Pareto Q-Learning with Reward Machines

Arnaud Lequen; Cl\'ement Legrand-Lixon; L\'eo Sauli\`eres

arxiv: 2606.19134 · v1 · pith:57TD53KFnew · submitted 2026-06-17 · 💻 cs.LG · cs.AI

Pareto Q-Learning with Reward Machines

Arnaud Lequen , Cl\'ement Legrand-Lixon , L\'eo Sauli\`eres This is my paper

Pith reviewed 2026-06-26 21:05 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords pareto q-learningreward machinesmulti-objective reinforcement learningnon-markovian rewardssample efficiencypareto front approximation

0 comments

The pith

PQLRM integrates Pareto Q-Learning with Reward Machines for sample-efficient multi-policy learning under non-Markovian rewards.

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

The paper introduces PQLRM to merge the set-based Pareto front approximation of Pareto Q-Learning with the factored automaton exploitation from Q-Learning with Reward Machines. This produces a multi-policy method that stays sample-efficient when rewards are history-dependent and encoded as reward machines rather than simple Markovian signals. A sympathetic reader would care because many real tasks involve multiple objectives with complex temporal structure that standard single-policy or Markovian methods cannot handle efficiently. The key gain is avoiding the computational cost of a full cross-product MDP while still approximating the Pareto front and recovering policies that QRM alone misses.

Core claim

PQLRM yields a multi-policy algorithm that remains sample-efficient under non-Markovian, RM-encoded rewards by maintaining sets of vector-valued Q-estimates while exploiting the factored automaton structure of the reward signal, converging faster than a naive PQL baseline on the cross-product MDP and synthesizing Pareto-optimal policies that QRM cannot.

What carries the argument

Maintenance of sets of vector-valued Q-estimates inside the factored automaton structure supplied by reward machines.

If this is right

PQLRM can approximate Pareto fronts in environments whose rewards are specified by automata without expanding the full product state space.
The method recovers multiple policies for tasks where single-policy QRM returns only one or none.
Sample efficiency gains hold for any reward machine whose automaton can be synchronized with the underlying MDP transitions.

Where Pith is reading between the lines

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

The same combination might extend to other set-based or distributional RL methods that track multiple value estimates.
Tasks with naturally occurring history dependence, such as navigation with memory of past goals, become more tractable under this framing.
One could test whether the approach scales when the number of objectives grows while the reward machine remains small.

Load-bearing premise

The factored automaton structure from reward machines can be directly exploited within the Pareto Q-Learning framework to preserve Pareto front approximation quality while gaining sample efficiency, without introducing new approximation errors that offset the claimed gains.

What would settle it

An experiment in which PQLRM applied to RM-specified tasks shows no faster convergence than naive PQL on the cross-product MDP and fails to recover any additional Pareto-optimal policies beyond those found by QRM.

Figures

Figures reproduced from arXiv: 2606.19134 by Arnaud Lequen, Cl\'ement Legrand-Lixon, L\'eo Sauli\`eres.

**Figure 2.** Figure 2: Progress of training over the number of training steps. For multi-policy algorithms (i.e. PQLRM and PQL), we report [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

We present Pareto Q-Learning with Reward Machines (PQLRM), a multi-objective reinforcement learning algorithm for tasks whose reward structure is specified by a set of reward machines (RMs). PQLRM combines Pareto Q-Learning (PQL), which maintains sets of vector-valued Q-estimates to approximate the Pareto front, with enhancements from Q-Learning with Reward Machines (QRM), which exploits the factored automaton structure of the reward signal. This yields a multi-policy algorithm that remains sample-efficient under non-Markovian, RM-encoded rewards. Experimental trials show that PQLRM converges faster than a naive PQL baseline applied to the cross-product MDP and can synthesize Pareto-optimal policies that QRM cannot.

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

PQLRM is a direct combination of Pareto Q-Learning and Reward Machines that targets multi-objective non-Markovian RL, but the abstract supplies no equations, update rules, or numbers to back the efficiency claims.

read the letter

The colleague should know that this paper merges two existing algorithms—Pareto Q-Learning for maintaining vector Q-sets and QRM for exploiting reward machine automata—to produce multiple policies under non-Markovian rewards. The stated advantage is that the factored RM structure avoids the full cross-product MDP while still approximating the Pareto front.

