificates (RAPCs) that identify states from which the stochastic reach-avoid specification is satisfiable at the required probability level. They then construct a contraction-based Bellman formulation that uses the certificates as a surrogate for the constraint, integrate this formulation into reinforcement learning so that expected cost is minimized subject to the probabilistic requirement, and prove that the resulting algorithms converge almost surely to locally optimal policies with respect to the surrogate objective. MuJoCo experiments show that the learned policies incur lower costs and achieve higher reach-avoid satisfaction rates than a

What carries the argument

Reach-avoid probability certificates (RAPCs) together with a contraction-based Bellman operator that acts as a surrogate for integrating the probabilistic reach-avoid requirement into cost optimization.

If this is right

Policies obtained from the surrogate objective achieve strictly lower expected cost than baselines while maintaining the required reach-avoid probability.
The contraction property of the Bellman operator guarantees almost-sure convergence of both value iteration and policy iteration variants to a local optimum of the combined objective.
The same certificate construction extends directly to continuous control tasks, as demonstrated by improved performance on standard MuJoCo benchmarks.

Where Pith is reading between the lines

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

The certificate approach could be lifted to other probabilistic temporal specifications by replacing the reach-avoid sets with appropriate winning regions.
In deployed systems the learned surrogate may reduce reliance on runtime shielding because the probability guarantee is already baked into the value function.
Accuracy of the certificates under function approximation remains the main practical bottleneck; tighter bounds on approximation error would strengthen the convergence claim.

Load-bearing premise

Reach-avoid probability certificates can be computed or approximated accurately enough during learning to serve as a reliable surrogate for the true probabilistic constraint in stochastic environments.

What would settle it

Repeated evaluation of the learned policy in held-out stochastic environments where the empirical frequency of satisfying the reach-avoid specification falls below the prescribed probability p, or where the achieved cost exceeds that of a simple baseline that ignores the constraint.

Figures

Figures reproduced from arXiv: 2605.11975 by Bai Xue, Jingduo Pan, Taoran Wu, Yiling Xue.

**Figure 1.** Figure 1: illustrates the deterministic benchmark environments. Across PointGoal and FixedWing, RAPCPO consistently achieves the lowest cumulative cost while maintaining higher or comparable reach rates relative to P P Oβ, Sauté, CPPO, and RESPO ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: Cumulative cost and reach rate on deterministic benchmarks. RAPCPO achieves lower cost with competitive or higher reach rates compared to baselines [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 5.** Figure 5: Cumulative cost and reach rate on stochastic benchmarks. RAPCPO achieves the best cost-success trade-off among all methods. 6.3. Ablation Study: Compensation Factor ϕ π γ and Hyperparameter p We isolate the auxiliary compensation factor on FrozenLake (Brockman et al., 2016), where the objective is to maximize the probability of safely reaching the goal [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 3.** Figure 3: Pendulum results. RAPCPO achieves lower cumulative cost by favoring longer, smoother trajectories. 6.2. Stochastic Minimum-Cost Reach-Avoid [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 6.** Figure 6: Impact of the compensation factor on cost and reach rate across tasks. Then, when both methods employ the compensation factor, we compare the fixed-γ Bellman formulation (8) with the enhanced Bellman formulation (9) for minimum costs reachavoid problem. Under the same iteration budget, the fixed-γ Bellman formulation (8) suffers from severe reward sparsity and achieves low reach rates. In contrast, the en… view at source ↗

**Figure 4.** Figure 4: illustrates the stochastic benchmark environments. Under action noise, RAPCPO consistently outperforms all baselines on both Safety Hopper and Safety HalfCheetah ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 7.** Figure 7: Ablation on the hyperparameter p in the Safety Hopper environment. Right: reach rate as a function of p. Left: additional cumulative cost as a function of p. The cost is plotted on a logarithmic scale due to its wide dynamic range. 7. Conclusion and Limitations This paper addressed the minimum-cost reach-avoid reinforcement learning problem in stochastic environments. We first proposed the Reach-Avoid Pr… view at source ↗

**Figure 8.** Figure 8: Effect of the auxiliary compensation factor on reach–avoid probability estimation. The compensation factor significantly reduces conservativeness. • RESPO: https://github.com/milanganai/milanganai.github.io/tree/main/ NeurIPS2023/code (No license) • CPPO: https://github.com/yingchengyang/CPPO (No license) • Sauté: https://github.com/huawei-noah/HEBO/tree/master/SIMMER (No license) D.4. Hyperparameters In e… view at source ↗

read the original abstract

We study stochastic minimum-cost reach-avoid reinforcement learning, where an agent must satisfy a reach-avoid specification with probability at least $p$ while minimizing expected cumulative costs in stochastic environments. Existing safe and constrained reinforcement learning methods typically fail to jointly enforce probabilistic reach-avoid constraints and optimize cost in the learning setting in stochastic environments. To address this challenge, we introduce reach-avoid probability certificates (RAPCs), which identify states from which stochastic reach-avoid constraints are satisfiable. Building on RAPCs, we develop a contraction-based Bellman formulation that serves as a principled surrogate for integrating reach-avoid considerations into reinforcement learning, enabling cost optimization under probabilistic constraints. We establish almost sure convergence of the proposed algorithms to locally optimal policies with respect to the resulting objective. Experiments in the MuJoCo simulator demonstrate improved cost performance and consistently higher reach-avoid satisfaction rates.

