Choosing a Suitable Acquisition Function for Batch Bayesian Optimization: Comparison of Serial and Monte Carlo Approaches

This paper compares serial and Monte Carlo batch acquisition functions for Bayesian optimization on synthetic and empirical datasets, concluding that the q-upper confidence bound (qUCB) is the most robust default choice for optimizing unknown black-box functions in up to six dimensions.

The Gist

Imagine you are a treasure hunter trying to find the single best spot on a massive, foggy island to dig for gold. You don't have a map, and you don't know what the terrain looks like. The only way to learn is to dig a hole, see how much gold you find, and then decide where to dig next.

However, there's a catch: Digging is expensive. It takes a lot of time, money, and energy. You can't just dig holes randomly all over the island. You need a smart strategy to find the "Gold Mine" with as few digs as possible.

This is exactly what Bayesian Optimization does for scientists and engineers. It's a smart computer algorithm that helps them find the best settings for complex experiments (like making new solar cells or drugs) without wasting resources.

The Problem: Digging One Hole vs. Digging a Group

Usually, you dig one hole at a time. But in the real world, sometimes you can dig a small group of holes (a "batch") at the same time for almost the same cost as digging just one. Maybe you have a team of 4 people, or a machine that can test 4 samples simultaneously.

The big question this paper answers is: When you have a team of 4 diggers, how should you tell them where to stand?

Should you:

1. The Serial Approach: Tell Digger #1 where to go, wait for the result, then tell Digger #2 where to go based on that result, and so on?
2. The Parallel (Monte Carlo) Approach: Send all 4 diggers out at once, guessing where the best spots are based on probability, without waiting for the first one to finish?

The Contenders: The Strategies

The authors tested three main strategies (called "Acquisition Functions") to see which one finds the gold fastest:

1. UCB/LP (The Serial Planner): This is like a cautious, step-by-step planner. It picks the first spot, then uses a "local penalty" to make sure the next diggers don't just stand right next to the first one. It forces them to spread out a bit but stay close to the promising area. It's deterministic (very precise, no guessing).
2. qUCB (The Confident Gambler): This is a parallel strategy. It sends the whole team out at once. It uses a "stochastic" (randomized) method to pick spots that are likely to be good, but it also keeps a little bit of "exploration" to make sure it doesn't miss a hidden pocket of gold nearby.
3. qlogEI (The Optimistic Dreamer): This strategy focuses heavily on finding the absolute best spot immediately. It's very optimistic but can get confused easily, especially if the terrain is tricky.

The Test Drives

To see which strategy works best, the authors ran simulations on three different "islands":

1. The "Needle in a Haystack" (Ackley Function): Imagine a huge field of hay (bad spots) with one tiny, sharp needle (the gold) sticking out. It's very hard to find.

   • Result: The Serial Planner (UCB/LP) and the Confident Gambler (qUCB) both found the needle quickly. The Optimistic Dreamer (qlogEI) got lost in the hay and gave up.

2. The "Fake Gold Mine" (Hartmann Function): Imagine a landscape with two hills. One is the real gold mine, but the other is a fake mine that looks almost exactly as good. It's easy to get tricked.

   • Result: Both the Serial Planner and the Confident Gambler did well. The Dreamer was slower and more likely to get tricked by the fake hill.

3. The "Noisy, Foggy Island" (Real Solar Cell Data): This was a simulation based on real experiments making solar cells. The data was messy (noisy), like trying to dig in the fog.

   • Result: When the fog got thick (high noise), the Serial Planner got confused and started digging in the wrong places. The Confident Gambler (qUCB), however, handled the noise beautifully. It kept finding good spots even when the data was messy. The "Dreamer" with noise adjustments didn't do much better than the Gambler.

The Verdict: Who Wins?

The paper concludes that if you are a scientist or engineer starting a new experiment and you don't know what the landscape looks like (is it a needle? a fake hill? is it noisy?), you should use qUCB.

Why?

• It's the Swiss Army Knife: It works well on almost every type of terrain.
• It's Noise-Resistant: It doesn't get confused when the data is messy or imperfect.
• It's Efficient: It finds the best results using the fewest number of expensive "digs" (experiments).

The Simple Takeaway

If you are trying to optimize a complex process (like making a better battery or a new medicine) and you can test a few things at once, don't overthink the strategy.

Don't try to be a perfect step-by-step planner (Serial) or an overly optimistic dreamer. Instead, use the Confident Gambler (qUCB). It's the most reliable tool to help you find the "Gold Mine" in the shortest time, even when the map is foggy and the terrain is tricky. It's the "default setting" you should use when you're flying blind.

Technical Summary

1. Problem Statement

Bayesian Optimization (BO) is a standard machine learning technique for optimizing expensive "black-box" functions where the relationship between inputs ($X$) and outputs ($y$) is unknown and costly to evaluate. In many scientific and engineering contexts (e.g., materials synthesis), it is efficient to test a batch of $q$ samples simultaneously rather than one at a time (serial optimization).

The core challenge addressed in this paper is the selection of an appropriate batch acquisition function when the landscape of the objective function is unknown. Specifically, the authors compare two distinct algorithmic approaches:

• Serial Approaches: Select points sequentially within a batch, often using deterministic numerical methods (e.g., Local Penalization). These struggle with high dimensions ($>5-6$) and can be computationally expensive.
• Parallel (Monte Carlo) Approaches: Select the entire batch simultaneously by integrating over a joint probability density function using stochastic sampling (e.g., Monte Carlo methods). These are better suited for high-dimensional spaces.

