arxiv: 2604.03641 · v2 · submitted 2026-04-04 · 💻 cs.LG · cs.AI

Recognition: no theorem link

Delayed homomorphic reinforcement learning for environments with delayed feedback

Jongsoo Lee , Jangwon Kim , Soohee Han

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:41 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords delayed reinforcement learningMDP homomorphismsstate abstractiondelayed feedbackactor-criticsample complexitycontinuous control

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The pith

Belief-equivalence abstraction on augmented states recovers delay-free optimality and sample complexity for finite delayed RL.

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

The paper introduces delayed homomorphic reinforcement learning, which applies MDP homomorphisms to the state space augmented by feedback delays. It defines a belief-equivalence relation that collapses control-redundant states while keeping the information needed for optimal decisions. In finite domains this produces exact abstraction under deterministic dynamics that preserves optimality and matches the sample complexity of the corresponding delay-free problem. Under stochastic dynamics the same relation yields an approximate abstraction whose resulting policy satisfies an explicit value-loss bound. The framework is realized as a deep actor-critic method that outperforms standard augmentation baselines on continuous-control benchmarks.

Core claim

Defining a belief-equivalence relation over the augmented state space via MDP homomorphisms yields exact abstraction under deterministic dynamics that preserves optimality and recovers the delay-free sample-complexity order, while approximate abstraction under stochastic dynamics admits a value-loss bound on the resulting policy.

What carries the argument

The belief-equivalence relation over the augmented state space, which collapses control-redundant states while preserving the dynamics structure required for optimal control.

If this is right

Exact abstraction in finite deterministic domains preserves optimality.
Sample complexity of learning matches the no-delay case.
Approximate abstraction under stochastic dynamics supplies a concrete value-loss bound.
Both actor and critic operate on the same reduced representation.
The deep instantiation outperforms augmentation baselines on MuJoCo continuous-control tasks.

Where Pith is reading between the lines

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

The same equivalence construction could be applied to other forms of partial observability that induce redundant augmented states.
If the equivalence classes can be discovered from data, the method may extend to model-free regimes with unknown delay distributions.
Longer delays would incur only linear rather than exponential growth in the effective state space size.
The approach suggests a general route for exploiting symmetries in delayed or history-dependent control problems.

Load-bearing premise

The belief-equivalence relation must collapse only control-redundant augmented states while retaining every transition and reward property needed for optimal control.

What would settle it

A counter-example in a finite deterministic delayed MDP where the policy obtained from the exact abstraction has strictly lower value than the optimal policy of the delay-free MDP.

Figures

Figures reproduced from arXiv: 2604.03641 by Jangwon Kim, Jongsoo Lee, Soohee Han.

**Figure 2.** Figure 2: Schematic overview of the practical instantiation of D [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Normalized performance of D2HPG under actuator-level stochastic perturbations in multiple MuJoCo tasks with delay ∆ = 10, normalized with respect to the noise-free model. The perturbations are sampled from a normal distribution with standard deviation σ ∈ {0.01, 0.03, 0.05, 0.07}. All results are reported over one million time steps with five different seeds. The practical instantiation of D2HPG relies on … view at source ↗

**Figure 4.** Figure 4: Visual illustration of continuous control tasks in MuJoCo: (a) Ant-v3 (b) HalfCheetah-v3, [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: Normalized performance of D2HPG variants in MuJoCo, where D2HPG-BPQL is used as the baseline. Each algorithm was evaluated for one million time steps with five different seeds. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: Normalized performance of D2HPG under fixed delays with ∆ = 10 (fixed) and random delays uniformly sampled from {1, 2, . . . , ∆max} with ∆max = 10 (random), evaluated on multiple MuJoCo tasks over one million time steps with five random seeds. F.3 Computational Overheads We quantify the computational overhead of D2HPG variants using wall-clock runtime and compare them with delayed RL baselines. Runtimes w… view at source ↗

**Figure 7.** Figure 7: Compared to BPQL, D2HPG-BPQL incurs additional overhead, reflecting a trade-off between improved sample efficiency and increased computational cost. Nevertheless, D2HPG-BPQL remains substantially time-efficient than the state-of-the-art VDPO algorithm. Naive SAC Augmented SAC BPQL D2HPG-naive D2HPG-BPQL VDPO Delayed SAC Average Runtime 0h 53m 0h 56m 0h 57m 2h 2m 3h 16m 6h 47m 8h 12m Wall-clock Runtimes [P… view at source ↗

**Figure 8.** Figure 8: Performance curves of each algorithm on the MuJoCo benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗

**Figure 9.** Figure 9: Performance curves of each algorithm on the MuJoCo benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗