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

RAPCs give a fresh surrogate for joint probabilistic reach-avoid and cost minimization in stochastic RL, but convergence is only shown for the surrogate with no bound on how approximation error affects the real constraint.

read the letter

The paper's main contribution is a new way to handle probabilistic reach-avoid constraints together with cost minimization in stochastic reinforcement learning. They introduce reach-avoid probability certificates that flag states where the constraint is feasible, then build a contraction-based Bellman operator around them to create a surrogate objective for learning. This pairing of certificates with the contraction formulation does not appear in the earlier constrained RL literature they cite, so that part is new. The experiments on MuJoCo tasks report better cost performance while keeping higher rates of satisfying the reach-avoid specs, which suggests the approach is at least practically workable. The main weakness is that the convergence is only to a local optimum of the surrogate. In real stochastic environments the certificates cannot be computed exactly and must be approximated, yet there is no analysis showing how approximation error affects the true probability of satisfying the original constraint. That leaves the safety claim resting on an unverified link. The work is aimed at researchers in safe and constrained reinforcement learning who deal with reach-avoid specifications. Someone looking for algorithmic ideas to combine probabilistic safety with cost optimization could get value from the RAPC construction, even if they have to fill in the approximation details themselves. I would recommend sending it for peer review. The idea is concrete and the experimental setup is standard, so referees can evaluate whether the missing bounds can be supplied or whether the surrogate is sufficient in practice.

Referee Report

2 major / 1 minor

Summary. The paper studies stochastic minimum-cost reach-avoid reinforcement learning, where an agent must satisfy a reach-avoid specification with probability at least p while minimizing expected costs. It introduces reach-avoid probability certificates (RAPCs) to identify states satisfying the probabilistic constraints, develops a contraction-based Bellman formulation as a surrogate objective integrating these constraints, establishes almost sure convergence of the proposed algorithms to locally optimal policies with respect to the surrogate, and reports improved cost performance with higher reach-avoid satisfaction rates in MuJoCo experiments.

Significance. If the surrogate objective reliably enforces the original probabilistic constraints and the convergence result holds under realistic approximations, the work would advance constrained RL by offering a principled contraction-based approach to joint cost optimization and probabilistic safety in stochastic settings, addressing limitations of existing methods. The MuJoCo results indicate potential practical utility, but significance depends on closing the gap between surrogate optimality and true constraint satisfaction.

major comments (2)

[Abstract and theoretical development] Abstract and theoretical development: The central claim is almost sure convergence to locally optimal policies 'with respect to the resulting objective' obtained from the contraction-based Bellman operator on RAPCs. However, RAPCs cannot be computed exactly in continuous stochastic environments and require approximation (via sampling or function approximation). No quantitative bound is derived on how approximation error in RAPC values propagates through the Bellman operator to the true reach-avoid probability of the converged policy. Without such an error-propagation analysis, convergence to a local optimum of the surrogate does not establish satisfaction of the original p-threshold constraint.
[Approach and assumptions] The weakest assumption in the approach is that RAPCs can be approximated accurately enough during learning to serve as a reliable surrogate. The manuscript provides no analysis or experiments quantifying the sensitivity of the learned policy's true reach-avoid probability to RAPC approximation error, which is load-bearing for translating the theoretical guarantee into a practical safety guarantee.

minor comments (1)

[Abstract] The abstract introduces RAPCs without a one-sentence intuition or definition; adding this would improve accessibility for readers unfamiliar with reach-avoid specifications.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed comments, which help clarify the scope of our contributions. We address each major point below, emphasizing that our theoretical results concern convergence to the surrogate objective while experiments provide empirical evidence of practical performance.

read point-by-point responses

Referee: [Abstract and theoretical development] Abstract and theoretical development: The central claim is almost sure convergence to locally optimal policies 'with respect to the resulting objective' obtained from the contraction-based Bellman operator on RAPCs. However, RAPCs cannot be computed exactly in continuous stochastic environments and require approximation (via sampling or function approximation). No quantitative bound is derived on how approximation error in RAPC values propagates through the Bellman operator to the true reach-avoid probability of the converged policy. Without such an error-propagation analysis, convergence to a local optimum of the surrogate does not establish satisfaction of the original p-threshold constraint.

