arxiv: 2605.01961 · v1 · submitted 2026-05-03 · 💻 cs.LG

Recognition: unknown

Multi-User Dueling Bandits: A Fair Approach using Nash Social Welfare

Maheed H. Ahmed , Mahsa Ghasemi

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:59 UTC · model grok-4.3

classification 💻 cs.LG

keywords multi-user dueling banditsNash social welfarefairnessregret boundsCondorcet winnerheterogeneous preferencesonline learning

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The pith

Using Nash Social Welfare to enforce fairness across users with heterogeneous preferences in dueling bandits produces a regret lower bound of order T to the two-thirds power.

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

The paper tries to establish the statistical price of fairness when learning from pairwise preferences that come from multiple users who disagree on the best arm. It does so by proving that any algorithm must suffer regret at least on the order of T to the two-thirds times the cube root of the smaller of the number of arms or users, and by giving algorithms whose upper bounds match this dependence on the time horizon. A sympathetic reader would care because many preference-based systems simply average all feedback, which can leave minority groups with systematically worse outcomes when tastes differ. The work defines each user's utility relative to their own Condorcet winner and then maximizes the product of those utilities rather than their sum. This yields both the lower-bound construction and two practical algorithms that first identify the per-user winners before committing to fair selections.

Core claim

We construct a hard instance to establish a regret lower bound of Ω(T^{2/3} min(K,D)^{1/3}) for a time horizon T, K arms, and D users, which, to the best of our knowledge, is the first result quantifying the cost of fairness in dueling bandits with heterogeneous preferences. We then present the Fair-Explore-Then-Commit and Fair-ε-Greedy algorithms with a Condorcet winner identification phase. We further derive their regret upper bounds that match the lower-bound dependence on T up to logarithmic factors.

What carries the argument

Nash Social Welfare objective applied to per-user utilities, each defined as the probability that a chosen arm beats that user's own Condorcet winner (an arm preferred to every other arm).

If this is right

The regret of any fair algorithm scales at least with the cube root of min(K, D) in addition to the T^{2/3} term.
The Fair-Explore-Then-Commit and Fair-ε-Greedy procedures achieve upper bounds whose leading term matches the lower bound in its T dependence.
A separate identification phase for each user's Condorcet winner is required before fair selection can begin.
When the number of users or arms grows, the fairness penalty increases, forcing more exploration overall.

Where Pith is reading between the lines

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

In applications such as collecting human feedback for model training, the extra exploration cost suggests allocating additional rounds of pairwise comparisons to protect smaller user groups.
The same product-based fairness objective could be applied to other sequential decision problems where one wants to avoid any single participant falling too far behind.
If some users share the same Condorcet winner the effective number of distinct winners drops, which might reduce the observed min(K, D) factor in practice.

Load-bearing premise

Each user has a Condorcet winner, an arm that beats every other arm according to that user's private preference ordering.

What would settle it

An algorithm that achieves regret growing strictly slower than T to the two-thirds power when run on the paper's constructed hard instance with heterogeneous Condorcet winners would contradict the lower bound.

Figures

Figures reproduced from arXiv: 2605.01961 by Maheed H. Ahmed, Mahsa Ghasemi.

**Figure 1.** Figure 1: Cumulative NSW Regret across 6 different problem settings varying in Users (D), Arms (K), and Horizon (T). The minimum preference gap is fixed at ∆ = 0.1. Metrics. We compute the utility of each user as the cumulative scores achieved up to round P t, i.e., uˆd(t) = t τ=1 0.5 × (sd(iτ ) + sd(jτ )). We assess performance using four key metrics: (1) Cumulative Regret: the cumulative difference between the o… view at source ↗

