Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration

Bayesian optimization leveraging Gaussian processes (GPs) has become a dominant framework for global optimization of expensive black-box functions. Early methods like Expected Improvement (EI) and Upper Confidence Bound (UCB) provided theoretical guarantees in sequential settings, but their efficiency diminishes in high-dimensional or noisy environments. Recent developments introduced parallel batch strategies, such as GP-BUCB and SM-UCB, aiming to accelerate convergence by querying multiple points simultaneously. However, these methods often suffer from dimension-dependent regret bounds and computational complexity issues, limiting their scalability. The evolution of these techniques reflects a need for algorithms that combine theoretical rigor with practical scalability, especially in large-scale applications like hyperparameter tuning, engineering design, and scientific simulations. Despite progress, achieving dimension-free regret bounds and computational efficiency remains an open challenge, motivating the development of new strategies that can adaptively balance exploration and exploitation in parallel settings.

The core challenge addressed is how to efficiently optimize an unknown, noisy function in high-dimensional spaces using batch queries. Traditional sequential methods are slow and sample-inefficient, especially when evaluations are costly. Existing parallel algorithms improve speed but often rely on dimension-dependent bounds and require initialization phases, which can be time-consuming and less robust. The problem becomes more complex when the function's evaluation cost is high, noise is significant, and the search space is large. Developing an algorithm that guarantees tight regret bounds independent of dimension, while effectively balancing exploration and exploitation in parallel, is critical for advancing practical optimization in scientific and industrial contexts.

This paper introduces the GP-UCB-PE algorithm, which innovatively combines Gaussian process UCB with pure exploration within each batch. Key innovations include: 1) a strategy that selects the first point via UCB to balance exploration and exploitation; 2) subsequent points chosen by maximizing information gain within a relevant region, enhancing exploration; 3) derivation of regret bounds proportional to \(\sqrt{K}\) with constants independent of dimension, overcoming the curse of dimensionality. The method eliminates the need for initialization phases common in prior work, simplifies implementation, and leverages Bayesian inference for adaptive confidence intervals. These innovations collectively enable more efficient, scalable, and theoretically grounded parallel optimization.

• �� Model the target function as a Gaussian process with a specified kernel (e.g., RBF, Matérn). • At each iteration, compute the posterior mean and variance based on previous observations. • Select the first query point by maximizing the UCB: \(x_0 = \arg\max_x \mu_t(x) + eta_t \sigma_t(x)\). • Define a relevant region \(R_t\) based on the lower confidence bound and the current estimate of the maximum. • For remaining \(K-1\) points, select locations that maximize the posterior variance within \(R_t\), using a greedy approach to approximate maximum information gain. • Update the posterior after observing noisy evaluations, and repeat. • Theoretical analysis employs information-theoretic measures like maximum information gain \(\gamma_{T}^{K}\) to derive regret bounds, showing improvement over prior methods.

Experiments involve synthetic functions (Himmelblau, Gaussian mixture, Mackey-Glass) and real-world tasks (tsunami modeling, Abalone dataset). Batch size set to 10, with initial samples of 20 points. Performance measured by cumulative regret over multiple runs (64 repetitions). Hyperparameters tuned via marginal likelihood maximization. Comparisons made against GP-BUCB and SM-UCB, focusing on convergence speed, robustness to noise, and computational efficiency. Results consistently show faster convergence and lower regret, especially in high-noise and high-dimensional scenarios. Additional ablation studies analyze the impact of batch size and kernel choice.

GP-UCB-PE outperforms baseline algorithms across all benchmarks, with regret bounds improving proportionally to \(\sqrt{K}\). In synthetic functions, it reduces cumulative regret by over 15% compared to GP-BUCB. On complex functions like Mackey-Glass, it converges 2-3 times faster. Empirical results confirm that the constants involved are dimension-free, validating theoretical predictions. The approach maintains robustness under noise, with stable performance even at high noise levels. Computationally, lazy variance updates enable practical deployment despite large kernel matrices, demonstrating a good balance between theoretical guarantees and real-world efficiency.

The algorithm is suitable for hyperparameter tuning in machine learning, engineering design optimization, and scientific simulations where evaluations are costly. Its parallel nature allows leveraging modern multi-core and distributed computing resources, significantly reducing total optimization time. The method's robustness to noise makes it applicable in experimental sciences, such as material discovery or biological data analysis. Future integration with sparse Gaussian processes could further extend its use to large-scale problems, enabling real-time decision-making in industrial settings.

Despite dimension-free bounds, the computational cost of kernel matrix inversion remains a bottleneck for very large datasets. Approximate inference methods can help but may compromise theoretical guarantees. The assumption of Gaussian noise limits robustness in non-Gaussian environments. Estimating the maximum information gain \(\gamma_{T}^{K}\) in practice is challenging, affecting the tightness of regret bounds. Further research is needed to address these scalability and robustness issues, especially for ultra-high-dimensional problems.