Batch Bayesian Inversions

• Batch Bayesian inversions are methods that evaluate multiple candidate points simultaneously to reduce computational time while preserving optimization quality.
• Dynamic batch selection adapts the Expected Improvement criterion using fictional outcomes to determine an optimal, variable batch size within a Gaussian Process framework.
• Empirical results demonstrate 6%-18.4% speedup in wall-clock time with near-optimal performance across complex engineering and simulation tasks.

Batch Bayesian inversions refer to methodologies in Bayesian optimization and Bayesian inference where multiple candidate points, experimental conditions, or inversion problems are evaluated simultaneously in each iteration—contrasting with traditional sequential approaches that select and process one evaluation at a time. The development of batch algorithms is motivated by scenarios where function evaluations (experiments, simulations, or forward model runs) are expensive but can be performed in parallel, allowing for substantial reductions in wall-clock time while minimizing the loss in statistical or optimization performance.

1. Foundations of Batch Bayesian Inversions

Batch Bayesian inversions extend the core ideas of Bayesian optimization to parallel or concurrent experimental design and inversion, leveraging probabilistic surrogate models—most commonly Gaussian Processes (GPs)—to balance exploration and exploitation. Rather than updating the surrogate after each individual function evaluation as in the fully sequential paradigm, batch methods propose a group of points to be evaluated together, then update the model based on the whole batch of new data.

The critical technical challenge is that, in general, the optimal set of batch points should account for the mutual information among them—the outcome at each point can reduce the expected benefit of evaluating the others due to redundancy or dependence. Direct joint optimization of acquisition functions over batches is computationally intractable in moderate-to-high dimensions or for larger batch sizes. Therefore, efficient batch selection schemes have been developed that approximate sequential decision-making as closely as possible, while taking advantage of parallel evaluations to reduce time to solution (Azimi et al., 2011).

2. Dynamic Batch Selection via Expected Improvement

A key contribution to batch selection in Bayesian optimization is the dynamic adaptation of batch size based on conditional independence of candidate points. In the approach of “Dynamic Batch Bayesian Optimization” (Azimi et al., 2011), the classic Expected Improvement (EI) acquisition function is adapted for batch selection through the following mechanism:

• The first point in the batch is selected as the maximizer of EI with respect to the current data.
• A “fictional” best-case observation (e.g., assigning a value $M$ larger than the current maximum) is temporarily added to the data at this first location, and the surrogate model is updated accordingly.
• For each successive candidate point, the expected change in the GP posterior mean, $E[|\Delta^*(\mu_z)|]$, if the new point were added, is used as a proxy for dependence: points are accepted into the batch if this change is less than a small threshold $\varepsilon$.
• This process is repeated until either no additional “independent” point is found or a pre-set maximum batch size is reached.

Mathematically, candidate z is accepted if

$E[|\Delta^*(\mu_z)|] \leq \varepsilon,$

meaning its acquisition value is nearly independent of the uncertain outcome at previously selected batch points.

This procedure leads to variable batch sizes, adjusting adaptively to the geometry of the surrogate’s predictive uncertainty and reducing the risk of regret penalties associated with naive or fixed-size batch approaches.

3. Theoretical Underpinnings and Algorithmic Structure

In the dynamic batch framework (Azimi et al., 2011), the key theoretical result is that the expected effect of “fictionally” sampling a point (i.e., optimistically assuming a best-case outcome) on the GP posterior mean at another point can be bounded in terms of the model’s predictive variance and the GP kernel structure. For $\mathcal{O}$ denoting past observed points and $x^*$ the selected candidate, the change in GP variance and mean after fictional sampling is given by explicit formulas involving kernel functions (see, e.g., Theorem 2). Specifically, the change in variance at $z$ is computed via

$$\Delta^*(\sigma_z) = \sigma_z^2 - \sigma_z^{*2} = (P A^{-1}B^T - k^*_z)m(PA^{-1}B^T - k^*_z)^T,$$

with kernel-derived terms $A, B, P, m, k^*_z$ built from the GP’s covariance and the data.

The stopping criterion for growing the batch is thus analytically computable within the GP framework, allowing for rigorous control of dependence among batch points.

4. Performance and Trade-offs: Speedup versus Regret

Empirical validation on multiple synthetic (Cosines, Rosenbrock, Hartman, Michalewicz, Shekel) and real-world (Fuel Cell and Hydrogen) benchmarks (Azimi et al., 2011) shows that dynamic batch Bayesian optimization yields substantial speedup in wall-clock time compared to fully sequential EI, with little to no loss in optimization performance (measured via regret, $M - \max(y_i)$). Reported speedups range from roughly 6% to 18.4%, with the regret remaining near-minimal. As the total number of evaluations increases, large batches (up to the maximum allowed) become more frequent, further improving computational efficiency.

The method’s main limitation is that when the acquisition function landscape is sharply peaked—so optimal next points are highly dependent—the adaptive algorithm reverts to sequential selection, reducing parallelism gain.

The trade-off is governed by threshold $\varepsilon$ and the fictional output value $M$; setting these appropriately is necessary to balance speed and solution quality.

5. Methodological Extensions and Related Approaches

Subsequent research extends the batch paradigm in several directions:

• Hybrid Sequential/Batch Policies: Algorithms combine sequential decision quality with batch efficiency, dynamically switching between modes based on the current model’s predictive uncertainty and a rigorous error bound that quantifies deviation from purely sequential acquisition (Azimi et al., 2012).
• Complex Acquisition Functions & Non-Gaussian Surrogates: Recent work integrates batch selection methods with more complex acquisition functions (such as Entropy Search or Thompson Sampling) or non-GP surrogates, extending applicability to high-dimensional and non-stationary regimes.
• Batch Bayesian Inversions Beyond Optimization: The batch selection logic has been ported to Bayesian inversion and numerical integration problems (see batch Bayesian quadrature (Wagstaff et al., 2018, Adachi et al., 2022)), where batches of quadrature nodes are selected based on penalization (e.g., Local Penalisation or Kernel Recombinations) or pseudo-observation updates (Kriging Believer) to maximize informativeness and efficiency.

6. Applications and Practical Implications

Batch Bayesian inversion techniques are particularly valuable in scientific and engineering contexts where:

• Experimental or simulation resources enable parallel (or distributed) evaluation.
• Each function, design, or inversion query is computationally costly.
• The user wishes to exploit available computational resources without significant accuracy loss.

Examples include engineering design optimization (e.g., computational fluid dynamics), hyperparameter tuning in machine learning, parallelized scientific experiments (such as fuel cell or chemical process optimization), and various inverse problems (e.g., geophysical imaging, where multiple inversion queries can be evaluated in parallel provided their outcomes are sufficiently independent).

Adoption of dynamic batch selection strategies enables maximal resource utilization without compromising the rigor of uncertainty quantification inherent to Bayesian analysis.

7. Summary and Future Outlook

Batch Bayesian inversions—characterized by adaptive, independence-aware batch selection—serve to close the gap between decision-theoretically optimal sequential policies and the practical constraints of parallel experimentation. By simulating sequential acquisition and admitting into each batch only those candidates whose outcomes are nearly conditionally independent (as measured through GP-based metrics), these methods realize significant speedup in experimental campaigns with minimal loss in performance.

The foundation laid by dynamic batch expected improvement (Azimi et al., 2011) has been widely extended in subsequent literature to more general acquisition functions, alternative surrogate models, and broader domains of Bayesian inversion. Batch approaches continue to be indispensable in contemporary scientific and engineering optimization, design, and inference workflows.