Estimation of Extreme Values and Associated Level Sets of a Regression Function via Selective Sampling

Area of Science:

• Optimization
• Statistical Inference
• Numerical Analysis

Background:

• Estimating function extrema is crucial in many scientific fields.
• Existing methods often require strong assumptions like differentiability or unimodality.
• Robust methods are needed for noisy, unconstrained functions.

Purpose of the Study:

• To propose a new, robust method for estimating function maxima/minima from noisy observations.
• To minimize regularity and shape constraints on the function.
• To provide performance guarantees for the proposed estimators.

Main Methods:

• Developed a novel estimation technique for function extrema.
• The method accommodates functions lacking differentiability or unique maxima.
• Performance bounds (upper and lower) were derived theoretically.

Main Results:

• The proposed method accurately estimates function locations and extreme values.
• Tight theoretical bounds validate the estimator's performance.
• The method is adaptive to unknown problem parameters and distributions.

Conclusions:

• The new method offers a flexible and robust approach to function extremum estimation.
• It overcomes limitations of traditional methods by relaxing regularity assumptions.
• The adaptive nature enhances its applicability across diverse problems.