The work does one thing cleanly: it identifies a practical gap where single-policy methods like QRM cannot return trade-off policies and where naive multi-objective methods lose sample efficiency on the expanded state space. If the full paper shows how the RM transitions are folded into the Pareto set updates without extra approximation error, that would be a useful incremental step for anyone already using reward machines.

The soft spots are straightforward. The abstract asserts faster convergence and policies that QRM cannot reach, yet it contains no quantitative results, no environment descriptions, no baseline details, and no pseudocode. Without those, it is impossible to judge whether the claimed gains are real or whether the combination preserves front quality. The central assumption—that the automaton structure plugs in without offsetting costs—remains untested in the provided text.

This paper is for researchers already working at the intersection of multi-objective RL and formal reward specifications. A reader who needs to handle vector rewards with structured non-Markovian signals would get the most from seeing the concrete implementation.

I would send it to peer review. The problem is relevant and the combination is reasonable; referees can request the missing algorithmic and experimental details.

Referee Report

2 major / 0 minor

Summary. The paper presents Pareto Q-Learning with Reward Machines (PQLRM), a multi-objective reinforcement learning algorithm that combines Pareto Q-Learning (PQL) for maintaining sets of vector-valued Q-estimates to approximate the Pareto front with enhancements from Q-Learning with Reward Machines (QRM) to exploit the factored automaton structure of RM-encoded rewards. It claims this produces a sample-efficient multi-policy algorithm for non-Markovian rewards that converges faster than naive PQL on the cross-product MDP and synthesizes Pareto-optimal policies unreachable by QRM.

Significance. If the claims hold, the work would meaningfully extend multi-objective RL to non-Markovian settings with automaton-specified rewards by preserving Pareto-front quality while improving sample efficiency, addressing a practical gap between single-policy RM methods and multi-policy Pareto methods.

major comments (2)

Abstract: the manuscript claims experimental superiority (faster convergence than naive PQL and policies beyond QRM) but supplies no quantitative results, error bars, baseline details, or experiment descriptions, leaving the central empirical claim without verifiable support.
Abstract: no equations, update rules, or description of how the factored RM automaton structure is integrated into the PQL vector-valued Q-maintenance are provided, preventing assessment of whether the claimed exploitation preserves front approximation quality without offsetting approximation errors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below.

read point-by-point responses

Referee: Abstract: the manuscript claims experimental superiority (faster convergence than naive PQL and policies beyond QRM) but supplies no quantitative results, error bars, baseline details, or experiment descriptions, leaving the central empirical claim without verifiable support.

Authors: The abstract serves as a high-level overview of the contributions. Detailed quantitative experimental results, including convergence curves with error bars, baseline comparisons, and experiment descriptions, are provided in the Experiments section of the full manuscript. To improve the abstract's informativeness and address this concern, we will revise it to include key quantitative highlights from our results. revision: yes
Referee: Abstract: no equations, update rules, or description of how the factored RM automaton structure is integrated into the PQL vector-valued Q-maintenance are provided, preventing assessment of whether the claimed exploitation preserves front approximation quality without offsetting approximation errors.

Authors: We acknowledge that the abstract does not include technical details such as equations or update rules. These are presented in Section 3 of the manuscript, where we describe the integration of the RM automaton structure into the Pareto Q-maintenance and provide the relevant update rules. We will revise the abstract to include a brief description of this integration to allow for better assessment of the approach's properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents PQLRM as a synthesis of two independently published algorithms (Pareto Q-Learning and Q-Learning with Reward Machines). The central claim—that the factored RM automaton structure can be exploited inside the PQL framework to retain Pareto-front quality while improving sample efficiency—is framed as a direct algorithmic combination rather than a reduction to any fitted parameter, self-defined quantity, or self-citation chain. No equations or update rules in the provided text equate a derived result to its own inputs by construction, and the experimental comparisons are described as external validation against baselines. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, invented entities, or ad-hoc axioms beyond standard RL background assumptions. The central claim rests on the unstated premise that RM structure transfers cleanly to Pareto sets.

