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arxiv: 2601.04392 · v2 · submitted 2026-01-07 · 💻 cs.LG · cs.AI· cs.RO· cs.SY· eess.SY· math.OC

Enhanced-FQL(λ), an Efficient and Interpretable RL with novel Fuzzy Eligibility Traces and Segmented Experience Replay

Mohsen Jalaeian-Farimani , Xiong Xiong , Luca Bascetta This is my paper

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

classification 💻 cs.LG cs.AIcs.ROcs.SYeess.SYmath.OC
keywords fuzzy reinforcement learningeligibility tracesexperience replaycontinuous controlQ-learninginterpretable RLCart-Pole
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The pith

Enhanced-FQL(λ) integrates fuzzified eligibility traces and segmented experience replay into fuzzy Q-learning to achieve sample-efficient continuous control with an interpretable rule base.

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

The paper introduces Enhanced-FQL(λ), a fuzzy reinforcement learning method that replaces neural architectures with an interpretable fuzzy rule base for continuous control tasks. It adds fuzzified eligibility traces for stable multi-step credit assignment via a fuzzified Bellman equation and uses segmented experience replay to boost sample efficiency. Theoretical analysis establishes convergence under standard assumptions. Experiments on the Cart-Pole benchmark show improved sample efficiency and reduced variance compared to n-step fuzzy TD and fuzzy SARSA(λ), while matching the tested DDPG baseline.

Core claim

Enhanced-FQL(λ) proves convergence for fuzzy Q-learning augmented by fuzzified eligibility traces and segmented experience replay, delivering competitive performance on Cart-Pole through an interpretable fuzzy rule base instead of neural networks.

What carries the argument

Fuzzified Eligibility Traces (FET) combined with Segmented Experience Replay (SER) inside the Fuzzified Bellman Equation (FBE) for fuzzy Q-learning.

If this is right

  • The algorithm converges under the same assumptions used for standard fuzzy TD methods.
  • Sample efficiency improves over n-step fuzzy TD and fuzzy SARSA(λ) on Cart-Pole.
  • Learning variance decreases relative to the compared fuzzy baselines.
  • Performance stays competitive with DDPG while using far fewer parameters.
  • The framework remains computationally compact for moderate-scale continuous control.

Where Pith is reading between the lines

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

  • Policies learned this way could be inspected and modified directly by inspecting the fuzzy rules rather than post-hoc explanation of a neural net.
  • The segmented replay mechanism might extend naturally to other memory-based fuzzy learners to cut storage costs.
  • If the rule base can be learned or adapted online, the method could apply to tasks where human-readable policies are required for certification.

Load-bearing premise

The fuzzy rule base is expressive enough to represent near-optimal policies for the target tasks.

What would settle it

A run on Cart-Pole where Enhanced-FQL(λ) fails to reach the performance level of the DDPG baseline despite a well-tuned fuzzy rule base.

Figures

Figures reproduced from arXiv: 2601.04392 by Luca Bascetta, Mohsen Jalaeian-Farimani, Xiong Xiong.

Figure 2
Figure 2. Figure 2: compares the speed of reaching a target per￾formance (return = -200). Enhanced-FQL(λ) achieves this with far fewer episodes, demonstrating its superior sample efficiency. VI. DISCUSSION The computed value function Qb⋆ represents the suboptimal fixed point within the chosen fuzzy rule base. While the proposed method involves an inherent bias-approximation trade-off; refining the state and action partitions—… view at source ↗
read the original abstract

This paper introduces a fuzzy reinforcement learning framework, Enhanced-FQL($\lambda$), that integrates novel Fuzzified Eligibility Traces (FET) and Segmented Experience Replay (SER) into fuzzy Q-learning with the Fuzzified Bellman Equation (FBE) for continuous control. The proposed approach employs an interpretable fuzzy rule base instead of complex neural architectures, while maintaining competitive performance through two key innovations: a fuzzified Bellman equation with eligibility traces for stable multi-step credit assignment, and a memory-efficient segment-based experience replay mechanism for enhanced sample efficiency. Theoretical analysis proves the proposed method convergence under standard assumptions. On the Cart--Pole benchmark, Enhanced-FQL($\lambda$) improves sample efficiency and reduces variance relative to $n$-step fuzzy TD and fuzzy SARSA($\lambda$), while remaining competitive with the tested DDPG baseline. These results support the proposed framework as an interpretable and computationally compact alternative for moderate-scale continuous control problems.

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 / 3 minor

Summary. The paper presents Enhanced-FQL(λ), which integrates Fuzzified Eligibility Traces (FET) and Segmented Experience Replay (SER) into fuzzy Q-learning based on the Fuzzified Bellman Equation for continuous control tasks. It claims a proof of convergence under standard assumptions and shows on the Cart-Pole benchmark that it achieves better sample efficiency and lower variance than n-step fuzzy TD and fuzzy SARSA(λ), while matching DDPG performance, positioning it as an interpretable and efficient alternative to deep RL methods.

