Learning Fair Pareto-Optimal Policies in Multi-Objective Reinforcement Learning

Peilang Li; Umer Siddique; Yongcan Cao

arxiv: 2606.18111 · v1 · pith:ZP3MM47Fnew · submitted 2026-06-16 · 💻 cs.LG · cs.AI

Learning Fair Pareto-Optimal Policies in Multi-Objective Reinforcement Learning

Umer Siddique , Peilang Li , Yongcan Cao This is my paper

Pith reviewed 2026-06-27 00:55 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords multi-objective reinforcement learningfairnessPareto-optimal policiesconvex coverage setgeneralized Gini welfare functionmulti-policy Q-learningnon-stationary policiesstochastic policies

0 comments

The pith

For concave piecewise-linear welfare functions like GGF, fair policies stay inside the convex coverage set of multi-objective RL.

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

The paper formalizes the problem of learning a set of Pareto-optimal policies that remain fair across all possible user preferences in multi-objective reinforcement learning. It proves that when welfare functions are concave and piecewise-linear, these fair policies lie within the convex coverage set obtained from linear scalarization. The work then gives three algorithms that integrate the generalized Gini welfare function with multi-policy multi-objective Q-learning, including versions that use state augmentation for non-stationary policies and extensions for stochastic policies. These methods are shown to produce diverse fair policies that adapt to different preferences where single-policy approaches cannot.

Core claim

For concave, piecewise-linear welfare functions such as the generalized Gini welfare function, fair policies remain inside the convex coverage set. Non-stationary policies that track accrued reward histories and stochastic policies both increase fairness by responding to historical imbalances. Three algorithms are introduced that combine the GGF scalarization with multi-policy multi-objective Q-learning to learn the required set of policies.

What carries the argument

The convex coverage set (CCS) as an approximated Pareto front that contains all fair policies for concave piecewise-linear welfare functions, integrated with the generalized Gini welfare function inside multi-policy multi-objective Q-learning.

If this is right

A single set of policies learned via the proposed algorithms suffices for any user preference expressible by the welfare function.
Non-stationary policies that carry reward history improve equity without leaving the CCS.
Stochastic policies further reduce historical inequities while preserving Pareto optimality.
The same CCS-based learning framework works across multiple domains when compared with prior MORL baselines.

Where Pith is reading between the lines

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

The result suggests that fairness constraints need not expand the policy search space beyond what linear scalarization already covers for this class of welfare functions.
In practice this could let deployed systems switch among pre-learned policies when user preferences shift rather than retraining from scratch.
A natural next test is whether the same containment extends to welfare functions that are concave but not piecewise linear.

Load-bearing premise

Welfare functions must be concave and piecewise-linear for the containment of fair policies inside the convex coverage set to hold.

What would settle it

An explicit counter-example of a concave piecewise-linear welfare function together with a fair policy that lies strictly outside the convex coverage set would falsify the central containment result.

Figures

Figures reproduced from arXiv: 2606.18111 by Peilang Li, Umer Siddique, Yongcan Cao.

**Figure 2.** Figure 2: Example of MOMDP where actions lead to different rewards. considered. However, considering historical data, i.e., racc, a1 yields a higher accrued episodic return of (10, 10) and a welfare score of 10. Similarly, a2 yields (15, 5) and 7 episodic return and welfare scores, respectively. Note that action a1 is a fairer choice in this case since it balances the two objectives, unlike action a2, which fails… view at source ↗

**Figure 3.** Figure 3: Left Figure: Point A Pareto-dominates B and is preferred to C by the Pigou–Dalton [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Performances of multi-policy MORL baselines and our methods in species conservation. [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Individual densities of Envelope, and our proposed methods during testing with unseen [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Performances of multi-policy MORL baselines and our methods in resource gathering. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: Performances of multi-policy MORL baselines and our proposed methods in the MPW. [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

read the original abstract

Fairness is an important aspect of decision-making in multi-objective reinforcement learning (MORL), where policies must ensure both optimality and equity across multiple, potentially conflicting objectives. While single-policy MORL methods can learn fair policies for fixed user preferences using welfare functions such as the generalized Gini welfare function (GGF), they fail to provide the diverse set of policies necessary for dynamic or unknown user preferences. To address this limitation, we formalize the fair optimization problem in multi-policy MORL, where the goal is to learn a set of Pareto-optimal policies that ensure fairness across all possible user preferences. Our key technical contributions are threefold: (1) We show that for concave, piecewise-linear welfare functions (e.g., GGF), fair policies remain in the convex coverage set (CCS), which is an approximated Pareto front for linear scalarization. (2) We demonstrate that non-stationary policies, augmented with accrued reward histories, and stochastic policies improve fairness by dynamically adapting to historical inequities. (3) We propose three novel algorithms, which include integrating GGF with multi-policy multi-objective Q-Learning (MOQL), state-augmented multi-policy MOQL for learning non-statoinary policies, and its novel extension for learning stochastic policies. We evaluate our algorithms across various domains and compare our methods against the state-of-the-art MORL baselines. The empirical results show that our methods learn a set of fair policies that accommodate different user preferences.

