Bayesian Optimization for Building Social-Influence-Free Consensus

Anurag Singh; Krikamol Muandet; Masaki Adachi; Michael A. Osborne; Siu Lun Chau; Wenjie Xu

REVIEW 5 major objections 5 minor 1 cited by

Bayesian Optimization for Building Social-Influence-Free Consensus

T0 review · 5 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Social Bayesian Optimization can reach a social-influence-free consensus mostly from cheap public votes by learning the social graph faster than the utilities themselves.

desk verdict Novel problem setup and clear algorithm, but the central cost-saving guarantee is undone by two algebra errors in the appendix; reject as written, worth a careful revision. read the letter →

arxiv 2502.07166 v1 pith:7DKD6VNL submitted 2025-02-11 cs.MA cs.GTcs.LGstat.ML

Masaki Adachi , Siu Lun Chau , Wenjie Xu , Anurag Singh , Michael A. Osborne , Krikamol Muandet This is my paper · ORCID

classification cs.MAcs.GTcs.LGstat.ML

keywords socialBayesianoptimizationconsensusbuildinginfluencedualvotinggraphconvolutionpreferencelearningno-regretgroupthink

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that group consensus can be found without paying for private votes most of the time, despite public votes being distorted by social influence. It first proves an impossibility theorem: with noisy feedback alone, no aggregation rule can be guaranteed to preserve the social-influence-free consensus unless everyone already agrees on the best option. To get around that, it introduces Social Bayesian Optimization, which uses a cheap public vote and an expensive private vote for the same pairwise comparison, models the influence as an unknown graph convolution $v = Au$, and estimates both the graph and the utilities from the two vote streams. The main guarantee is that when the influence graph is identifiable, the cumulative regret $R_T/T$ and the fraction of private votes $|Q^u_T|/T$ both go to zero as $T\to\infty$, because the graph estimate converges at a faster rate than the utility estimates. If the guarantee holds, a facilitator can mostly run show-of-hands votes and still converge to the consensus of the true aggregated preferences.

What carries the argument

The load-bearing identity is the graph convolution $$v(\cdot,:)^{\top} = A u(\cdot,:)^{\top},$$ with $A$ a row-stochastic, strictly positive adjacency matrix; the algorithm treats private utilities $u$ and public utilities $v$ as linked by this unknown $A$. The rest of the machinery is optimistic MAP: confidence sets of utility functions whose log posterior lies near the MAP estimate, an acquisition function that maximizes the upper confidence bound of social-utility improvement, and a stopping criterion $w_t^u(x_t,x_{t-1})$ that stops querying private votes once the graph is sufficiently well estimated.

What would settle it

Run SBO on synthetic agents whose true influence is nonlinear, for example $v(x,i) = \sigma(\sum_j A_{ij}u(x,j))$, while keeping utilities in the same RKHS family; if the private-vote fraction $|Q^u_T|/T$ fails to vanish or the cumulative regret stops converging under the same identifiability condition, the linear-convolution premise is falsified.

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Extended reading notes

Core claim

The central claim is that the influence-free consensus $x^\star$ is reachable without sustained private voting. Under the dual voting design, public votes follow the corrupted utility $v$, private votes follow the truthful utility $u$, and the link is the linear convolution $v(\cdot,:)^{\top} = A u(\cdot,:)^{\top}$ with an unknown row-stochastic adjacency matrix $A$. The paper proves that if $A$ is identifiable from the votes, SBO has no-regret cumulative regret and its private-vote complexity satisfies $|Q^u_T|/T\to0$, because the graph identification error decays like $|Q^u_t|^{-1/2}$, faster than the pointwise utility error $|Q^u_t|^{-1/4}$. This is the precise sense in which cheap public votes can replace expensive private votes after an initial learning phase.

