Batch Bayesian Optimization

• Batch Bayesian Optimization is a framework that selects several candidate points simultaneously to evaluate expensive black-box functions, balancing exploration and exploitation.
• It employs a Gaussian Process surrogate with acquisition functions like UCB and uses strategies such as explicit distance exploration with Sobol sequences to maintain diversity.
• This approach reduces wall-clock time with empirical speedups (up to 4.5×) compared to sequential methods, making it highly efficient for parallel evaluations.

Batch Bayesian Optimization (Batch BO) is a family of methodologies in Bayesian optimization in which, at each iteration, multiple (batch) candidate points are selected simultaneously for parallel evaluation of an expensive, typically black-box, objective function. Batch BO enables more efficient use of parallel computational or experimental resources and can dramatically reduce total wall-clock optimization time compared to strictly sequential approaches. The core challenge in Batch BO is to select batch points that collectively maximize expected improvement or information gain while maintaining sufficient diversity to avoid redundant or non-informative evaluations.

1. Foundations of Batch Bayesian Optimization

Batch BO operates on the standard Bayesian optimization framework: given a black-box function $$f : \mathcal{X} \subseteq \mathbb{R}^d \rightarrow \mathbb{R}$$ and a statistical surrogate model (most commonly a Gaussian process, GP), the algorithm iteratively selects input locations to evaluate so as to efficiently maximize $f$ using as few queries as possible. In each batch iteration $t$, the agent selects a batch $\mathbf{x}_t = (x_{t,1}, \dots, x_{t,B})$ of $B$ points, observes corresponding (possibly noisy) outcomes $y_{t,i} = f(x_{t,i}) + \epsilon_{t,i}$, and updates its GP posterior.

The batch setting introduces a key coupling: since all batch points are chosen simultaneously and evaluated in parallel, the values at $x_{t,1},\dots,x_{t,B}$ are unknown when selecting any $x_{t,i}$ for $i>1$, precluding the greedy selection logic of sequential BO. Informative batch selection must therefore address the joint effect of all points in the batch on the posterior while balancing exploration (diversification, information gain) and exploitation (focusing on promising regions).

2. Classical Batch BO Algorithms and Their Bottlenecks

The most classical batch BO methods are direct multi-point extensions of acquisition functions, such as the parallel expected improvement (q-EI) and parallel upper confidence bound (q-UCB):

• q-EI:

$$\alpha_{q-\mathrm{EI}}(X_q \mid \mathcal{D}) = \mathbb{E}_{y_1\ldots y_q \mid \mathcal{D}} \left[\max(0, \max_i y_i - f_\mathrm{best})\right]$$

• q-UCB:

$$\alpha_{q-\mathrm{UCB}}(X_q \mid \mathcal{D}) = \sum_{i=1}^q \mu(x_i) + \kappa \sigma(x_i)$$

Jointly maximizing these batch objectives over all $q$ points is computationally expensive, incurring a complexity of $O(C_d \cdot B \cdot T N^2)$ or worse for $T$ rounds and $N$ total data points, where $C_d$ is the cost of a single global optimization in $d$ dimensions. When objective evaluations are relatively cheap (e.g., 1–10 seconds), this BO overhead dominates total wall time (Nguyen et al., 2018).

Heuristic approximations, such as greedily filling the batch sequentially ("Constant Liar", "Kriging Believer", hallucinated UCB), simplify the batch construction but can introduce bias or redundant points, especially as batch size grows. Other methods estimate design rewards by penalizing the acquisition function near already-picked points (Local Penalization), but calculation of local Lipschitz constants is challenging in high dimensions.

3. Efficient Batch Selection via Distance Exploration (UCB-DE)

To improve efficiency, especially for less expensive function evaluations, UCB-DE proposes a two-component batch selection:

1. Exploitation: The first point $x_{t,1}$ is selected as the maximizer of a single-point acquisition (typically UCB):

$$x_{t,1} = \arg\max_{x \in \mathcal{X}} \mu_t(x) + \kappa_t \sigma_t(x)$$

2. Explicit Space-Filling Exploration: The remaining $B-1$ points are filled by maximizing the minimum squared Euclidean distance to all previously observed points, formally

$$g(x, \mathcal{D}) = \min_{x_i \in \mathcal{D}} \|x - x_i\|_2^2$$

$$x_\mathrm{DE} = \arg\max_{x \in \mathcal{X}} g(x, \mathcal{D})$$

This exploration is efficiently implemented using a precomputed low-discrepancy Sobol sequence $S$ over $\mathcal{X}$ (with $M \approx 10 \cdot T \cdot B$ points). The algorithm sequentially selects, as each DE candidate, the Sobol point farthest from existing observed locations (Nguyen et al., 2018). This sidesteps the need for additional global optimization or acquisition maximization for each exploration point.