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

The paper formalizes fair multi-policy MORL using GGF and claims CCS containment, then gives three MOQL extensions, but the containment step is the part that needs checking.

read the letter

The paper formalizes the multi-policy version of fair optimization in MORL and proposes three algorithm variants built on multi-policy MOQL. The central new claim is that for concave piecewise-linear welfare functions such as GGF, every fair policy stays inside the convex coverage set. They then introduce GGF-MOQL, a state-augmented version that tracks reward history for non-stationary policies, and a stochastic extension. The motivation is clear: single-policy methods cannot handle shifting or unknown user preferences, so a covering set of fair policies is the practical target.

The algorithmic ideas are straightforward extensions of existing MOQL work and address a real gap when preferences are dynamic. Using accrued reward history to adapt to past inequities is a reasonable move, and the stochastic variant adds another lever for fairness.

The soft spot is the CCS containment result. It rests on concavity and piecewise linearity guaranteeing that the nonlinear optimum can still be recovered by some linear scalarization. If the attainable reward geometry or the GGF breakpoints do not admit the required supporting hyperplane, a fair policy can sit outside the CCS. In that case the subsequent claim that the learned set covers all preferences no longer follows. The abstract presents the result as shown, but the geometry condition is load-bearing and not obviously automatic.

Empirical claims are stated without numbers, baselines, or variance details, so it is hard to gauge how much the new variants actually improve fairness or coverage.

This is for researchers already working in MORL who need to handle fairness with uncertain preferences. The problem formalization and algorithm sketches are concrete enough that a reader in that subfield would get usable ideas. The paper engages the existing literature directly and is coherent on its own terms, so it deserves a serious referee even if the theory section will need more explicit checks on the containment conditions.

Referee Report

2 major / 2 minor

Summary. The paper formalizes the fair multi-policy MORL problem and claims three contributions: (1) for concave piecewise-linear welfare functions such as GGF, every fair policy lies inside the convex coverage set (CCS) of policies optimal for some linear scalarization; (2) non-stationary and stochastic policies improve fairness via history augmentation; (3) three MOQL-based algorithms (GGF-integrated, state-augmented, and stochastic variants) learn covering sets of fair policies. Empirical results across domains are reported to show that the learned sets accommodate varying user preferences better than prior MORL baselines.

Significance. If the CCS-containment result is correct, the work supplies a theoretically grounded reduction that lets multi-policy MORL restrict its search to an approximated Pareto front while still guaranteeing fairness for any concave piecewise-linear welfare function; this would be a useful bridge between single-policy welfare optimization and preference-agnostic policy sets. The algorithmic extensions for non-stationary and stochastic policies are natural and the empirical demonstration, if properly controlled, would support practical adoption.

major comments (2)

[§3] §3 / Theorem on CCS containment: the central claim that every GGF-optimal policy remains inside the CCS rests on concavity and piecewise-linearity guaranteeing a supporting hyperplane; the manuscript must supply the full derivation (including the precise geometric condition on the attainable reward set) and discuss or exhibit cases where the breakpoints of the GGF or non-convexity of the reward set would place a fair policy strictly outside the CCS, as this directly determines whether restricting search to the CCS is sufficient.
[§5] §5 (empirical evaluation): the claim of superiority over state-of-the-art MORL baselines is load-bearing for the algorithmic contribution, yet the reported results lack error bars, explicit dataset generation details, hyper-parameter tables, and per-preference welfare scores; without these the reader cannot verify that the learned CCS covers arbitrary user preferences rather than overfitting the tested welfare functions.

minor comments (2)

Notation for the CCS and the welfare function should be introduced once with a single consistent symbol set rather than re-defined across sections.
[§5] The abstract states 'we evaluate our algorithms across various domains' but the experimental section should include a table listing environment names, objective counts, and episode lengths for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments. We address each major point below and will revise the manuscript to strengthen both the theoretical presentation and the empirical reporting.

read point-by-point responses

Referee: [§3] §3 / Theorem on CCS containment: the central claim that every GGF-optimal policy remains inside the CCS rests on concavity and piecewise-linearity guaranteeing a supporting hyperplane; the manuscript must supply the full derivation (including the precise geometric condition on the attainable reward set) and discuss or exhibit cases where the breakpoints of the GGF or non-convexity of the reward set would place a fair policy strictly outside the CCS, as this directly determines whether restricting search to the CCS is sufficient.