Significance. Should the theoretical convergence be rigorously established and the empirical advantages confirmed with proper controls, this contribution would be significant for the field of interpretable reinforcement learning. It provides a way to incorporate multi-step learning and efficient replay into fuzzy systems without resorting to black-box neural networks, potentially aiding applications where transparency is required.

major comments (2)
  1. Theoretical Analysis section: The claim that the method converges under standard assumptions requires explicit verification that the Fuzzified Eligibility Traces preserve the contraction property of the Bellman operator and that Segmented Experience Replay maintains the necessary ergodicity or sampling conditions for convergence with probability 1. Without this re-derivation, the extension of standard fuzzy Q-learning convergence arguments remains unverified and is central to the paper's theoretical contribution.
  2. Experimental Evaluation section (Cart-Pole results): The improvements in sample efficiency and variance reduction are presented relative to baselines, but the manuscript does not specify the number of independent runs, confidence intervals, or statistical tests used. This undermines the strength of the empirical claims supporting the method's advantages.
minor comments (3)
  1. Abstract: The abstract introduces acronyms like FET, SER, and FBE without expanding them on first use, which may confuse readers unfamiliar with the framework.
  2. Method Description: The definition of the Fuzzified Eligibility Traces could include a clearer mathematical formulation, perhaps as an equation following the standard eligibility trace update but fuzzified.
  3. Related Work: Missing references to recent works on fuzzy RL or eligibility traces in continuous control to better position the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to strengthen both the theoretical and empirical contributions.

read point-by-point responses

  1. Referee: Theoretical Analysis section: The claim that the method converges under standard assumptions requires explicit verification that the Fuzzified Eligibility Traces preserve the contraction property of the Bellman operator and that Segmented Experience Replay maintains the necessary ergodicity or sampling conditions for convergence with probability 1. Without this re-derivation, the extension of standard fuzzy Q-learning convergence arguments remains unverified and is central to the paper's theoretical contribution.

    Authors: We agree that an explicit re-derivation is required. In the revised manuscript we will expand the Theoretical Analysis section with a detailed proof showing that the Fuzzified Eligibility Traces preserve the contraction property of the Bellman operator (by verifying that the fuzzification operator remains a non-expansive mapping) and that Segmented Experience Replay satisfies the ergodicity and sampling conditions needed for convergence with probability 1. The proof will extend the standard fuzzy Q-learning arguments by explicitly accounting for the effects of FET and SER. revision: yes

  2. Referee: Experimental Evaluation section (Cart-Pole results): The improvements in sample efficiency and variance reduction are presented relative to baselines, but the manuscript does not specify the number of independent runs, confidence intervals, or statistical tests used. This undermines the strength of the empirical claims supporting the method's advantages.

    Authors: We acknowledge the omission. In the revision we will state that all Cart-Pole results are averaged over 10 independent runs with distinct random seeds, include 95% confidence intervals, and report paired t-test p-values to establish statistical significance of the observed improvements in sample efficiency and variance reduction relative to the baselines. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation chain is self-contained with independent extensions.

full rationale

The paper introduces FET and SER as novel algorithmic components added to the Fuzzified Bellman Equation within fuzzy Q-learning. The convergence claim is stated under standard assumptions without any quoted reduction of the proof to a self-citation, fitted parameter, or redefinition of inputs as outputs. No self-definitional loops, fitted-input predictions, or ansatz smuggling via citation appear in the provided text. The central theoretical and empirical claims retain independent content and do not collapse to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on standard RL convergence assumptions plus two newly introduced algorithmic mechanisms whose independent validation is not supplied in the abstract.

axioms (1)
  • domain assumption Standard assumptions for convergence of fuzzy Q-learning hold when eligibility traces and segmented replay are incorporated
    Invoked for the theoretical analysis mentioned in the abstract.
invented entities (2)
  • Fuzzified Eligibility Traces (FET) no independent evidence
    purpose: Stable multi-step credit assignment in fuzzy setting
    New component introduced to extend fuzzy Q-learning.
  • Segmented Experience Replay (SER) no independent evidence
    purpose: Memory-efficient experience reuse for improved sample efficiency
    New replay mechanism proposed in the paper.

pith-pipeline@v0.9.0 · 5498 in / 1237 out tokens · 58583 ms · 2026-05-16T16:07:57.564423+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    A., Veness, J., Bellemare, M

    Mnih, V ., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., ... & Hassabis, D. (2015). Human-level control through deep reinforcement learning. nature, 518(7540), 529-533

    work page 2015

  2. [2]

    Fard, O., & Jalaeian-Farimani, M., (2025)

    Khalili Amirabadi, R., S. Fard, O., & Jalaeian-Farimani, M., (2025). Towards optimal control of HPV model using safe reinforcement learning with actor–critic neural networks. Expert Systems with Ap- plications, 264, 125783

    work page 2025

  3. [3]

    Fard, O., (2025)

    Khalili Amirabadi, R., Jalaeian-Farimani, M., & S. Fard, O., (2025). LSTM-empowered reinforcement learning in Bi-level optimal control for nonlinear systems with uncertain dynamics. ISA Transactions, 2025 Nov 20:S0019-0578(25)00645-7

    work page 2025

  4. [4]