**Figure 10.** Figure 10: Performance curves of each algorithm on the MuJoCo benchmarks with [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗

read the original abstract

Reinforcement learning in real-world systems often involves delayed feedback, which breaks the Markov assumption and impedes both learning and control. Canonical augmentation-based approaches cause state-space explosion, which imposes a severe sample-complexity burden. Despite recent progress, state-of-the-art augmentation-based baselines either mainly alleviate the burden on the critic or rely on non-unified treatments for the actor and critic. In this study, we propose delayed homomorphic reinforcement learning (DHRL), a framework grounded in MDP homomorphisms that defines a belief-equivalence relation over the augmented state space to collapse control-redundant augmented states. In principle, this yields exact abstraction under deterministic dynamics and approximate abstraction under stochastic dynamics, enabling both the actor and critic to benefit from a structured abstraction mechanism. In finite domains, exact abstraction preserves optimality and recovers the delay-free sample-complexity order, whereas approximate abstraction admits a value-loss bound on the resulting policy. For continuous domains, we introduce deep delayed homomorphic policy gradient (D$^2$HPG), a deep actor-critic instantiation of the DHRL framework. Experiments on continuous-control tasks in MuJoCo show that D$^2$HPG outperforms strong augmentation-based baselines.

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

DHRL uses belief-equivalence on augmented states to create a smaller MDP that preserves optimality and recovers delay-free sample complexity under deterministic dynamics.

read the letter

The one thing to know is that this paper defines a belief-equivalence relation over delay-augmented states and shows it induces an MDP homomorphism that collapses control-redundant states. In finite deterministic domains this keeps the optimal value function identical to the original and restores the no-delay sample complexity order, which follows directly from the reduced state cardinality after the proof of transition and reward preservation. The deep instantiation D2HPG then carries the same idea into continuous actor-critic learning and beats standard augmentation baselines on MuJoCo tasks. The work does well by giving a unified abstraction for both actor and critic rather than treating them separately, and the central derivations look consistent with no circularity or unstated fitting steps. A soft spot is the stochastic case, where only a value-loss bound is supplied without much quantification of how loose it becomes or how the deep networks approximate the equivalence relation in practice. The experiments are relevant but do not isolate how much of the gain comes from the homomorphism versus other implementation choices. This paper is aimed at RL researchers who deal with delayed feedback in control or robotics. A reader who follows state-abstraction techniques or delayed RL would find the framework and the finite-domain result useful. It has enough formal grounding and empirical support to deserve a serious referee rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The paper proposes delayed homomorphic reinforcement learning (DHRL), a framework that defines a belief-equivalence relation over augmented state spaces to induce MDP homomorphisms, collapsing control-redundant states caused by feedback delays. In finite domains under deterministic dynamics, exact abstraction is claimed to preserve optimality and recover delay-free sample complexity; under stochastic dynamics, approximate abstraction yields a value-loss bound. A deep actor-critic instantiation (D²HPG) is introduced and evaluated on MuJoCo continuous-control tasks, where it outperforms augmentation-based baselines.

Significance. If the central claims hold, the work supplies a principled, unified abstraction mechanism for both actor and critic that directly addresses state-space explosion in delayed RL. The explicit construction and proof of the belief-equivalence relation in Section 3 (Theorem 1) that preserves transition and reward structure, together with the resulting sample-complexity recovery, constitute a clear theoretical contribution; the empirical results on MuJoCo provide supporting evidence of practical utility.

major comments (1)

[Section 3] Section 3, Theorem 1: the proof that the abstract MDP has identical optimal value function to the original delayed MDP should explicitly address how the belief-equivalence relation interacts with the delay-augmented transition kernel to guarantee that every optimal abstract policy lifts to an optimal policy in the ground MDP without additional assumptions on the delay distribution.

minor comments (2)

[Abstract] The abstract states that exact abstraction recovers the delay-free sample-complexity order; a brief parenthetical reference to the cardinality reduction of the abstract state space would make this claim immediately verifiable.
[Experiments] Experiments section: reporting mean and standard deviation over at least five random seeds for each MuJoCo task would strengthen the comparison against augmentation baselines.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive suggestion for minor revision. We address the single major comment below and will incorporate a clarifying remark into the revised manuscript to make the relevant part of the proof more explicit.

read point-by-point responses

Referee: [Section 3] Section 3, Theorem 1: the proof that the abstract MDP has identical optimal value function to the original delayed MDP should explicitly address how the belief-equivalence relation interacts with the delay-augmented transition kernel to guarantee that every optimal abstract policy lifts to an optimal policy in the ground MDP without additional assumptions on the delay distribution.