Load-bearing premise

The debiasing step collapses if social influence is not exactly a fixed linear mixture of the agents' true utilities, written $v = Au$, that is recoverable from the noisy public and private vote pairs the algorithm collects; nonlinear, time-varying, or unidentifiable influence would break the graph-learning speedup.

Editorial extensions

If this is right

With an identifiable influence graph, SBO is no-regret: the average error between its consensus estimate and the true social optimum vanishes as $T\to\infty$.
The number of expensive private votes grows sublinearly, so a facilitator can eventually stop one-on-one interviews and rely on public votes debiased by the learned graph.
If the graph is known and invertible, only public votes are needed, giving $|Q^u_T|=0$, but estimation noise is amplified by up to $\|A^{-1}\|\le n$.
If the graph is not identifiable, the guarantee reverts to private-votes-only behavior with linear private-vote complexity, so identifiability is exactly what buys the cost saving.
The decay parameter $q$ trades convergence speed against the number of private votes, and the experiments set $q=1/2$ as a practical balance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The model's symmetry in $u$ and $v$ leaves implicit a reverse application: if public discussion corrects individual mistakes rather than corrupting them, the same graph convolution could turn cheap public votes into the signal and private votes into a calibration check.
A natural stress test beyond the paper is to replace the linear graph convolution with a nonlinear or time-varying influence mechanism and observe whether the faster graph-learning rate and vanishing private-vote fraction still hold.
The identification-rate gap between $|Q^u_t|^{-1/2}$ and $|Q^u_t|^{-1/4}$ suggests the economic value of the method comes from recovering a low-dimensional linear structure, so the practical benefits should shrink as the influence process becomes more complex.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper introduces a genuinely new problem—consensus-building under social influence via dual public/private votes with an unknown influence graph—and the SBO algorithm is clearly described and sensibly motivated. The writing is good, and the experiments, while only synthetic, cover several plausible real-world tasks. That is real value.

The theoretical core, however, does not hold as written. Two load-bearing algebraic mistakes invert the main claims. First, the private-vote sample complexity derivation in Appendix E.6.2 yields |Qu_T| = O(T^{4q} ...), not the O(T^q) reported in Theorem 3.5 and Table 1. The appendix even states that sublinearity requires q ≤ 1/4, while the main text claims 0 < q < 1 and the experiments set q = 1/2. With q = 1/2, the proven bound is T^2, which gives no private-vote savings. This directly contradicts the paper's headline promise that SBO provably relies less on costly private votes.

Second, the graph identification bound in Appendix E.8.3 has the strong-convexity constant m on the wrong side. From MER ≥ (m/2)||Â - A||^2 and MER ≤ O(n^2/|Qu|), you get ||Â - A||^2 ≤ O(n^2/(m |Qu|)), not O(m n^2/|Qu|). With their chosen m ≈ 1/n^2, the correct bound grows with n^2 instead of vanishing at the claimed rate. The asymptotic graph-identification argument collapses. There is also a smaller problem: the impossibility theorem's proof uses dictatorial graphs with zero entries, which violate the strict positivity of Eq. (3). That one is probably patchable by taking a limit, but it is not valid as written.

The exponential-family likelihood and graph prior are standard, and the citations look appropriate. The problem is not circularity; the flaws are concrete mathematical errors. Because these errors sit at the heart of the paper's guarantees, I would not cite it as a reliable result right now. I would, however, send it to referees rather than desk-reject: the problem formulation is novel enough that a careful revision could be publishable, and reviewers can point to exactly what needs fixing.

Referee Report

5 major / 5 minor

Summary. The paper proposes Social Bayesian Optimization (SBO), a dual-voting algorithm for collective decision-making under social influence. It models social influence as an unknown row-stochastic graph convolution v = Au, combines cheap public votes with expensive private votes, and claims (i) an impossibility theorem for groupthink-proof aggregation, (ii) a no-regret bound on cumulative regret, and (iii) sublinear sample complexity of private votes. The theoretical results are supported by several appendices and evaluated on synthetic and real-world tasks.

