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Lower Bounds on the Noiseless Worst-Case Complexity of Efficient Global Optimization.
Wenjie Xu1,2, Yuning Jiang1, Emilio T Maddalena1
1Automatic Control Laboratory, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland.
Journal of Optimization Theory and Applications
|May 13, 2024
Summary
This study establishes a unified lower bound for efficient global optimization, crucial for expensive black-box functions. The findings demonstrate this bound closely matches existing upper bounds for common kernels, indicating near-optimal performance.
Area of Science:
- Machine Learning
- Optimization Theory
Background:
- Efficient global optimization (EGO) is vital for expensive black-box functions.
- Existing research often provides kernel-specific complexity bounds.
Purpose of the Study:
- To derive a unified lower bound for EGO oracle complexity.
- To compare this lower bound with existing upper bounds for common kernels.
Main Methods:
- Analysis of EGO worst-case oracle complexity.
- Derivation of a unified lower bound using metric entropy in reproducing kernel Hilbert spaces.
Main Results:
- A unified lower bound for EGO oracle complexity is established.
- This bound nearly matches the upper bound for squared exponential and Matérn kernels.
- The matching is within factors of dimension (d) and logarithmic terms.
Conclusions:
- The derived lower bound is nearly optimal for commonly used kernels.
- This unified approach simplifies complexity analysis in EGO.
- The results provide theoretical insights into the efficiency of EGO algorithms.
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