Authors: We agree that a quantitative error-propagation bound would provide a stronger link between surrogate optimality and the original probabilistic constraint. Our manuscript establishes almost-sure convergence specifically to locally optimal policies with respect to the surrogate objective induced by the contraction Bellman operator on RAPCs; this is the precise statement of the theoretical result. The surrogate is constructed so that, under exact RAPCs, its optima satisfy the reach-avoid specification. We do not derive an explicit propagation bound because obtaining tight, non-vacuous bounds in general continuous stochastic MDPs remains an open technical challenge. The contraction property nevertheless guarantees stability of the iterates. Our MuJoCo experiments, conducted under realistic function approximation, show that the learned policies consistently achieve higher reach-avoid satisfaction rates than baselines while improving cost. In revision we will add a dedicated paragraph in the discussion section that explicitly states the scope of the convergence guarantee and identifies error-bound analysis as an important direction for future work. revision: partial
Referee: [Approach and assumptions] The weakest assumption in the approach is that RAPCs can be approximated accurately enough during learning to serve as a reliable surrogate. The manuscript provides no analysis or experiments quantifying the sensitivity of the learned policy's true reach-avoid probability to RAPC approximation error, which is load-bearing for translating the theoretical guarantee into a practical safety guarantee.

Authors: We acknowledge that the reliability of RAPC approximations is a central modeling assumption. The current manuscript does not contain a dedicated sensitivity study that systematically varies RAPC estimation error and measures the resulting change in true reach-avoid probability. The design of the contraction-based surrogate, however, embeds the probabilistic certificates directly into the objective, and the empirical evaluation on stochastic MuJoCo tasks demonstrates that the resulting policies maintain elevated constraint satisfaction rates under the function-approximation regime used in practice. To strengthen the practical interpretation, we will add a new subsection in the experiments that reports results under controlled variations of RAPC sampling budget (hence approximation quality) and tabulates the observed true reach-avoid probabilities. revision: yes

Circularity Check

0 steps flagged

No circularity; convergence derived from standard contraction on explicitly constructed surrogate

full rationale

The paper defines RAPCs as states satisfying the probabilistic reach-avoid condition, then constructs a contraction-based Bellman operator as an explicit surrogate objective. Almost-sure convergence is claimed for policies optimal w.r.t. this surrogate via standard arguments. No equation reduces the claimed result to a fitted parameter renamed as prediction, no self-citation is load-bearing for the core derivation, and the surrogate is not defined circularly in terms of its own outputs. The approximation issue for RAPCs affects correctness of constraint satisfaction but does not create a definitional loop in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the method appears to rest on standard MDP assumptions plus the new certificates.

axioms (1)

domain assumption The environment is a stochastic Markov decision process whose transition probabilities allow well-defined reach-avoid probabilities.
Required for the definition of RAPCs and the probabilistic constraint.

invented entities (1)

Reach-Avoid Probability Certificates (RAPCs) no independent evidence
purpose: Identify states from which the probabilistic reach-avoid specification is satisfiable.
New construct introduced to make the constraint tractable inside the learning loop.

pith-pipeline@v0.9.0 · 5444 in / 1265 out tokens · 47391 ms · 2026-05-13T07:23:59.690351+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Define the operator B^π[V](x) := max{ h(x), min{ g(x), γ E[V(x')] } } ... V(x)=B^π[V](x) admits a unique bounded solution V^π_{g,h} and B^π is a γ-contraction
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

surrogate optimization problem (RAPCRL): min_π E[ V^π_c(x) 1_{Xp} + V^π_{g,h}(x)/ϕ^π_γ(x) 1_{X∖Xp} ]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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Proofs A.1

11 Stochastic Minimum-Cost Reach-Avoid Reinforcement Learning A. Proofs A.1. The proof for lemma 4.4 Proof.LetB(R n)denote the space of all bounded functions mapping fromR n toR. That is, B(Rn) = V:R n →R|sup x∈Rn |V(x)|<∞ . It is a standard result that the space B(Rn) of bounded functions, endowed with the supremum norm ∥V∥ ∞ = supx∈Rn |V(x)|, constitute...

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Given thatγ∈(0,1), the operatorB π is a contraction mapping on the Banach space(B(R n),∥ · ∥ ∞)

=γ∥V 1 −V 2∥∞.(Expectation of constant is constant) Since the inequality |Bπ[V1](x)−B π[V2](x)| ≤γ∥V 1 −V 2∥∞ holds for all x∈R n, we can take the supremum over x on the left side: ∥Bπ[V1]−B π[V2]∥∞ = sup x∈Rn |Bπ[V1](x)−B π[V2](x)| ≤γ∥V 1 −V 2∥∞. Given thatγ∈(0,1), the operatorB π is a contraction mapping on the Banach space(B(R n),∥ · ∥ ∞). By the Banac...

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Definition B.1(Absorbing Markov Decision Process).The Absorbing Markov Decision Process is defined on f with an added absorbing states

that facilitates the computation of gradients. Definition B.1(Absorbing Markov Decision Process).The Absorbing Markov Decision Process is defined on f with an added absorbing states. We define the transition functionf ′ r with the absorbing state as f ′ r(x, a) = ( f(x, a),ifg(x)≥V π g,h(x)≥h(x), s,otherwise. (34) Denote byd ′ π(x)the stationary distribut...

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Recognizing this, our method simplifies the formulation by defining g directly on the original state x, thereby avoiding the need for state augmentation

employs state augmentation, defining g based on an augmented state ˜x, the function g(˜x)in their work effectively depends only on the original state component x. Recognizing this, our method simplifies the formulation by defining g directly on the original state x, thereby avoiding the need for state augmentation. Apart from this change, the remaining ex...

work page 2024