Convergence rates of efficient global optimization algorithms (1101.3501v3)

Abstract: Efficient global optimization is the problem of minimizing an unknown function f, using as few evaluations f(x) as possible. It can be considered as a continuum-armed bandit problem, with noiseless data and simple regret. Expected improvement is perhaps the most popular method for solving this problem; the algorithm performs well in experiments, but little is known about its theoretical properties. Implementing expected improvement requires a choice of Gaussian process prior, which determines an associated space of functions, its reproducing-kernel Hilbert space (RKHS). When the prior is fixed, expected improvement is known to converge on the minimum of any function in the RKHS. We begin by providing convergence rates for this procedure. The rates are optimal for functions of low smoothness, and we modify the algorithm to attain optimal rates for smoother functions. For practitioners, however, these results are somewhat misleading. Priors are typically not held fixed, but depend on parameters estimated from the data. For standard estimators, we show this procedure may never discover the minimum of f. We then propose alternative estimators, chosen to minimize the constants in the rate of convergence, and show these estimators retain the convergence rates of a fixed prior.

Asymptotic Behavior of Expected Improvement Algorithms in Efficient Global Optimization

In the domain of efficient global optimization, a critical challenge involves minimizing an unknown function $f$ while utilizing as few observations of $f(x)$ as feasible. This problem falls under the purview of continuum-armed-bandit problems characterized by noiseless data and a focus on minimizing simple regret. A prevalent approach to tackle this problem is through expected improvement (EI) algorithms, notable within Bayesian optimization frameworks. This paper presents an in-depth theoretical exploration of the asymptotic behavior of these algorithms.

Key Insights and Results

The authors first establish that the implementation of expected improvement algorithms necessitates a decision on the Gaussian-process (GP) prior, which in turn determines the associated function space via its reproducing-kernel Hilbert space (RKHS). A critical property of EI is its convergence on the minimum of any function in its RKHS when a fixed GP prior is chosen. The work extends prior studies by providing convergence rates that are optimal for functions exhibiting low smoothness. Additionally, a modified EI algorithm is proposed, achieving near-optimal rates for smoother functions.

When addressing the practical scenario of sequentially estimated priors from data, a stark claim is made: standard estimators have the potential to indefinitely avoid the function's true minimum. To counter this, the paper proposes alternative parameter estimation strategies aimed at minimizing constants within the convergence rate, while still preserving the convergence characteristics of a fixed prior.

Implications and Future Directions

The theoretical contributions concerning convergence rates offer significant implications for optimizing functions of varied smoothness within RKHS settings. These results underscore the importance of selecting appropriate priors and parameter estimation techniques, particularly in real-world applications where computational cost is a concern.

The findings provoke further investigation into robust parameter estimation techniques that remain efficient across diverse function classes. Future research could potentially extend these strategies to noisy data scenarios or adapt them to more general function spaces beyond RKHS. Additionally, integrating these insights with advancements in Bayesian optimization might improve optimization strategies across applied fields such as machine learning and engineering design.

Speculations on AI Developments

In the broader landscape of AI, the insights from this paper could be pivotal for developing more sophisticated and adaptive optimization algorithms. As models increasingly rely on complex, high-dimensional optimization challenges, understanding and enhancing algorithmic convergence can lead to more efficient learning and decision-making processes. Moreover, addressing the interplay between exploration and exploitation in optimization parallels ongoing challenges in reinforcement learning, offering a potential avenue for cross-pollination of theoretical advancements.

In conclusion, while the paper sets a groundwork for understanding the efficacy of expected improvement algorithms under specific conditions, it also invites continued exploration into adaptive and robust optimization methodologies within artificial intelligence.