Significance. The problem of reaching a consensus free of social influence is timely and practically relevant, and the dual public/private voting mechanism is an interesting formulation. If the theoretical claims were correct, the paper would make a useful contribution to preferential Bayesian optimization and social-choice-inspired algorithm design. The authors also provide a wide experimental evaluation across four real-world tasks. However, the central theoretical guarantees are not established as written: the private-vote sample-complexity proof contradicts the paper's own tables, the impossibility proof uses matrices outside the admissible model, and the graph-identification proof contains an algebraic error. These issues are load-bearing for the main claims.

major comments (5)

[Appendix E.6.2, Eqs. (106)-(110), Table 1, Table 2] The proof of the private-vote sample complexity in Appendix E.6.2 derives |Qu_T| = O(T^{4q} (γuu'_T)^2 log T N/δ), and explicitly concludes that sublinearity requires q ≤ 1/4. However, Table 1 and Table 2 report |Qu_T| = O(T^q (γvv'_T)^2 log T N/δ), and Section 3.4 and the experiments set q = 1/2. With q = 1/2, the derived bound is O(T^2), which is not sublinear and gives no private-vote savings. This directly contradicts the central claim that SBO reduces reliance on costly private votes while maintaining no-regret.
[Appendix C.2, proof of Theorem 3.3] The impossibility proof considers the set G_dictatorial = {G_i | A_jk = I{i=k}} whose adjacency matrices have zero off-diagonal entries. This violates the strict positivity requirement A_ij > 0 in Eq. (3) for the social-influence model. The proof therefore establishes impossibility for a class of graphs outside the model, not for the model asserted in the paper. The theorem may be repairable by a limiting argument with ε-perturbations, but as written the proof is not sound.
[Appendix E.8.3, Eqs. (131)-(140)] The derivation of the graph-identification rate contains an algebra error. From MER ≥ (m/2)||Â - A||^2 and MER ≤ O(n^2/|Qu_t|), the correct conclusion is ||Â - A||^2 ≤ O(n^2/(m|Qu_t|)), not O(m n^2/|Qu_t|). The paper's subsequent choice m ≈ 1/n^2 yields, with the correct formula, ||Â - A|| = O(n^2/√|Qu_t|) rather than the claimed O(1/√|Qu_t|). The claimed faster graph-learning rate and the stated q = 1/2 optimality are therefore not established.
[Section 3.4, condition (c); Lemma B.5] Condition (c) assumes A is identifiable from votes, and Lemma B.5 proves identifiability from full-rank exact utility pairs (u(xτ, :), v(xτ, :)). The algorithm, however, collects noisy Bernoulli comparisons at adaptively chosen pairs (x_t, x_{t-1}); it is not shown that such data identify A, nor that the adaptive query process guarantees the full-rank condition. The identifiability assumption is load-bearing for both the graph-estimation theorem and the debiasing step, and the paper does not bridge this gap.
[Appendix E.5.2, Assumptions E.7 and E.8] The regret bound in Theorem 3.5 relies on Assumptions E.7 and E.8, which assert pointwise inequalities of the optimistic estimate and a Lipschitz property of sorting-induced weight differences. These assumptions are stated as postulates rather than derived from the model; the justification for E.7 via monotonicity does not imply the needed pointwise dominance at both x_t and x_{t-1}. Since Lemma E.11 and the main regret bound depend on these assumptions, the no-regret claim is conditional on assumptions that remain unverified.

minor comments (5)

[Abstract] The word 'findigns' should be 'findings'.
[Assumption 2.2] 'Bradly-Terry' is a typo; the standard name is Bradley-Terry.
[Table 5] 'Tiknohov' should be 'Tikhonov'.
[Section 5 and Figure 4] The figure captions are missing the legend for some curves; consider adding a consistent legend across Figures 4-8.
[Appendix H.2.1] The social-influence matrix for Thermal comfort repeats in EnergyTrading; if this is intentional, state it explicitly to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the regret and private-vote bounds are derived from explicit assumptions and compared against the independently defined consensus; the proof contains a separate algebraic error, but that is a correctness concern, not a circularity.