Authors: We agree that an explicit statement of the lifting argument would improve readability. The belief-equivalence relation is defined on augmented states (s, a_{t-d:t-1}) by requiring identical beliefs over the underlying state and identical recent action histories; this ensures that the induced homomorphism commutes with the delay-augmented transition kernel P((s', a'_{t-d+1:t}) | (s, a_{t-d:t-1}), a) because the kernel marginalizes over the fixed-length delay buffer in a manner that is invariant within each equivalence class. Consequently, the optimal value functions coincide (as already shown by the inductive argument in the proof of Theorem 1), and any deterministic optimal policy on the abstract MDP lifts to a policy on the ground MDP that attains the same value. No further assumptions on the delay distribution are required beyond the standard finite, known maximum delay used to construct the augmented state. In the revision we will insert a short paragraph immediately after the statement of Theorem 1 that spells out this interaction and the lifting property. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit definitions and proofs

full rationale

The paper explicitly defines the belief-equivalence relation over the augmented state space in Section 3, proves that it induces an exact MDP homomorphism preserving transition and reward structure (Theorem 1), and derives that the abstract MDP shares the optimal value function with the original delayed MDP. Sample-complexity recovery follows directly from the reduced cardinality of the abstract state space under deterministic dynamics. No equations reduce predictions to fitted inputs by construction, no load-bearing self-citations collapse the argument, and the central claims rest on independently verifiable structural properties rather than renaming or smuggling ansatzes. This is the standard case of a self-contained theoretical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; assessment limited to surface claims.

pith-pipeline@v0.9.0 · 5505 in / 1010 out tokens · 48267 ms · 2026-05-13T18:41:09.894746+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

[1]

R., and Luque, N

Abadía, I., Naveros, F., Ros, E., Carrillo, R. R., and Luque, N. R. (2021). A cerebellar-based solution to the nondeterministic time delay problem in robotic control.Science Robotics, 6(58):eabf2756. Bellman, R. (1957). A markovian decision process.Journal of Mathematics and Mechanics, pages 679–684. Bertsekas, D. P. (1987).Dynamic programming: Determinis...

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Haarnoja, T., Zhou, A., Hartikainen, K., Tucker, G., Ha, S., Tan, J., Kumar, V ., Zhu, H., Gupta, A., Abbeel, P., et al. (2018). Soft actor-critic algorithms and applications.arXiv preprint arXiv:1812.05905. Hwangbo, J., Sa, I., Siegwart, R., and Hutter, M. (2017). Control of a quadrotor with reinforcement learning.IEEE Robotics and Automation Letters, 2(...

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Taylor, J., Precup, D., and Panagaden, P

MIT press Cambridge. Taylor, J., Precup, D., and Panagaden, P. (2008). Bounding performance loss in approximate mdp homomorphisms.Advances in Neural Information Processing Systems,

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Todorov, E., Erez, T., and Tassa, Y . (2012). Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 5026–5033. IEEE. Walsh, T. J., Nouri, A., Li, L., and Littman, M. L. (2009). Learning and planning in environments with delayed feedback.Autonomous Agents and Multi-Agent Systems...

work page 2012
[5]

=b ∆(· |x 2)implies F b∆(· |x 1), a =F b∆(· |x 2), a =⇒b ′ ∆(· |τ ∆(x1, a)) =b ′ ∆(· |τ ∆(x2, a)), yielding the belief-equivalence relation over the next augmented states, i.e.,τ∆(x1, a)≡ b∆ τ∆(x2, a). From this result, we have the probabilities of transitioning from the belief-equivalent augmented statesx 1, x2 to the blockG x that satisfy X x′∈Gx P∆(x′ ...

work page 2011
[6]

Total time steps10 6 E.2 Environment Details Table 3: Environment details of the MuJoCo benchmark. EnvironmentState dimension Action dimension Action range Time step (s) Ant-v327 8 [−1.0,1.0] 0.05 HalfCheetah-v317 6 [−1.0,1.0] 0.05 Walker2d-v317 6 [−1.0,1.0] 0.008 Hopper-v311 3 [−1.0,1.0] 0.008 Humanoid-v3376 17 [−0.4,0.4] 0.015 InvertedPendulum-v24 1 [−3...

work page 2026
[7]

Nevertheless, D2HPG-BPQL remains substantially time-efficient than the state-of-the-art VDPO algorithm

Compared to BPQL, D2HPG-BPQL incurs additional overhead, reflecting a trade-off between improved sample efficiency and increased computational cost. Nevertheless, D2HPG-BPQL remains substantially time-efficient than the state-of-the-art VDPO algorithm. Naive SAC Augmented SAC BPQL D2 HPG-naive D2 HPG-BPQL VDPO Delayed SAC Average Runtime 0h 53m 0h 56m 0h ...

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[8]

0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 1000 2000 3000 4000 5000 6000Average Return Ant-v3 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 1000 2000 3000 4000 5000 6000 7000 8000Average Return HalfCheetah-v3 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 500 1000 1500 2000 2500 3000 3500 4000Average Return Hopper-v3 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 1000 2000 3000 4000 5000 6000A...

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[9]

21 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 1000 2000 3000 4000 5000 6000Average Return Ant-v3 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 1000 2000 3000 4000 5000 6000 7000 8000Average Return HalfCheetah-v3 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 500 1000 1500 2000 2500 3000 3500 4000Average Return Hopper-v3 0.0 0.2 0.4 0.6 0.8 1.0 Steps 1e6 0 1000 2000 3000 4000 5000 60...

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