Optimizer

As explained in Bayesian Optimization in Action

An optimizer is an input location at which an objective function attains its maximum or minimum over the domain under consideration.

In black-box optimization, the optimizer is the input location where the objective reaches its highest value. If x* is optimal, then f* = f(x*) ≥ f(x) for every x in the domain D. section 6.2.1

A Gaussian-process belief about the objective induces a probability distribution over x*. That distribution may be multimodal, place mass at domain endpoints, and be difficult to model as dimensionality increases. section 6.2.1

Useful consequences of treating the optimizer as a random variable include:

• Sampling functions from the Gaussian process and recording each sampled maximum provides a simple estimate of the optimizer distribution. section 6.2.1
• Entropy-search policies target uncertainty about the optimizer, while the optimal value f* offers a simpler one-dimensional alternative. section 6.2.1

Figure 6.10 GP belief (top) and the distribution of the function optimizer x* (bottom). The distribution of the optimizer is non-Gaussian and fairly complicated, posing a challenge for modeling and decision-making.

Visualizes a Gaussian-process belief alongside the complicated, non-Gaussian distribution it induces over the optimizer. section 6.2.1

How Bayesian Optimization in Action covers it

How deep it goes

Defines the optimizer and explains its GP-induced distribution, but stops short of deriving entropy-search computations.

Examples

Uses binary search, GP function samples, and candidate locations near −5, 0, and 2 to illustrate optimizer uncertainty.

What it assumes

Assumes familiarity with objective functions, domains, and Gaussian-process beliefs.

What you can do after

Evaluate/understand how objective uncertainty becomes a distribution over possible maximizing inputs.

How it connects

Connects the optimizer to Bayesian optimization, Gaussian processes, entropy search, acquisition functions, and the optimal value.

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Related concepts

• optimal valueproduces · An optimal value is the objective-function value attained by a solution that optimizes the specified criterion.
• Gaussian processproduced by · A Gaussian process is a probability distribution over functions in which every finite collection of function values has a joint Gaussian distribution.
• optimizerproduced by · An optimizer is an algorithm or procedure that adjusts model parameters or other decisions to improve an objective function.

This explainer was generated by Manning's knowledge graph from the text of Bayesian Optimization in Action. The section links point to the parts of the book it draws on.