full rationale

The central derivation chain is not circular. The target consensus x* is defined in Eq. (1) as the argmax of A[u(x,·)] over an independently specified aggregation function and truthful utilities, while the algorithm only receives noisy Bernoulli votes (Assumption 2.2) and models influence as v = Au (Eq. 3). Cumulative regret RT and private-vote count |Qu_T| are measured against this ground-truth x*, not against any quantity inserted by the algorithm. The impossibility theorem (Theorem 3.3) is proved directly from the definition of groupthink-proofness and does not assume the algorithm's conclusion. The regret and graph-identification bounds rest on explicit assumptions (identifiability condition (c), RKHS boundedness, Bradley-Terry feedback); these are assumptions, not the target result. The likelihood-ratio confidence-set machinery cited from Xu et al. [2024a] is general preferential-BO theory and does not assume the social-influence-free consensus target, so it serves as independent support even though one current author is also an author of that prior work. The paper's Appendix E.6.2 contains a notable non-circular correctness issue: solving Eqs. (106)-(110) yields |Qu_T| = O(T^{4q} ...), not the O(T^q ...) reported in Table 1 and Table 2, and with q = 1/2 that would not establish sublinear private-vote complexity. This is a proof gap or over-claim, not a case of a prediction reducing to its inputs by construction. No load-bearing step is definitionally identical to, or fitted from, the quantity it claims to predict.

Assumptions & free parameters 5 free parameters · 6 assumptions · 1 invented entities

The central machinery rests on a linear, row-stochastic, strictly positive social graph plus the assumption that it is identifiable from votes. These are domain assumptions rather than derived facts, and the identifiability assumption is stronger than what is proved. The graph prior hyperparameters delta_A, xi, kappa_i, and the stopping-rate q are hand-set and some appear in the theoretical rate directly.

free parameters (5)

rho (GSF weight) = 0.1, 0.5, 1.0 in experiments
Chooses the tradeoff between utilitarian and egalitarian aggregation. The facilitator provides it a priori, but it directly changes the objective being optimized.
q (decay rate) = 0.5
Controls the private-vote stopping threshold in Algorithm 1. The paper claims q = 1/2 is optimal, but its own proof of |Qu_T| requires q <= 1/4 for a sublinear bound.
delta_A = 0.01
Minimum-entry threshold in the graph prior, used to enforce strict positivity of A.
xi (Tikhonov parameter) = 1/2 delta_A^2 n^2
Regularization strength in the graph prior; combined with kappa_i it controls the strong-convexity constant in the graph identification proof.
kappa_i (Dirichlet concentration) = 1 + 1/n^2(2 delta_A^2 - 1) in Table 5; 1 + delta_A^2/n^2 - 2 xi delta_A^2 in Appendix E.8
Concentration parameters of the row-wise Dirichlet prior on the graph. The two stated forms are inconsistent, and Appendix E.8 sets them to force a strong-convexity constant that yields the desired graph rate.

assumptions (6)

domain assumption Pairwise feedback follows the Bradley-Terry model: P(1=1) = sigma(u(x,i) - u(x',i)).
Assumption 2.2 is the likelihood model that links unobserved utilities to observed votes.
domain assumption Public votes reflect corrupted utilities v = A u, while private votes reflect true utilities u.
Assumption 3.4 plus Eq. (3). This linear convolution is the entire mechanism that makes debiasing possible.
domain assumption The social graph A is row-stochastic and strictly positive: sum_j A_ij = 1 and A_ij > 0 for all i,j.
Eq. (3). Strict positivity is used in the graph prior and Lemma B.3, but it also rules out the dictatorial graphs used in the impossibility proof.
ad hoc to paper A is identifiable from votes.
Condition (c) in Section 3.4. Lemma B.5 only establishes identifiability from full-rank exact utility pairs, not from the binary pairwise comparisons the algorithm queries.
domain assumption u and v lie in RKHS spaces with known norm bounds.
Assumption A.6. Standard in Bayesian optimization but strong for human utilities.
ad hoc to paper Assumptions E.7 and E.8: the optimistic acquisition is pointwise above the truth at x_t and below at x_{t-1}, and the sorting-induced weight differences are Lipschitz.
These assumptions are introduced specifically to make the regret decomposition in Lemma E.11 go through; the paper calls them 'justifiable' but does not derive them from more basic properties.