axioms (2)

standard math Standard Q-learning convergence assumptions hold in the cross-product MDP induced by reward machines
Implicit in any Q-learning extension described in the abstract
domain assumption The automaton structure of reward machines can be exploited without degrading Pareto front approximation
Required for the claimed sample-efficiency gains over naive cross-product PQL

pith-pipeline@v0.9.1-grok · 5643 in / 1377 out tokens · 27829 ms · 2026-06-26T21:05:24.706120+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 3 canonical work pages

[1]

What regularized auto-encoders learn from the data-generating distribution.J

Kristof Van Moffaert and Ann Now. Multi-objective reinforcement learning using sets of pareto dominating policies , journal =. 2014 , url =. doi:10.5555/2627435.2750356 , timestamp =

work page doi:10.5555/2627435.2750356 2014
[2]

Felten, Florian and Alegre, Lucas N. and Now. A Toolkit for Reliable Benchmarking and Research in Multi-Objective Reinforcement Learning , booktitle =
[3]

Machine learning , volume=

Q-learning , author=. Machine learning , volume=. 1992 , publisher=

1992
[4]

and Valenzano, Richard and McIlraith, Sheila A

Toro Icarte, Rodrigo and Klassen, Toryn Q. and Valenzano, Richard and McIlraith, Sheila A. , title =. Proceedings of the 35th International Conference on Machine Learning (ICML) , year =
[5]

Learning all optimal policies with multiple criteria , booktitle =

Leon Barrett and Srini Narayanan , editor =. Learning all optimal policies with multiple criteria , booktitle =. 2008 , url =. doi:10.1145/1390156.1390162 , timestamp =

work page doi:10.1145/1390156.1390162 2008
[6]

Evolutionary computation , volume=

HypE: An algorithm for fast hypervolume-based many-objective optimization , author=. Evolutionary computation , volume=. 2011 , publisher=

2011
[7]

Journal of Artificial Intelligence Research , volume=

A survey of multi-objective sequential decision-making , author=. Journal of Artificial Intelligence Research , volume=
[8]

Autonomous Agents and Multi-Agent Systems , volume=

A practical guide to multi-objective reinforcement learning and planning , author=. Autonomous Agents and Multi-Agent Systems , volume=. 2022 , publisher=

2022
[9]

and de Jong, Edwin D

Wiering, Marco A. and de Jong, Edwin D. , booktitle=. Computing Optimal Stationary Policies for Multi-Objective Markov Decision Processes , year=
[10]

Sutton and Andrew G

Richard S. Sutton and Andrew G. Barto , title =. 2018 , url =

2018
[11]

The deterministic part of the seventh International Planning Competition , journal =

Carlos Linares L. The deterministic part of the seventh International Planning Competition , journal =. 2015 , url =. doi:10.1016/J.ARTINT.2015.01.004 , timestamp =

work page doi:10.1016/j.artint.2015.01.004 2015
[12]

Agent57: Outperforming the

Adri. Agent57: Outperforming the. Proceedings of the 37th International Conference on Machine Learning (ICML) , year =
[13]

Spatial-Temporal Traffic Flow Control on Motorways Using Distributed Multi-Agent Reinforcement Learning , journal =

Kre. Spatial-Temporal Traffic Flow Control on Motorways Using Distributed Multi-Agent Reinforcement Learning , journal =
[14]

IEEE Consumer Communications and Networking Conference (CCNC) , year =

Babatunji Omoniwa and Boris Galkin and Ivana Dusparic , title =. IEEE Consumer Communications and Networking Conference (CCNC) , year =
[15]

Real Robot Challenge 2022: Learning Dexterous Manipulation from Offline Data in the Real World , booktitle =

Nico G. Real Robot Challenge 2022: Learning Dexterous Manipulation from Offline Data in the Real World , booktitle =. 2023 , publisher =

2022
[16]