    Event-triggered dynamic seed invasive weed optimization (ET-DSIWO): a nature-inspired approach for non- stationary optimization

    Jalaeian-Farimani, M., Khalili Amirabadi, R., Esmaeili Ranjbar, M., & Samadzadeh, S., (2025). Event-triggered dynamic seed invasive weed optimization (ET-DSIWO): a nature-inspired approach for non- stationary optimization. Nonlinear Dynamics, 113(20), 27611–7636

    work page 2025

  5. [5]

    & Miao, Q

    Wang, X., Wang, S., Liang, X., Zhao, D., Huang, J., Xu, X., ... & Miao, Q. (2022). Deep reinforcement learning: A survey. IEEE Transactions on Neural Networks and Learning Systems, 35(4), 5064-5078

    work page 2022

  6. [6]

    (2016, March)

    Van Hasselt, H., Guez, A., & Silver, D. (2016, March). Deep reinforce- ment learning with double q-learning. In Proceedings of the AAAI conference on artificial intelligence (V ol. 30, No. 1)

    work page 2016

  7. [7]

    (2016, June)

    Wang, Z., Schaul, T., Hessel, M., Hasselt, H., Lanctot, M., & Freitas, N. (2016, June). Dueling network architectures for deep reinforcement learning. In International conference on machine learning (pp. 1995- 2003). PMLR

    work page 2016

  8. [8]

    & Silver, D

    Hessel, M., Modayil, J., Van Hasselt, H., Schaul, T., Ostrovski, G., Dabney, W., ... & Silver, D. (2018, April). Rainbow: Combining improvements in deep reinforcement learning. In Proceedings of the AAAI conference on artificial intelligence (V ol. 32, No. 1)

    work page 2018

  9. [9]

    H., Abdulkadir, S

    Sumiea, E. H., Abdulkadir, S. J., Alhussian, H. S., Al-Selwi, S. M., Alqushaibi, A., Ragab, M. G., & Fati, S. M. (2024). Deep deterministic policy gradient algorithm: A systematic review. Heliyon, 10(9)

    work page 2024

  10. [10]

    (2018, July)

    Fujimoto, S., Hoof, H., & Meger, D. (2018, July). Addressing function approximation error in actor-critic methods. In International confer- ence on machine learning (pp. 1587-1596). PMLR

    work page 2018

  11. [11]

    (2018, July)

    Haarnoja, T., Zhou, A., Abbeel, P., & Levine, S. (2018, July). Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In International conference on machine learning (pp. 1861-1870). Pmlr

    work page 2018

  12. [12]

    Fard, O., (2025)

    Khalili Amirabadi, R., Jalaeian-Farimani, M., & S. Fard, O., (2025). Self-Organizing Dual-Buffer Adaptive Clustering Experience Re- play (SODASER) for Safe Reinforcement Learning in Optimal Control. Available at SSRN: https://ssrn.com/abstract=5191427 or http://dx.doi.org/10.2139/ssrn.5191427

    work page doi:10.2139/ssrn.5191427 2025

  13. [13]

    Wu, H., Zhang, J., Wang, Z., Lin, Y ., & Li, H. (2022). Sub-A VG: Overestimation reduction for cooperative multi-agent reinforcement learning. Neurocomputing, 474, 94-106

    work page 2022

  14. [14]

    Farzanegan, B., & Jagannathan, S. (2025). Explainable and safety aware deep reinforcement learning-based control of nonlinear discrete- time systems using neural network gradient decomposition. IEEE Transactions on Automation Science and Engineering

    work page 2025

  15. [15]

    F., & You, Z

    Juang, C. F., & You, Z. B. (2024). Reinforcement learning of an interpretable fuzzy system through a neural fuzzy actor-critic Frame- work for Mobile Robot Control. IEEE Transactions on Fuzzy Systems, 32(6), 3655-3668

    work page 2024

  16. [16]

    Lu, J., Ma, G., & Zhang, G. (2024). Fuzzy machine learning: A comprehensive framework and systematic review. IEEE Transactions on Fuzzy Systems, 32(7), 3861-3878

    work page 2024

  17. [17]

    (2025, June)

    Jalaeian-Farimani, M., Nikkhouy, D., Rastegarmoghaddam, M., Samadzadeh, S., Mozaffari, S., & Alirezaee, S. (2025, June). Fuzzy Q-Learning with Fuzzified Bellman Equation for Unmanned Ground Vehicle Navigation. In 2025 9th International Conference on Robotics and Automation Sciences (ICRAS) (pp. 300-304). IEEE

    work page 2025

  18. [18]

    Wang, D., Yuan, Z., Liu, A., Lin, Q., & Qiao, J. (2025). Model- Free Neuro-Fuzzy Q-Learning Control With Swarm Intelligence. IEEE Transactions on Fuzzy Systems

    work page 2025

  19. [19]

    Cuesta-Solano, R., Moya-Albor, E., Brieva, J., & Ponce, H. (2024). A Vision-based Robotic Navigation Method Using an Evolutionary and Fuzzy Q-Learning Approach. Journal of Artificial Intelligence and Technology, 4(4), 363-369

    work page 2024