**Figure 2.** Figure 2: Fairness metrics under clustered user preferences (D = 10, K = 10). We vary the clustering fraction ρ (majority size). The utilitarian baseline exhibits high inequality (high Gini, low Min Welfare) as the majority cluster grows, whereas fair algorithms remain robust. ness metrics, specifically minimum user welfare and the Gini coefficient, our algorithms perform significantly better than the utilitarian a… view at source ↗

read the original abstract

Learning from human preference data is becoming a useful tool, from fine-tuning large language models to training reinforcement learning agents. However, in most scenarios, the model is trained on the average preference of all human evaluators, which, under large variations of preferences, can be unfair to minority groups. In this work, we consider fairness in dueling bandits, a standard framework for online learning from preference data. We assume that each user has a (potentially distinct) Condorcet winner, which is an arm preferred to every other arm. Using these user-specific Condorcet winners as reference points, we evaluate and score arms according to their performance relative to the corresponding winner. To promote fairness across heterogeneous users, we adopt the well-established Nash Social Welfare objective, which maximizes the product of user utilities, thereby inherently penalizing inequality and preventing the marginalization of any single user. Within this framework, we construct a hard instance to establish a regret lower bound of $\Omega(T^{2/3}\min(K,D)^\frac{1}{3})$ for a time horizon $T$, $K$ arms, and $D$ users, which, to the best of our knowledge, is the first result quantifying the cost of fairness in dueling bandits with heterogeneous preferences. We then present the Fair-Explore-Then-Commit and Fair-$\epsilon$-Greedy algorithms with a Condorcet winner identification phase. We further derive their regret upper bounds that match the lower-bound dependence on $T$ up to logarithmic factors.

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

This paper gives the first lower bound quantifying the cost of fairness in multi-user dueling bandits with heterogeneous Condorcet winners, plus two algorithms whose upper bounds match it up to logs.

read the letter

The main takeaway is that they prove a regret lower bound of Ω(T^{2/3} min(K,D)^{1/3}) under the Nash Social Welfare objective and supply two algorithms that achieve matching upper bounds. The setup uses per-user Condorcet winners as reference points and scores arms relative to each user's winner to avoid averaging preferences across heterogeneous users. Nash Social Welfare is applied to the product of those utilities, which penalizes any user being left with low utility. They build a hard instance to show the lower bound and then give Fair-Explore-Then-Commit and Fair-ε-Greedy, each with an explicit Condorcet-winner identification phase before the main exploration or commitment step. The upper bounds recover the T^{2/3} dependence on the time horizon. The assumption that every user has a Condorcet winner is stated up front and used consistently to define the utilities, so the analysis stays internally coherent. The hard-instance construction and the adaptation of standard bandit techniques to the multi-user fair case appear to be the actual new pieces relative to the single-user literature. A minor limitation is that the abstract and claims give no empirical checks or simulations, so we do not yet see how the algorithms behave on real preference data or when the Condorcet assumption is only approximately true. The work is aimed at people working on preference-based online learning and fairness constraints in bandits or RLHF. Anyone who cares about regret analysis with explicit fairness objectives will find the matching bounds useful. It is worth sending to a serious referee because the central claim is new, the technical approach is direct, and the internal logic holds together without obvious circularity or free parameters.

Referee Report

2 major / 2 minor

Summary. The manuscript studies fairness in multi-user dueling bandits under the assumption that each user has a (possibly distinct) Condorcet winner. It adopts the Nash Social Welfare objective to evaluate arms relative to each user's winner, constructs a hard instance yielding a regret lower bound of Ω(T^{2/3} min(K,D)^{1/3}), and presents the Fair-Explore-Then-Commit and Fair-ε-Greedy algorithms (each with an explicit Condorcet-winner identification phase) whose upper bounds match the lower bound's dependence on T up to logarithmic factors.

Significance. If the bounds are correct, the work supplies the first explicit quantification of the price of fairness in heterogeneous-preference dueling bandits. The matching T^{2/3} dependence (up to logs) together with the principled use of Nash Social Welfare and the two concrete algorithms constitute a substantive contribution to online learning from preference data, with direct relevance to applications such as RLHF.

major comments (2)

[Lower-bound section / abstract] The lower-bound construction (abstract and the dedicated lower-bound section) is load-bearing for the central claim that the result quantifies the cost of fairness. The manuscript must explicitly verify that the constructed hard instance preserves both the per-user Condorcet-winner property and the Nash Social Welfare utility definition; any deviation would invalidate the claimed Ω(min(K,D)^{1/3}) dependence.
[Algorithm sections and regret analysis] The upper-bound proofs for both algorithms rely on the sample complexity of the Condorcet-winner identification phase. The manuscript should state the precise number of rounds (or pulls) allocated to identification and show how this term is absorbed into the overall O(T^{2/3} polylog) regret without introducing an extra additive factor that would break the claimed matching.

