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Provable Pluralistic Alignment: Multi-Party RLHF under Offline Human Feedback
Huiying Zhong, Tianwei Gao, Zhiwei Steven Wu, Linjun Zhang, Weijie J. Su, Zhun Deng
TL;DR
Offline RLHF commonly fits a single reward model, while pluralistic alignment must preserve potentially conflicting party preferences and still choose one collective policy. This paper jointly estimates party-specific rewards and optimizes explicit welfare objectives, with finite-sample guarantees under stated assumptions; it also treats general pairwise preferences using a pessimistic von Neumann winner.
Problem
Standard RLHF typically fits one reward model to pooled comparisons, implicitly assuming feedback from different users can share a common reward despite persistent, potentially conflicting preferences.
Source: §1 · p. 1Method
The paper jointly estimates party-specific rewards under a shared low-rank representation and uses pessimistic optimization for explicit social-welfare objectives.
Source: §1 · p. 3Results
The paper establishes nonasymptotic bounds for party-specific reward estimation and collective-policy suboptimality under realizability, diversity, and offline-coverage conditions.
Source: §1 · p. 4Takeaways & Limitations
The framework makes preference aggregation an explicit design choice rather than an implicit consequence of pooling feedback.
Source: §1 · p. 3In the reward-representable regime, each party's reward is assumed to be linear in known features with a parameter in a bounded set.
Source: §3 · p. 10
Abstract
from arXiv · showhide
Pluralistic alignment requires learning from feedback that reflects persistent and potentially conflicting stakeholder preferences while ultimately selecting a single collective policy. We study this problem in offline reinforcement learning from human feedback (RLHF), where the party associated with each comparison is observed. Under a shared low-rank linear reward model, we jointly estimate party-specific rewards and perform pessimistic policy optimization under Nash, Utilitarian, and Egalitarian social-welfare objectives. We establish nonasymptotic bounds for party-specific reward estimation and the resulting policy suboptimality under offline coverage conditions. We further consider general pairwise preferences that need not admit a scalar reward representation and may exhibit cycles. In this setting, we construct a pessimistic von Neumann winner policy and derive corresponding performance guarantees. Under these models, our results provide a unified finite-sample solution to a central challenge in pluralistic alignment: learning from limited, heterogeneous, and potentially cyclic feedback, and producing a single policy with explicit collective-welfare guarantees. Our framework thereby makes preference aggregation an explicit and statistically analyzable design choice rather than an implicit consequence of pooling human feedback.
1 Introduction
The paper studies how to learn one collective policy from offline comparisons while preserving observed party identities and disagreement. It combines party-specific preference learning with explicit welfare aggregation, and situates this approach among work on single-reward learning, personalization, and social choice.
1 Introduction
- Pooling can erase disagreement: in the example, it cannot identify compromise option B, while retaining party labels and applying Nash or Egalitarian aggregation selects B.The selected policy gives both parties reward 1/2.Source: §1 · p. 2
- The framework jointly learns unknown party-specific rewards from offline comparisons and selects one collective policy through an explicit aggregation rule.Unlike personalization, it does not produce a separate policy for each party.Source: §1 · p. 4
- Under a shared low-rank linear reward model, the paper gives finite-sample bounds for party-reward estimation and collective-policy suboptimality under realizability, diversity, and offline-coverage conditions.The analysis propagates estimation uncertainty through pessimistic Utilitarian, Nash, and Egalitarian objectives.Source: §1 · p. 4
- For general, potentially cyclic preferences, the paper constructs a pessimistic von Neumann winner from offline party-indexed comparisons and establishes finite-sample guarantees.The extension also analyzes ex-post efficiency under additional structural conditions.Source: §1 · p. 4
1.1 Related Work
- Prior offline RLHF theory mainly studies a single latent preference model, whereas this work retains party identities and propagates joint reward-estimation uncertainty through collective welfare objectives.The paper frames this as coupling preference estimation and aggregation rather than treating them as separate single-reward stages.Source: §1.1 · p. 5
- The analysis assumes observed, persistent party identities and non-strategic feedback, alongside the stated preference-model and offline-coverage conditions.The paper distinguishes these guarantees from complementary questions such as incentive compatibility and active elicitation.Source: §1.1 · p. 7
2 Preliminaries
The preliminaries define the offline comparison setting, where party-indexed feedback supports learning one collective policy. They then separate learning party-specific values from choosing a social-welfare rule to aggregate them.