Authors: We agree that the current presentation would benefit from an expanded derivation. In the revised manuscript we will include the complete proof of the CCS-containment result, explicitly stating the supporting-hyperplane argument that follows from concavity and piecewise linearity of the GGF together with the precise geometric condition on the attainable reward set. We will also add a dedicated paragraph analyzing potential violations: we show that any departure from concavity or piecewise linearity (or a non-convex reward set) can indeed place a fair policy outside the CCS, but that these cases lie outside the assumptions of our theorem; under the stated conditions the containment holds. This addition will clarify the scope of the reduction to the CCS. revision: yes
Referee: [§5] §5 (empirical evaluation): the claim of superiority over state-of-the-art MORL baselines is load-bearing for the algorithmic contribution, yet the reported results lack error bars, explicit dataset generation details, hyper-parameter tables, and per-preference welfare scores; without these the reader cannot verify that the learned CCS covers arbitrary user preferences rather than overfitting the tested welfare functions.

Authors: We accept that the current empirical section is insufficiently detailed for independent verification. In the revised manuscript we will (i) report mean performance with standard-error bars across multiple random seeds, (ii) provide explicit descriptions of dataset generation procedures and environment parameterizations, (iii) include a comprehensive hyper-parameter table, and (iv) add tables of per-preference welfare scores evaluated on a wider grid of user preference vectors. These changes will demonstrate that the learned policy sets generalize across preferences rather than overfitting the specific welfare functions used during training. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central technical claim is a mathematical containment result: for concave piecewise-linear welfare functions such as GGF, fair policies remain inside the CCS. This is stated as an independent derivation from standard MORL concepts rather than a redefinition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The three proposed algorithms follow from this result and from the introduction of non-stationary and stochastic policies, but none of the steps reduce by construction to the inputs via the enumerated circularity patterns. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on domain assumptions about welfare function properties and standard RL convergence; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Welfare functions such as GGF are concave and piecewise-linear
Invoked to establish that fair policies remain in the CCS.

pith-pipeline@v0.9.1-grok · 5791 in / 1201 out tokens · 29004 ms · 2026-06-27T00:55:42.125096+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

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equal treatment of equals

16 Supplementary Materials The following content was not necessarily subject to peer review. 8 Proofs of Technical Analysis In this section, we provide formal proofs of our technical analysis in detail. For better legibility, we first recall the equations and results that we need for our proofs. ∀ω∈Ω,max π∈Π ϕ(V(π)),(3) whereΩis the set of valid preferenc...

2008
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This domain presents a unique challenge centered around the acquisition of three types of resources: gold, gems, and stones, thereby establishing a multi-objective framework withK=

10.2 Resource Gathering In this scenario of resource gathering, we consider a5×5grid world domain inspired from (Barrett & Narayanan, 2008). This domain presents a unique challenge centered around the acquisition of three types of resources: gold, gems, and stones, thereby establishing a multi-objective framework withK=

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arXiv

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2023

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Debmalya Mandal and Jiarui Gan. Socially fair reinforcement learning.arXiv preprint arXiv:2208.12584,

arXiv

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Scalable multi-objective reinforcement learning with fairness guarantees using lorenz dominance

Dimitris Michailidis, Willem Röpke, Diederik M Roijers, Sennay Ghebreab, and Fernando P Santos. Scalable multi-objective reinforcement learning with fairness guarantees using lorenz dominance. arXiv preprint arXiv:2411.18195,

arXiv

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arXiv

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Mathieu Reymond, Eugenio Bargiacchi, and Ann Nowé. Pareto conditioned networks.arXiv preprint arXiv:2204.05036,

arXiv

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Fairness in preference-based reinforcement learning

Umer Siddique, Abhinav Sinha, and Yongcan Cao. Fairness in preference-based reinforcement learning. InICML 2023 Workshop The Many Facets of Preference-Based Learning,

2023

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equal treatment of equals

16 Supplementary Materials The following content was not necessarily subject to peer review. 8 Proofs of Technical Analysis In this section, we provide formal proofs of our technical analysis in detail. For better legibility, we first recall the equations and results that we need for our proofs. ∀ω∈Ω,max π∈Π ϕ(V(π)),(3) whereΩis the set of valid preferenc...

2008

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This domain presents a unique challenge centered around the acquisition of three types of resources: gold, gems, and stones, thereby establishing a multi-objective framework withK=

10.2 Resource Gathering In this scenario of resource gathering, we consider a5×5grid world domain inspired from (Barrett & Narayanan, 2008). This domain presents a unique challenge centered around the acquisition of three types of resources: gold, gems, and stones, thereby establishing a multi-objective framework withK=

2008