The paper seeks to determine which approach offers the best balance of convergence speed, accuracy, and robustness to noise for real-world experimental optimization.

2. Methodology

The authors conducted a comprehensive comparative study using three distinct test cases representing different types of optimization landscapes:

A. Test Functions:

1. Ackley Function (6D): Represents a "needle-in-haystack" problem. It is highly heterogeneous with a sharp global maximum surrounded by a vast, flat region of low values.
2. Hartmann Function (6D): Represents a "false optimum" problem. It contains a secondary local maximum with a value nearly degenerate to the global maximum, making it easy for algorithms to converge on the wrong peak.
   • Variations: Tested in both noiseless conditions and with Gaussian noise (ranging from 1% to 20% of the peak-to-peak amplitude).
3. Empirical Perovskite Solar Cell (PCE) Model (4D): A regression model built from real experimental data regarding the power conversion efficiency of flexible halide perovskite solar cells. This model includes real-world noise, systematic errors, and a "flat plateau" landscape near the optimum.

B. Acquisition Functions Compared:

• UCB/LP (Serial): Upper Confidence Bound with Local Penalization. Uses deterministic quasi-Newtonian methods.
• qUCB (Parallel/MC): Monte Carlo Upper Confidence Bound.
• qlogEI (Parallel/MC): Monte Carlo log Expected Improvement.
• qlogNEI (Parallel/MC): Noise-integrated version of qlogEI, designed specifically for noisy environments.

C. Experimental Setup:

• Surrogate Model: Gaussian Process Regression (GPR) with an ARD Matern 5/2 kernel.
• Initialization: 99 distinct initial training sets generated via Latin Hypercube Sampling (LHS) to ensure statistical robustness.
• Batch Size: $q = 4$.
• Iterations: Capped at 50 iterations per run (totaling 224 data points including initialization).
• Metrics: Instantaneous Regret (IR) and Cumulative Regret (CR) for both the objective value ($y$) and the input vector ($X$).

3. Key Results

A. Ackley Function (Needle-in-Haystack):

• Performance: Both UCB/LP and qUCB significantly outperformed qlogEI.
• Convergence: UCB/LP and qUCB converged to the global maximum within ~15 iterations.
• Mechanism: UCB/LP (deterministic) showed slightly better final accuracy in modeling the sharp peak because its local penalization strategy aggressively exploits the region once a promising point is found. qUCB (stochastic) scattered points more broadly, which was slightly less efficient for the sharp peak but still highly effective.
• Failure Case: qlogEI failed to model the function correctly without domain-narrowing techniques (TuRBO). Even with TuRBO, it remained inferior to UCB/LP and qUCB.

B. Hartmann Function (False Optimum):

• Noiseless: All functions performed adequately, but UCB/LP and qUCB were superior to qlogEI. Both successfully avoided the false maximum in roughly 70% of runs.
• With Noise:
  • Monte Carlo superiority: All Monte Carlo functions (qUCB, qlogEI, qlogNEI) significantly outperformed the serial UCB/LP, especially at high noise levels (20%).
  • Robustness: UCB/LP became highly sensitive to initial conditions and failed to model the global maximum accurately under high noise.
  • qUCB vs. qlogNEI: While qlogNEI was designed for noise, it offered no significant advantage over qUCB. qUCB demonstrated the best overall performance in modeling the objective value under noise.

C. Empirical PCE Model (Real-World Data):

• Landscape Challenge: The landscape featured a broad, flat plateau near the maximum, meaning many different input vectors yield near-optimal efficiency.
• Performance: qUCB found input conditions yielding near-optimal PCE values in fewer iterations (30–40) compared to UCB/LP and qlogNEI (50).
• Sensitivity: qUCB was less sensitive to initial conditions than the other methods.
• Conclusion: In real-world scenarios where the exact global maximum ($X_{max}$) is less critical than achieving a high objective value ($y_{max}$) within experimental uncertainty, qUCB was the most efficient.

4. Key Contributions

1. Direct Comparison: Provided the first rigorous, large-scale comparison (99 initial conditions per function) between serial (deterministic) and parallel (Monte Carlo) batch acquisition functions across diverse landscape types.
2. Noise Robustness Analysis: Demonstrated that while serial methods (UCB/LP) excel in noiseless, sharp-peak scenarios, Monte Carlo methods (specifically qUCB) are far more robust to experimental noise and false optima.
3. Empirical Validation: Validated synthetic findings on a real-world materials science problem (perovskite solar cells), bridging the gap between theoretical benchmarks and practical application.
4. Noise Integration: Showed that explicitly integrating noise into the acquisition function (qlogNEI) does not necessarily outperform standard Monte Carlo UCB (qUCB) in batch settings.

5. Significance and Recommendations

The paper concludes that for batch Bayesian optimization in materials synthesis and engineering experiments (typically 4–6 dimensions) where the landscape and noise characteristics are unknown:

• Recommended Default: qUCB (Monte Carlo Upper Confidence Bound).
• Rationale: qUCB offers the best trade-off between:
  • Reliability: Consistent performance across "needle-in-haystack," "false optimum," and flat-plateau landscapes.
  • Noise Immunity: Superior convergence in noisy environments compared to serial methods.
  • Efficiency: Minimizes the number of expensive experimental iterations required to reach a confident optimum.
  • Simplicity: Does not require complex noise-integration modifications (like qlogNEI) to handle real-world data.

The authors advise against using qlogEI for high-dimensional problems and suggest that while UCB/LP is effective for specific noiseless, high-precision tasks, qUCB is the safer, more versatile default choice for general experimental campaigns.