Since the objective function is unknown, the Bayesian strategy is to treat it as a random function and place a prior over it. The prior captures our beliefs about the behaviour of the function. After gathering the function evaluations, which are treated as data, the prior is updated to form the posterior distribution over the objective function. The posterior distribution, in turn, is used to construct an acquisition function (often also referred to as infill sampling criteria) that determines what the next query point should be.
Examples of acquisition functions include probability of improvement, expected improvement, Bayesian expected losses, upper confidence bounds (UCB), Thompson sampling and mixtures of these. They all trade-off exploration and exploitation so as to minimize the number of function queries. As such, Bayesian optimization is well suited for functions that are very expensive to evaluate.
The maximum of the acquisition function is typically found by resorting to discretization or by means of an auxiliary optimizer.
The approach has been applied to solve a wide range of problems, including learning to rank, interactive animation, robotics, sensor networks, automatic algorithm configuration, automatic machine learning toolboxes, reinforcement learning, planning, visual attention, architecture configuration in deep learning, static program analysis, experimental particle physics, etc.
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- BayesOpt, NIPS workshop on Bayesian Optimization (BayesOpt).
- GPyOpt, Python open-source library for Bayesian Optimization based on GPy.
- Bayesopt, an efficient implementation in C/C++ with support for Python, Matlab and Octave.
- Spearmint, a Python implementation focused on parallel and cluster computing.
- Hyperopt, a Python implementation for hyperparamenter optimization.
- SMAC, a Java implementation of random-forest-based Bayesian optimization for general algorithm configuration.
- pybo, a Python implementation of modular Bayesian optimization.
- Bayesopt.m, a Matlab implementation of Bayesian optimization with or without constraints.
- MOE MOE is a Python/C++/CUDA implementation of Bayesian Global Optimization using Gaussian Processes.
- SigOpt SigOpt offers Bayesian Global Optimization as a SaaS service focused on enterprise use cases.
- Metaopt, a Python implementation of hyperparameter optimization focused on parallel and cluster computing.
- Mind Foundry OPTaaS offers Bayesian Global Optimization via web-services with flexible parameter constraints.