invented entities (1)

Latent social influence graph A
purpose: Models how each agent's true utility is mixed with neighbors' utilities to produce the publicly observed, influenced utility.
The graph is inferred from the same public/private vote pairs that define it. The paper offers no external measurement, survey, or falsifiable prediction that would validate A outside the model.

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Cite this review

Pith. "Pith review of Bayesian Optimization for Building Social-Influence-Free Consensus." pith.science (2026). https://pith.science/paper/7DKD6VNL

@misc{pith2026250207166,
  author       = {Pith},
  title        = {Pith review of: Bayesian Optimization for Building Social-Influence-Free Consensus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7DKD6VNL}},
  note         = {Machine review of arXiv:2502.07166}
}

read the original abstract

We introduce Social Bayesian Optimization (SBO), a vote-efficient algorithm for consensus-building in collective decision-making. In contrast to single-agent scenarios, collective decision-making encompasses group dynamics that may distort agents' preference feedback, thereby impeding their capacity to achieve a social-influence-free consensus -- the most preferable decision based on the aggregated agent utilities. We demonstrate that under mild rationality axioms, reaching social-influence-free consensus using noisy feedback alone is impossible. To address this, SBO employs a dual voting system: cheap but noisy public votes (e.g., show of hands in a meeting), and more accurate, though expensive, private votes (e.g., one-to-one interview). We model social influence using an unknown social graph and leverage the dual voting system to efficiently learn this graph. Our theoretical findigns show that social graph estimation converges faster than the black-box estimation of agents' utilities, allowing us to reduce reliance on costly private votes early in the process. This enables efficient consensus-building primarily through noisy public votes, which are debiased based on the estimated social graph to infer social-influence-free feedback. We validate the efficacy of SBO across multiple real-world applications, including thermal comfort, team building, travel negotiation, and energy trading collaboration.

Figures

Figures reproduced from arXiv: 2502.07166 by the authors.

**Figure 1.** A dual voting system: the difference in public and private votes can identify the social influence. Once identified, social-influence-free consensus can be estimated only from noisy public votes, thereby reducing the total cost. requires maximizing the aggregated preference—a task that presents another significant challenge. Classical economics often assumes that utility functions [Fishburn, 1968] are given analytic… view at source ↗

**Figure 2.** Pareto frontier corrupted by the social influence [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The dotted and solid lines represent the ground truth u, v and estimated utilities u, ˜ v˜, respectively, with the shaded area indicating the confidence interval. Dots mark the queried points, where |Qu t | = 10 and |Qv t | = 20. Since utility values are not directly observable, the dots are for visual guidance. The acquisition function α(·, xt−1) represents the upper confidence bound of improvement from the previou… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Robustness analysis was conducted using the function shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Real-world experiments with varying number of agents n and aggregation rule ρ completely independent; (d) SBO (ours): Assumes v = Au with the graph prior p(A). See Appendix H for experimental setup details for reproducibility. Along with cumulative regret and query co…

**Figure 6.** Figure 6: Simple regret, cumulative regret, and cumulative queries on different social influence graph. [PITH_FULL_IMAGE:figures/full_fig_p049_6.png]