2 Preliminaries
- The preliminaries define [n] as {1, 2, ..., n} and specify the vector and matrix norms used in the paper.They also define the matrix-induced norm and singular-value notation.Source: §2 · p. 7
2.1 Offline Data Collection and Comparison Model
- The contextual-bandit formulation models prompt-response evaluation with horizon H = 1 and no transitions, and the framework is later extended to general MDPs.The learner outputs one policy for all parties, rather than separate personalized policies.Source: §2.1 · pp. 7–8
- Offline data contain party-specific prompts, candidate-response pairs, and binary preference labels; feedback is pre-collected because obtaining it is resource-intensive and time-consuming.A party may represent an individual or a stakeholder group whose feedback providers contribute to the same dataset.Source: §2.1 · p. 8
2.2 Social Welfare Function
- Party value is aggregated using a chosen welfare objective, separating statistical preference learning from the normative choice of collective policy.The section focuses on Utilitarian, Egalitarian, and Nash welfare, which balance party values differently.Source: §2.2 · pp. 8–9
- Utilitarian welfare emphasizes average party value, Egalitarian welfare maximizes the worst-off party’s value, and Nash welfare balances aggregate value with sensitivity to lower-valued parties.The objectives assign equal normative weight to parties; other externally specified weights are possible but are not analyzed.Source: §2.2 · p. 9
3 Reward-Representable Regime
This section learns party-specific linear rewards from offline comparisons under a shared low-rank representation and selects a deterministic policy through pessimistic social-welfare optimization. It establishes finite-sample reward and policy guarantees, including approximate efficiency and extensions to alternative welfare objectives and trajectory feedback.
3 Reward-Representable Regime
- The reward-representable regime models party-indexed comparisons with BTL preferences over linear rewards and assumes their parameters share a low-rank representation.A fixed positive normalization makes policy values uniformly positive, and welfare guarantees are conditional on that normalization.Source: §3 · pp. 10–11
3.1 Pessimistic Nash Welfare Algorithm
- The algorithm jointly estimates the shared representation and distinct party rewards, builds party-specific confidence sets, and maximizes Nash welfare under the least favorable plausible rewards.This pessimism is intended to guard policy selection against estimation error from limited offline coverage.Source: §3.1 · p. 11
3.2 Sub-optimality
- Under the stated identifiability and feature-design assumptions, the pessimistic Nash policy has finite-sample suboptimality guarantees when offline data provide sufficient coverage.The upper-bound analysis extends the estimation-error improvement from shared low-rank learning; a lower bound identifies unavoidable representation-dimension dependence and a remaining gap.Source: §3.2 · pp. 13–15
3.3 Efficiency and Welfare Guarantees
- The Nash welfare guarantee implies approximate Pareto efficiency: the learned policy cannot substantially improve one party without worsening another, and it follows the approximate Pigou-Dalton principle.The efficiency statement is established with high probability under the same conditions as the reward-estimation guarantee.Source: §3.3 · p. 15
3.4 Comparison with Various Social Welfare Functions
- Pessimistic Utilitarian and Egalitarian variants inherit the same statistical rate while optimizing average and worst-off-party welfare, respectively.For MDPs, the framework instead uses trajectory-based preferences and values with a trajectory occupancy measure, and retains a suboptimality upper bound.Source: §3.4 · pp. 16–17
- Generation follows the transition kernel, with details deferred to Appendix A.3.Source: §3.4 · p. 17
4 Extension to General Pairwise-Preference Model
The general pairwise-preference model handles preferences that may be nontransitive or cyclic without requiring scalar rewards. It uses party-indexed comparisons to construct a pessimistic von Neumann winner, with approximate preference and conditional efficiency guarantees.