Pseudocode: Distance Exploration via Sobol Sequence

for i in range(2, B+1): m_star = argmax_m min_{x_j in D} ||s_m - x_j||^2 x_t[i] = s_m_star D.append(x_t[i])

After the batch is evaluated, all points and outputs are used to update the GP, and the procedure repeats for $T$ rounds.

4. Computational Complexity and Empirical Performance

Denote $N = T \cdot B$ the total number of points. BUCB or q-EI requires $B$ global optimization steps per batch (cost: $O(C_d T^3 B^3)$), while UCB-DE limits this to one global optimization per batch ($O(C_d T N^2)$), with the DE procedure costing $O(B \cdot M \cdot N)$, typically negligible for moderate $d$ and $B$ (Nguyen et al., 2018).

Empirically on both synthetic benchmarks (Alpine2, Hartmann6, Sobol-10d) and real-world problems (aluminum simulator, BNMC hyperparameter tuning, PES acquisition optimization), UCB-DE matches or beats BUCB and other batch BO baselines in objective value per number of evaluations, while requiring only 1/3 to 1/6 of the wall time.

Batch BO Empirical Results (summarized):

Method	Time to 95% Opt	Speedup vs BUCB
UCB-DE	100 s	4.5×
BUCB	450 s	1×
LP	600 s	0.8×
CL	800 s	0.6×

This demonstrates near-linear scaling with batch size $B$ for UCB-DE, as opposed to superlinear complexity for BUCB.

5. Implementation Recommendations and Limitations

• When to use: UCB-DE is especially suitable when function evaluations are not highly expensive (typ. 1–100s) and the overhead of BO can dominate evaluation time.
• Batch size: Should match available parallel resources; UCB-DE's cost grows only mildly in $B$.
• Sobol sequence size: $M \approx 10 T B$ offers a trade-off between accuracy and search overhead; can be increased further if negligible in application context.
• Acquisition function: Although UCB is standard for exploitation, any single-point criterion (EI, Thompson Sampling) may be used.
• Limitations: DE assumes a Euclidean space; for non-isotropic or manifold-constrained inputs, the distance metric must be adapted to preserve space-filling properties. Applicability may be reduced for expensive function evaluations, where marginal acquisition improvements are more valuable than BO overhead reduction.

6. Comparative Perspective: Space-Filling and Exploitation–Exploration Trade-Offs

UCB-DE operationalizes the exploitation–exploration trade-off by always including a single globally optimized exploitation point and using explicit geometric diversity to fill out the batch. This approach contrasts with:

• Sequential hallucinated selection ("constant liar", BUCB, GP-BUCB), which simulates data from unobserved points and maximizes the acquisition in updated fictitious posteriors.
• Batch-wise acquisition maximization (q-EI, q-KG), which directly optimize vector-valued acquisitions but are computationally intractable for large $B$ or higher dimension.
• Clustering-based approaches (KMBBO), which sample from acquisition landscapes and cluster to promote diversity.

The full-batch DE exploration is most beneficial when space coverage is important, i.e., risk of redundant queries is high due to low marginal information gain in closely spaced points, or when surrogate parameter uncertainty is large. In regimes where the surrogate is already sharply peaked, batch diversity may have diminishing returns compared to more targeted (but costlier) batch acquisition maximization.

7. Extensions and Generalizations

• UCB-DE's decoupling of the exploitation and exploration components enables plug-in replacement of both: the exploitation step may use other acquisition functions, and the DE fill can be extended to non-Euclidean or mixed-categorical domains by substituting appropriate distance or kernel metrics.
• Expansion to adaptive batch size selection could be considered for further gains, but is not explored in this method.
• Empirical evidence suggests that for less expensive functions, UCB-DE yields a robust, parameter-light, and highly efficient approach to batch BO (Nguyen et al., 2018). Its utility may be reduced if evaluations are very costly, high-precision surrogate-guided query selection is paramount, or if space-filling exploration is not aligned with optimization goal.

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UCB-DE, as presented, exemplifies an efficient design for batch BO under low-to-moderate black-box function evaluation costs, explicitly trading off efficient global search, computational overhead, and parallel resource utilization. Its geometric exploration, inspired by design of experiments and quasi-random sequences, distinguishes it from classical batch BO approaches reliant on expensive joint acquisition maximization.