minor comments (2)

[Abstract and regret theorems] Clarify whether the upper bounds also recover the min(K,D)^{1/3} dependence or only the T^{2/3} term; the abstract mentions only the T dependence.
[Notation and preliminaries] Ensure uniform notation for the number of users (D) and arms (K) across all theorems, lemmas, and algorithm pseudocode.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation for minor revision. The comments highlight important points for rigor in the lower-bound construction and clarity in the algorithmic analysis. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation without altering the core results.

read point-by-point responses

Referee: [Lower-bound section / abstract] The lower-bound construction (abstract and the dedicated lower-bound section) is load-bearing for the central claim that the result quantifies the cost of fairness. The manuscript must explicitly verify that the constructed hard instance preserves both the per-user Condorcet-winner property and the Nash Social Welfare utility definition; any deviation would invalidate the claimed Ω(min(K,D)^{1/3}) dependence.

Authors: We agree that explicit verification strengthens the lower-bound argument. The hard instance is constructed by partitioning arms and users such that each user has a unique Condorcet winner (with pairwise preference probabilities strictly greater than 1/2 against all other arms for that user), and the Nash Social Welfare is computed exactly as the product of per-user utilities relative to their respective winners. To address the comment, we will insert a short lemma immediately preceding the regret calculation that formally confirms both properties hold throughout the instance construction, including under the chosen gap parameters that yield the Ω(min(K,D)^{1/3}) factor. This addition is purely expository and does not affect the bound. revision: yes
Referee: [Algorithm sections and regret analysis] The upper-bound proofs for both algorithms rely on the sample complexity of the Condorcet-winner identification phase. The manuscript should state the precise number of rounds (or pulls) allocated to identification and show how this term is absorbed into the overall O(T^{2/3} polylog) regret without introducing an extra additive factor that would break the claimed matching.

Authors: We thank the referee for this clarification request. In Fair-Explore-Then-Commit, the identification phase allocates O(K log(KD T)) total pulls (using a tournament-style or pairwise comparison procedure with union bound over D users and failure probability 1/T). In Fair-ε-Greedy the identification is performed in an initial phase of the same order. Because this term is O(log T), it is absorbed into the existing polylogarithmic factors of the O(T^{2/3} polylog T) regret bound; the dominant terms arise from the subsequent exploration and commitment phases, which are calibrated to match the lower-bound exponent. We will add an explicit statement of the identification sample complexity together with a short paragraph in the regret analysis showing the absorption, ensuring no extraneous additive term appears. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the explicit assumption that each user has a distinct Condorcet winner, constructs an independent hard instance to obtain the Ω(T^{2/3} min(K,D)^{1/3}) lower bound, and then applies standard explore-then-commit and ε-greedy analysis to produce matching upper bounds for the two algorithms. The Nash Social Welfare objective is invoked as a pre-existing fairness criterion rather than being derived from the paper's own regret expressions or fitted quantities. No equation reduces a claimed prediction to a parameter fitted from the same data, no self-citation chain bears the central result, and the regret scalings are obtained through separate lower-bound construction and upper-bound analysis rather than by re-labeling inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of per-user Condorcet winners and standard concentration inequalities for regret analysis; no new entities are postulated and no parameters are fitted to data.

axioms (2)

domain assumption Each user possesses a Condorcet winner (an arm strictly preferred to all others)
Stated explicitly in the abstract as the reference point for scoring arms and constructing the hard instance.
standard math Standard regret analysis tools (concentration bounds, exploration-exploitation trade-offs) apply to the multi-user setting
Invoked implicitly when deriving both lower and upper bounds.

pith-pipeline@v0.9.0 · 5576 in / 1446 out tokens · 37444 ms · 2026-05-10T14:59:51.148962+00:00 · methodology

discussion (0)

Reference graph

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