4 Extension to General Pairwise-Preference Model
- Preferences are represented directly by party-specific pairwise comparisons, allowing stochastic, nontransitive, or cyclic preferences that need not admit scalar rewards.The analysis focuses on finite state and action spaces and uses state-wise preference matrices.Source: §4 · p. 17
4.1 Pessimistic Von Neumann Winner Algorithm
- The algorithm aggregates party-specific preference estimates with equal weights, forms entrywise confidence bounds, then solves a zero-sum matrix game using the pessimistic payoff matrix.The resulting state-wise max-min strategies induce a randomized collective policy.Source: §4.1 · pp. 18–19
4.2 Efficiency Guarantees
- With probability at least 1 − δ, the learned strategy has worst-case population preference margin at least −ε, statewise and over the policy space.The error decreases at rate n^-1/2, grows with action-space size, and depends logarithmically on party and state counts apart from concentratability.Source: §4.2 · pp. 19–20
- When each party’s preferences are transitive, the learned strategy is τ-approximate ex-post efficient with probability at least 1 − δ.The guarantee makes Pareto-dominated alternatives unlikely to be selected; the paper states that transitivity is necessary for this result.Source: §4.2 · p. 20
5 Numerical Illustrations
The simulations compare party-aware learning with pooled baselines in reward-representable and general pairwise settings. Across both settings, the proposed methods show lower suboptimality patterns than pooling; the paper also identifies theoretical and practical boundaries for extending the framework.
5 Numerical Illustrations
- The simulations are stylized contextual-bandit illustrations of heterogeneous preferences and unequal party sample sizes.The reward-representable experiment uses 24 parties, while the general pairwise experiment uses one state, three actions, and two parties.Source: §5 · p. 20, §5.1 · p. 20, §5.2 · p. 21
5.1 Reward-Representable Model
- Across the reported sample sizes, the joint low-rank method achieves lower Nash welfare suboptimality than independent and pooled baselines.Both pooling baselines retain a substantial gap in the constructed setting with three clusters of heterogeneous preferences.Source: §5.1 · p. 21
- The Nash-targeted policy also has lower Utilitarian and Egalitarian suboptimality than pooling-obs, with a more balanced party-level regret profile.The reported ratios are below one for these comparisons.Source: §5.1 · p. 21
5.2 General Pairwise Preference model
- In the general pairwise simulation, game-value suboptimality decreases toward zero with more samples, while the observation-pooling baseline retains a non-vanishing gap.The experiment uses pairwise outcomes only; neither method receives the underlying rewards, features, or reward parameters.Source: §5.2 · p. 21
6 Conclusion
- The conclusion characterizes the guarantees as conditional on realizability, diversity, normalization, and offline-coverage conditions.The paper describes the general pairwise method as providing an approximate max-min preference guarantee and limited probability on Pareto-dominated alternatives under additional structural conditions.Source: §6 · p. 22
7 Discussion
- For the general pairwise model, linear programming can compute the matrix-game step efficiently in tabular cases.The reward-based constrained max-min optimization is described as computationally heavy, though gradient ascent-descent is a possible implementation approach.Source: §7 · p. 23
- Extending shared multi-party reward learning beyond linear models while retaining finite-sample confidence guarantees remains open.The discussion also identifies uncertainty about latent or overlapping party membership and evolving preferences, and scalable methods for practical LLM alignment, as future work.Source: §7 · p. 23
A Further Details on Methods
The appendix clarifies how welfare objectives relate to individual utility properties, connects the pairwise and reward-based models, and extends the reward-based setting to MDPs. It also records parameter and feature-difference notation used in technical proofs.
A.1 Social Welfare Function Details
- Nash welfare is singled out for independence of individual utility scales, Utilitarian welfare for independence of individual zeros, and Egalitarian welfare reduces to maximizing the minimum party utility.These properties distinguish how the objectives respond to party-specific utility transformations.Source: §A.1 · p. 29
A.2 Connections Between Two Models
- With reward-based preferences and one or two parties, the general model’s von Neumann winner aligns with the Utilitarian solution.For two parties, the average pairwise preference comparison favors an action pair when the sum of the parties’ rewards favors it.Source: §A.2 · p. 30
A.3 Generalization to MDPs
- The MDP extension models preferences over trajectories and uses trajectory-based occupancy in its value function and coverage coefficient.The transition kernels are known, allowing the occupancy measure for a policy to be calculated; the stated suboptimality scales with the concentratability coefficient.Source: §A.3 · pp. 30–31
B Technical Details for Reward-Based Model
- The technical section defines vectorized parameter and feature-difference notation for party-specific reward estimates.It also specifies the matrix-induced norm, orthogonal-matrix notation, inner product, and singular-value convention used in the proofs.Source: B Technical Details for Reward-Based… · p. 32
B.1 Proof of Theorem 3.3
The proof material establishes concentration results for shared low-rank reward estimation and uses confidence sets to support the policy suboptimality argument. The supplied proof passages state the assumptions and proof structure, but do not include the full theorem bounds.