Terry and John U

Mark Towers and Ariel Kwiatkowski and Jordan K. Terry and John U. Balis and Gianluca De Cola and Tristan Deleu and Manuel Goul. Gymnasium: A Standard Interface for Reinforcement Learning Environments , journal =. 2024 , doi =

2024

[1] [1]

What regularized auto-encoders learn from the data-generating distribution.J

Kristof Van Moffaert and Ann Now. Multi-objective reinforcement learning using sets of pareto dominating policies , journal =. 2014 , url =. doi:10.5555/2627435.2750356 , timestamp =

work page doi:10.5555/2627435.2750356 2014

[2] [2]

Felten, Florian and Alegre, Lucas N. and Now. A Toolkit for Reliable Benchmarking and Research in Multi-Objective Reinforcement Learning , booktitle =

[3] [3]

Machine learning , volume=

Q-learning , author=. Machine learning , volume=. 1992 , publisher=

1992

[4] [4]

and Valenzano, Richard and McIlraith, Sheila A

Toro Icarte, Rodrigo and Klassen, Toryn Q. and Valenzano, Richard and McIlraith, Sheila A. , title =. Proceedings of the 35th International Conference on Machine Learning (ICML) , year =

[5] [5]

Learning all optimal policies with multiple criteria , booktitle =

Leon Barrett and Srini Narayanan , editor =. Learning all optimal policies with multiple criteria , booktitle =. 2008 , url =. doi:10.1145/1390156.1390162 , timestamp =

work page doi:10.1145/1390156.1390162 2008

[6] [6]

Evolutionary computation , volume=

HypE: An algorithm for fast hypervolume-based many-objective optimization , author=. Evolutionary computation , volume=. 2011 , publisher=

2011

[7] [7]

Journal of Artificial Intelligence Research , volume=

A survey of multi-objective sequential decision-making , author=. Journal of Artificial Intelligence Research , volume=

[8] [8]

Autonomous Agents and Multi-Agent Systems , volume=

A practical guide to multi-objective reinforcement learning and planning , author=. Autonomous Agents and Multi-Agent Systems , volume=. 2022 , publisher=

2022

[9] [9]

and de Jong, Edwin D

Wiering, Marco A. and de Jong, Edwin D. , booktitle=. Computing Optimal Stationary Policies for Multi-Objective Markov Decision Processes , year=

[10] [10]

Sutton and Andrew G

Richard S. Sutton and Andrew G. Barto , title =. 2018 , url =

2018

[11] [11]

The deterministic part of the seventh International Planning Competition , journal =

Carlos Linares L. The deterministic part of the seventh International Planning Competition , journal =. 2015 , url =. doi:10.1016/J.ARTINT.2015.01.004 , timestamp =

work page doi:10.1016/j.artint.2015.01.004 2015

[12] [12]

Agent57: Outperforming the

Adri. Agent57: Outperforming the. Proceedings of the 37th International Conference on Machine Learning (ICML) , year =

[13] [13]

Spatial-Temporal Traffic Flow Control on Motorways Using Distributed Multi-Agent Reinforcement Learning , journal =

Kre. Spatial-Temporal Traffic Flow Control on Motorways Using Distributed Multi-Agent Reinforcement Learning , journal =

[14] [14]

IEEE Consumer Communications and Networking Conference (CCNC) , year =

Babatunji Omoniwa and Boris Galkin and Ivana Dusparic , title =. IEEE Consumer Communications and Networking Conference (CCNC) , year =

[15] [15]

Real Robot Challenge 2022: Learning Dexterous Manipulation from Offline Data in the Real World , booktitle =

Nico G. Real Robot Challenge 2022: Learning Dexterous Manipulation from Offline Data in the Real World , booktitle =. 2023 , publisher =

2022

[16] [16]

Terry and John U

Mark Towers and Ariel Kwiatkowski and Jordan K. Terry and John U. Balis and Gianluca De Cola and Tristan Deleu and Manuel Goul. Gymnasium: A Standard Interface for Reinforcement Learning Environments , journal =. 2024 , doi =

2024