**Figure 7.** Figure 7: Simple regret, cumulative regret, and cumulative queries on different aggregation function. [PITH_FULL_IMAGE:figures/full_fig_p049_7.png]

**Figure 8.** Figure 8: Computation time on real-world tasks. We report the computation time on [PITH_FULL_IMAGE:figures/full_fig_p050_8.png]

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Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

In a strong sense , u(x, :) ≻ u(x′, :) for x ∈ Xwhenever there exist i, j∈ V such that (a) ∀k ∈ V \{i, j} u(x, k) = u(x′, k), (b) u(x, i) + u(x, j) = u(x′, i) + u(x′, j), and (c) |u(x′, i) − u(x′, j)| > |u(x, i) − u(x, j)|

work page

[2] [2]

In a weak sense , we only require that u(x, :) ⪰ u(x′, :). Intuitively, given a utility vector u(x, :), if an agent with a higher utility transfers ≤ 1/2 of its excess utility to another worse-off agent, the aggregate function A should prefer the transferred utility over the original for the fairness. A.4 ASSUMPTIONS Assumption A.6 (Bounded norm). For eac...

work page 2023

[3] [3]

TripAdvisor: Three colleagues (two influencers and a follower) are deciding on a hotel for their upcoming group retreat, using the TripAdvisor website. One prefers a luxurious hotel with the highest ranking, another seeks a budget-friendly option with reasonable reviews, and the third prioritizes the hotel with the highest overall review score. The facili...

work page 2023

[4] [4]

Boots", ’Ankle socks’ (agent 2)

EnergyTrading: Three firms (a large corporation, a niche startup, and a joint venture) form a strategic collaboration to enhance their energy trading business. Their profits rely on a machine learning model that predicts day-ahead market prices, which in turn depends on a demand dataset. Due to the scarcity of such data, they jointly invest in a market re...

work page 2020

[5] [5]

v35i10.17103

URL https://doi.org/10.1609/aaai. v35i10.17103. Majid Abdolshah, Alistair Shilton, Santu Rana, Sunil Gupta, and Svetha Venkatesh. Multi-objective Bayesian optimisation with preferences over ob- jectives. In Advances in Neural Information Processing Systems (NeurIPS) , volume 32, 2019. URL https://proceedings.neurips. cc/paper_files/paper/2019/file/ a7b7e4...

doi:10.1609/aaai

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v38i13.29364

URL https://doi.org/10.1609/aaai. v38i13.29364. Masaki Adachi, Brady Planden, David Howey, Michael A Osborne, Sebastian Orbell, Natalia Ares, Krikamol Muandet, and Siu Lun Chau. Looping in the hu- man: Collaborative and explainable Bayesian optimiza- tion. In International Conference on Artificial In- telligence and Statistics (AISTATS) , pages 505–513. P...

arXiv doi:10.48550/arxiv.2410.10452

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Eric Hans Lee, Valerio Perrone, Cedric Archambeau, and Matthias Seeger

URL https://proceedings.mlr.press/ v206/mikkola23a.html. Eric Hans Lee, Valerio Perrone, Cedric Archambeau, and Matthias Seeger. Cost-aware Bayesian optimization. arXiv preprint arXiv:2003.10870, 2020. URL https: //doi.org/10.48550/arXiv.2003.10870. Alessio Benavoli, Dario Azzimonti, and Dario Piga. Learn- ing choice functions with Gaussian processes. In ...

arXiv doi:10.48550/arxiv.2003.10870 2003

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Carl D Meyer

URL https://proceedings.mlr.press/ v206/hong23b.html. Carl D Meyer. Matrix analysis and applied linear algebra. SIAM, 2023. Bernhard Schölkopf, Ralf Herbrich, and Alex J Smola. A generalized representer theorem. In International Confer- ence on Computational Learning Theory (COLT), pages 416–426. Springer, 2001. URL https://doi.org/ 10.1007/3-540-44581-1_...

doi:10.1007/3-540-44581-1_27 2023