B.1 Proof of Theorem 3.3
- The estimation proof combines covariance concentration, a low-rank decomposition, an ε-net argument, and high-probability bounds.The excerpt also invokes a source-data guarantee and a target-data guarantee under the stated sample-size condition.Source: §B.1 · pp. 32, 35, 37
- Under Assumptions 3.1 and 3.2 and n ≫ d + log(M/δ), the proof establishes a covariance bound for every party.The supplied result states that ∥Σm − Σ∗∥ ≤ 0.1∥Σ∗∥ under this sample-size condition.Source: §B.1 · p. 32
B.2 Proof of Theorem 3.4
- The policy proof decomposes suboptimality and uses confidence-set coverage to bound the estimation contribution.The argument also uses the pessimistic policy’s maximization of its pessimistic objective.Source: §B.2 · p. 41
B.3 Proof of Theorem 3.5
The proof constructs a family of contextual-bandit instances to establish a lower bound, then extends the construction to multiple parties using perfectly coupled datasets.
B.3 Proof of Theorem 3.5
- For each constructed instance, the optimal policy selects either a1 or a2, and the feature-based coverage quantity is bounded.The instances are indexed by τ in {−1,1}^|S|.Source: §B.3 · p. 42
- Applying Assouad’s lemma yields an instance attaining the lower bound for the single-party case M = 1.The proof is adapted from Theorem 3.10 of Zhu et al. [2023].Source: §B.3 · p. 42
- The multi-party lower-bound construction reuses one dataset for all parties, so the learner sees n distinct comparisons rather than Mn; this coupling is exclusive to the proof.Elsewhere in the paper, party-specific datasets are generated independently.Source: §B.3 · p. 42
B.4 Proof of Theorem 3.7
These proofs establish an approximate Pigou-Dalton property for the selected policy, detail the welfare-case arguments, and extend reward-estimation bounds to the MDP setting.
B.4 Proof of Theorem 3.7
- With n ≫ d + log(M/δ), the selected policy satisfies a τ-approximate Pigou-Dalton principle with probability at least 1 − δ.The principle bounds the welfare increase from reducing the gap between two parties’ values while retaining their total value.Source: §B.4 · pp. 43–44
B.5 Proofs in Section 3.4
- The Egalitarian proof uses a sorted-values inequality to compare corresponding ordered party values, while the Utilitarian proof proceeds by summing party contributions.The excerpt presents the sorted-values lemma as the main difference in the Egalitarian argument.Source: §B.5 · pp. 44–45
B.6 Proof of Theorem A.3
- Under Assumptions 3.1 and 3.2 and n ≫ d + log(M/δ), Theorem B.8 gives a simultaneous high-probability bound for each party in the MDP setting.The feature-difference condition in Assumption 3.2 is changed to a cumulative difference over the trajectory.Source: §B.6 · p. 45
C.1 Proof of Theorem 4.1
The proof of Theorem 4.1 uses concentration inequalities to control empirical comparison preferences and establish conditions on a policy distribution.
C.1 Proof of Theorem 4.1
- The proof introduces McDiarmid and binomial concentration inequalities to control deviations in the comparison data.The cited inequalities are attributed to Vershynin [2018] and Xie et al. [2021].Source: §C.1 · p. 47
- For each party, state, and action pair, the proof applies concentration to binary preference responses observed in that party’s dataset.The argument treats the zero-observation case separately.Source: §C.1 · p. 47
- The proof states that the policy-space distribution must satisfy specified conditions, then separates the analysis by state to verify them.The excerpt does not provide the conditions’ full expressions.Source: §C.1 · pp. 49–50
C.2 Proof of Theorem 4.3
The proof of Theorem 4.3 examines how transitivity affects guarantees for a pessimistic von Neumann winner, including a deterministic-preference example where a dominated item remains in the outcome.
C.2 Proof of Theorem 4.3
- The argument for Theorem 4.3 follows a result similar to Fishburn [1984] and then considers an example to demonstrate the necessity of transitivity.The appendix also identifies its experimental-detail section as supplementary to Section 5.Source: §C.2 · pp. 50–51, D Experimental Details · p. 51
- The proof derives a bound on the probability assigned to item b in terms of ε and the pairwise preference quantity N*_s(a,b).The bound follows from the stated inequality ε ≥ bps(b)^2 N*_s(a,b).Source: §C.2 · pp. 50–51
- With deterministic preferences, the von Neumann winner is (1/3,1/3,1/3), even though item b is Pareto dominated by item a; the prior theorem’s result therefore requires transitivity.The proof states that without transitivity, preferences may be inconsistent.Source: §C.2 · p. 51
D.1.1 Dataset Construction.
