Consensus-Based Bayesian Optimization

• Consensus-Based Bayesian Optimization is a framework where multiple agents collaboratively optimize black-box functions via consensus protocols and distributed experimentation.
• It employs adaptive acquisition functions and consensus protocols, such as uniform-decay and leader-driven scheduling, to balance collective learning and individual personalization.
• Empirical evaluations and regret analyses demonstrate that CBO achieves accelerated convergence, robust scalability, and efficient optimization in heterogeneous settings.

Consensus-Based Bayesian Optimization (CBO), an extension of classical Bayesian Optimization (BO), is a framework where multiple agents collaboratively optimize black-box objective functions by reaching consensus on experimental decisions. The methodology leverages distributed experimentation, information sharing via consensus protocols, and transitional collaboration–personalization schemes to accelerate optimal design in both homogeneous and heterogeneous settings. CBO techniques, including the Collaborative Bayesian Optimization via Consensus (CBOC) and Social Bayesian Optimization (SBO) variants, provide rigorous theoretical guarantees and have demonstrated empirical efficacy in complex optimization domains (Yue et al., 2023, Adachi et al., 11 Feb 2025).

1. Problem Formulation and Notation

Consensus-based Bayesian optimization addresses multi-agent scenarios with $K \geq 2$ clients, each with a latent black-box objective $f_k: \mathbb{R}^D \to \mathbb{R}$. The goal for each client is to identify $$x_k^* = \arg\max_{x \in X \subset \mathbb{R}^D} f_k(x)$$ given a strictly bounded experimental budget $T$.

Each client maintains its local dataset $D_k^{(t)} = \{X_k^{(t)}, y_k^{(t)}\}$, where after $t$ rounds $N_k^{(t)} = N_k^{(0)} + t$. Learning and inference proceed via Gaussian Process (GP) surrogates for each $f_k$, with GP kernel, noise, and hyperparameters learned or specified a priori:

• $f_k(x) \sim \mathcal{GP}(0, K_k(x, x')),$ where $$K_k(x, x') = u_k^2 \exp(-\lVert x-x' \rVert^2 / (2\ell_k^2))$$ and $\varepsilon_k \sim \mathcal{N}(0, v_k^2)$.

Given $D_k^{(t)}$, the GP posterior mean and variance at $x$ are:

$$\mu_k^{(t)}(x) = K(x, X_k^{(t)})[K(X_k^{(t)}, X_k^{(t)}) + v_k^2 I]^{-1} y_k^{(t)}$$

$$\sigma_k^2{}^{(t)}(x) = K(x,x) - K(x, X_k^{(t)})[K(X_k^{(t)}, X_k^{(t)}) + v_k^2 I]^{-1}K(X_k^{(t)}, x)$$

(Yue et al., 2023).

2. Consensus Protocols and Collaboration Mechanisms

Consensus in CBO is achieved by clients exchanging sampling proposals and aggregating these collaboratively. At each round, every client maximizes a utility/acquisition function (e.g., Expected Improvement, EI) to propose a local sampling point $x_k^{(t)}$. The collection of proposals forms $$x_C^{(t)} = [x_1^{(t)};\dots;x_K^{(t)}] \in \mathbb{R}^{K D}$$.

The core consensus mechanism employs a doubly-stochastic matrix $W^{(t)}$ ($$W^{(t)}\mathbf{1} = \mathbf{1},\; (W^{(t)})^\top \mathbf{1} = \mathbf{1}$$, $W_{ij}^{(t)}\geq 0$), yielding the consensus update:

$$x_C^{(t, \text{new})} = (W^{(t)} \otimes I_D) x_C^{(t)}$$

so that

$$x_k^{(t, \text{new})} = \sum_{j=1}^K W_{kj}^{(t)} x_j^{(t)}$$

for each agent (Yue et al., 2023).

Transitional weighting schedules interpolate from full collaboration to independence, allowing strong early-stage knowledge sharing and late-stage personalization:

• Uniform-decay: $W^{(0)}_{ij} = 1/K$, followed by diag/off-diag updates such that $W^{(T)} \to I$.
• Leader-driven: At each $t$, one client is the leader (highest reward), with $W^{(t)}$ reweighted accordingly, always preserving doubly-stochasticity (Yue et al., 2023).

3. Acquisition and Sampling Strategies

CBO methods are acquisition-agnostic. The canonical example is Expected Improvement (EI):

$$\text{EI}_k^{(t)}(x) = \mathbb{E}[(f_k(x) - y_k^{*(t)})^+] = \sigma_k^{(t)}(x) \varphi(z_k) + (\mu_k^{(t)}(x) - y_k^{*(t)}) \Phi(z_k)$$

where $y_k^{*(t)} = \max y_k^{(t)}$ and $$z_k = (\mu_k^{(t)}(x) - y_k^{*(t)}) / \sigma_k^{(t)}(x)$$ (Yue et al., 2023).

Consensus is integrated at the sampling stage:

$$x_k^{(t, \text{new})} = \sum_j W_{kj}^{(t)} x_j^{(t)}$$

There is flexibility to use other acquisition functions including UCB, KG, PI, or qEI. The consensus protocol operates independently of the acquisition function (Yue et al., 2023).

