Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration (1304.5350v3)

Abstract: In this paper, we consider the challenge of maximizing an unknown function f for which evaluations are noisy and are acquired with high cost. An iterative procedure uses the previous measures to actively select the next estimation of f which is predicted to be the most useful. We focus on the case where the function can be evaluated in parallel with batches of fixed size and analyze the benefit compared to the purely sequential procedure in terms of cumulative regret. We introduce the Gaussian Process Upper Confidence Bound and Pure Exploration algorithm (GP-UCB-PE) which combines the UCB strategy and Pure Exploration in the same batch of evaluations along the parallel iterations. We prove theoretical upper bounds on the regret with batches of size K for this procedure which show the improvement of the order of sqrt{K} for fixed iteration cost over purely sequential versions. Moreover, the multiplicative constants involved have the property of being dimension-free. We also confirm empirically the efficiency of GP-UCB-PE on real and synthetic problems compared to state-of-the-art competitors.

• The paper introduces the GP-UCB-PE algorithm that blends UCB with pure exploration to reduce cumulative regret in costly and noisy function evaluations.
• It provides a theoretical guarantee showing a √K reduction in regret compared to sequential methods with dimension-free constants.
• Empirical results on synthetic and real-world data confirm its improved performance over state-of-the-art approaches like GP-BUCB and SM-UCB.

Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration

The paper "Parallel Gaussian Process Optimization with Upper Confidence Bound and Pure Exploration" introduces an algorithm designed to efficiently find the maximum of an unknown function which is costly and noisy to evaluate. This task is a common problem in real-world applications, particularly in high-dimensional input spaces where computational resources are highly consumed by evaluating costly functions. The authors propose a new approach that extends upon existing Bayesian optimization techniques by leveraging parallel evaluations to reduce the cumulative regret of the optimization process.

Gaussian Process Upper Confidence Bound and Pure Exploration Algorithm

The paper presents the \textsf{GP-UCB-PE} algorithm which blends the Upper Confidence Bound (UCB) strategy with Pure Exploration within the context of parallel batch evaluations. This combined approach is implemented to optimize the trade-off between exploration, learning about the unknown function, and exploitation, making use of the gathered knowledge to guide further evaluations. The proposed algorithm addresses both the exploration and exploitation trade-offs in a more efficient manner by using batches of evaluations while minimizing the cost of iterations.

Theoretical Contributions

The paper provides theoretical guarantees on the cumulative regret of the \textsf{GP-UCB-PE} algorithm. Specifically, it establishes upper bounds on the regret in scenarios where function evaluations are conducted in parallel with a fixed batch size $K$. The theoretical analysis demonstrates that the regret decays approximately by an order of $\sqrt{K}$ when compared to sequential evaluation strategies, without suffering from the constraints imposed by dimensionality, as the multiplicative constants involved are dimension-free.

Empirical Validation

The authors empirically validate the proposed \textsf{GP-UCB-PE} algorithm through experiments using both synthetic and real-world data. The experiments showcase its robustness and efficiency in finding the maximum of complex functions. It is compared favorably against existing state-of-the-art algorithms, notably \textsf{GP-BUCB} and \textsf{SM-UCB}, asserting its competence in both synthetic (e.g., Gaussian Process samples) and real-world datasets (e.g., tsunami wave run-up modeling).

Practical and Theoretical Implications

The parallelization approach introduced by \textsf{GP-UCB-PE} offers significant implications for practical applications where evaluating functions is computationally expensive or constrained by external factors. The enhanced exploration strategy promises effective use of available computational resources, thus achieving reduced cumulative regret and improved performance. Theoretically, the results provide a roadmap for applying such parallelization techniques to other Bayesian optimization problems, potentially extending its applicability to various domains within artificial intelligence and machine learning.

Future Directions

The proposed \textsf{GP-UCB-PE} algorithm is a key step towards achieving efficient parallel optimization in scenarios constrained by evaluation costs. Future research can further explore extensions of this algorithm to other frameworks like Maximum Expected Improvement (MEI), potentially broadening its application scope. Additionally, the paper lays the groundwork for future investigations into refining greedy strategies for sequential optimization without necessitating initial exploration phases, thereby accelerating convergence to the optimal solution.

In conclusion, this paper presents a valuable contribution to the field of Gaussian Process optimization by introducing a parallelized approach that effectively balances the cost and benefits of exploration and exploitation. The \textsf{GP-UCB-PE} algorithm not only enhances practical applications but also strengthens the theoretical foundations for future advancements in parallel optimization strategies.