The experiment creates heterogeneous party preferences with a shared three-dimensional reward structure embedded in a 24-dimensional comparison space. It compares joint low-rank pessimistic Nash optimization with independent and pooled baselines.
D.1.1 Dataset Construction.
- The experiment uses 3 states, 9 actions, and 24 parties, whose heterogeneous rewards share a hidden three-dimensional representation in a 24-dimensional feature space.The learner does not observe the representation used to generate the data.Source: §D.1.1 · pp. 51–52
- Three party clusters favor specialist actions, while a compromise action rewards all clusters and creates a collective trade-off.The remaining actions carry no true reward signal, and one action serves as the common comparison reference.Source: §D.1.1 · pp. 52–53
- Twenty-one nuisance coordinates provide orthogonal comparison directions and full-rank ambient covariance without affecting true rewards.The joint estimator can exploit the shared three-dimensional representation, unlike an unrestricted estimator operating in all 24 dimensions.Source: §D.1.1 · p. 53
D.1.2 Learning methods and baselines.
- The method jointly estimates the shared representation and party-specific coefficients, then selects the policy maximizing the geometric mean of party-specific worst-case values.Pessimistic values are optimized over the intersection of each confidence ellipsoid and parameter ball.Source: §D.1.2 · pp. 54–55
- The baselines contrast unrestricted party-level estimation with pooling that weights either each observation equally or each party equally.The independent baseline estimates an unrestricted parameter for each party; the two pooling baselines discard party identities.Source: §D.1.2 · p. 54
D.1.3 Evaluation Metric.
The evaluation measures welfare suboptimality, party-regret dispersion, confidence-set coverage, and estimation error. The joint estimator has lower reported error than the independent estimator, while the selected confidence-set multiplier yields high empirical coverage.
D.1.3 Evaluation Metric.
- The evaluation reports Nash welfare suboptimality and compares Nash-targeted policies under Utilitarian and Egalitarian objectives.It also measures differences between parties’ average classical regrets as a partial indicator of how evenly regret is distributed.Source: §D.1.3 · pp. 55–56
D.1.4 Confidence-set coverage.
- For K = 2.4, aggregate marginal coverage is 99.96%, mean simultaneous coverage is 99.15%, and minimum simultaneous coverage at an individual sample size is 98%.Simultaneous coverage requires the confidence event to hold for all 24 parties in the same repetition.Source: §D.1.4 · p. 56, Figure 3 · p. 56
D.1.5 Estimation accuracy and efficiency.
- At n = 150, joint-estimator mean error is 0.399 versus 0.622 for the independent estimator; at n = 3000, errors are 0.119 versus 0.225.The joint estimator has lower error throughout the reported sample-size range.Source: §D.1.5 · p. 57, Figure 4 · p. 57
D.2 Further Details for the General Pairwise-Preference Experiment (Section 5.2)
The experiment tests whether party-balanced aggregation preserves a collective preference winner when parties provide unequal numbers of offline comparisons. Pooling instead selects a different winner, while the full confidence bonus achieves simultaneous coverage in the reported instance.
D.2 Further Details for the General Pairwise-Preference Experiment (Section 5.2)
- The one-state experiment has two parties and three actions; the BTL model generates preferences, but methods observe only pairwise comparison outcomes.Party 2 supplies twice as many comparisons as Party 1, with comparisons distributed evenly among action pairs.Source: §D.2 · pp. 57–58
- Pooling comparisons makes a2 the Condorcet winner instead of a1, producing suboptimality 0.02098 under the equally weighted target.Party-balanced normalization preserves the intended equal party weights.Source: §D.2 · p. 58
- The full confidence bonus covers in every repetition at every sample size, while coverage is at least 99.4% for c = 0.5 and falls below 95% for c ≤ 0.3.The algorithm and main experiment use c = 1; the multiplier is diagnostic, not an algorithmic hyperparameter.Source: §D.2 · pp. 58–59, Figure 5 · p. 59