4. Algorithmic Workflow and Pseudocode

A prototypical CBOC iteration is:

1. Consensus step: Compute $x_k^{(t, \text{new})}$ by aggregating local proposals via $W^{(t)}$.
2. Experiment: Query $y_k(x_k^{(t, \text{new})})$ and update $D_k^{(t+1)}$.
3. GP update: Update surrogate using $D_k^{(t+1)}$.
4. Acquisition maximization: Find $x_k^{(t+1)} = \arg\max_x \text{EI}_k^{(t+1)}(x)$.
5. Consensus update: Adjust $W^{(t+1)}$ following the uniform-decay or leader-driven schedule.

All clients perform these steps in parallel. Acquisition optimization remains the principal computational bottleneck; the consensus step itself scales as $O(KD)$ per agent (Yue et al., 2023).

5. Regret Analysis and Theoretical Guarantees

In the homogeneous case ($f_1=\dots=f_K$), under squared-exponential kernels and any sequence of doubly-stochastic $W^{(t)}$, CBO maintains classical Bayesian Optimization regret bounds:

$$R_{k, T} = \mathcal{O} \left( \sqrt{T (\log T)^{D+4}} \right)$$

where $R_{k,T}$ is client $k$'s cumulative regret (Yue et al., 2023). The bound stems from GP posterior concentration, acquisition function analysis, and information-theoretic arguments. The consensus step does not harm these rates as long as $W^{(t)}$ is properly controlled and client iterates remain bounded.

For Social Bayesian Optimization, in collective decision-making with feedback corrupted by social influence, regret and the number of expensive "private" votes required are sublinear (in $T$) provided the graph is estimable and appropriate decay is set for private vote acquisition. The social graph can be estimated as $\lVert \hat{A}_t - A \rVert = O(|Q_T^u|^{-1/2})$, and utility estimation converges as $O(|Q_T^u|^{-1/4})$ (Adachi et al., 11 Feb 2025).

6. Empirical Validation

Simulation Benchmarks

CBOC was evaluated on Levy, Shekel, Branin, Ackley, and Hartmann functions with both homogeneous and heterogeneous agent objectives. The primary baseline comparisons were Individual BO and federated BO (FedBO):

Task	K	CBOC-L Gap	Individual Gap	FedBO Gap
Levy-2 (homog.)	5	≈0.993	0.931	0.990
Levy-4 (heterog.)	10	≈0.984	0.933	0.940
Shekel-10 (homog.)	20	≈0.592	0.335	0.535

CBOC demonstrated uniform acceleration, with performance gain amplifying as $K$ increased. The method rapidly directed clients into their optimal basins, especially in heterogeneous settings (Yue et al., 2023).

Real-World Case Study

In a collaborative sensor design task using three FEA simulators, CBOC-L achieved higher mean $\theta_{\max}$ ($0.85\pm0.03$) and faster convergence compared to non-collaborative runs ($0.72\pm0.07$), with about 30% reduction in convergence time and lower variance (Yue et al., 2023).

Social Consensus Applications

SBO was validated across tasks including thermal comfort, team-building, hotel selection, and energy trading. Notably, sublinear growth in private votes (often terminating after $10$–$20$ rounds) was observed, and regret metrics improved upon both private-only and oracle-A baselines. SBO demonstrated robustness to aggregator parameters and various social graph structures (Adachi et al., 11 Feb 2025).

7. Practical Implementation Considerations

• Consensus matrix ($W^{(t)}$): Use uniform-decay if no prior information, or leader-driven if trust/reliability differences exist. Doubly-stochasticity is critical.
• Communication frequency: Minimal consensus update per experiment is sufficient; skipping communication between some steps is permissible if $W^{(t)}$ varies slowly.
• Scalability: Storage overhead is $O(K^2)$; use sparse communications if needed.
• Privacy: Only share proposals $x_k^{(t)}$; retain private reward/observed data and GP hyperparameters.
• Acquisition: Any acquisition function integrates directly; consensus operates agnostic to the choice.
• Stopping: Global budget $T$ or local EI threshold.
• Reproducibility: Source code is available in the referenced open-source repository (Yue et al., 2023).

8. Extensions: Social Influence and Collective Consensus

The SBO framework generalizes consensus-based BO to settings with social influence, capturing both true and observed (graph-convolved) utilities. A dual-voting system (public and private votes) enables identification and debiasing of the social graph, enabling efficient social-influence-free consensus optimization with provable regret and sample complexity guarantees (Adachi et al., 11 Feb 2025).

SBO employs kernelized GP surrogates, upper-confidence–bound acquisitions, and decouples costly private queries as the social graph becomes well-estimated, lowering the practical burden of high-fidelity feedback.

Comparison of CBOC and SBO Approaches

Feature	CBOC (Yue et al., 2023)	SBO (Adachi et al., 11 Feb 2025)
Collaboration protocol	Doubly-stochastic consensus	Dual-voting, social-graph estimation
Objective(s)	Personalized per client	Aggregated social consensus
Surrogate model	Local GPs	GP over utilities (with graph)
Acquisition	EI (agnostic)	UCB-style over social aggregator
Empirical context	Simulation, sensor design	Decision-making, voting tasks

CBO methods formalize a principled class of distributed, collaborative BO techniques. They allow efficient transfer of sampling knowledge, adaptively balance knowledge sharing and personalization, and are extensible to consensus under nontrivial social dynamics, all supported by information-theoretic regret analysis and scalable empirical evidence (Yue et al., 2023, Adachi et al., 11 Feb 2025).