Bayesian optimization: implicit information about the optimum in a Gaussian process

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 18:53

Summary:

This video plots the (approximate) distribution of global minima of the posterior of a function modeled as a Gaussian process of which three samples have been observed.

First, the video reviews how to obtain said posterior. Next, we review how to obtain realizations of it, and the code to repeat them and obtain the minimum of each one (this is the basic of the ”Thompson sampling” heuristic in Bayesian optimization). With this code, six thousand minimums are obtained and three histograms are plotted:

• Marginal of $x^{\ast }$, probability density that in a given $x^{\ast }$ there is a global minimum of some realization;

• Marginal of $y^{\ast }$, probability density that a certain $y^{\ast }$ is the optimal value (global minimum) that achieves some realization;

• “joint” histogram approximating the joint density function over $(x^{\ast },y^{\ast })$ representing the probability that some realization has the global minimum at $x^{\ast }$ and its value is precisely $y^{\ast }$.

This is the implicitly available information about the location of the optimum. The goal of Bayesian optimization is to refine that information through sampling (explore) and/or to hit the optimum of my unknown underlying function as quickly as possible (exploit).

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