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# Bayesian Optimization and Hyperparameter Tuning
[Bayesian optimization](https://en.wikipedia.org/wiki/Bayesian_optimization) is a strategy for optimizing black-box functions. Generally, Bayesian optimization is useful when the function you want to optimize is not differentiable, or each function evaluation is expensive. As such, it is a natural candidate for hyperparameter tuning.

## Bayesian optimization
The outline of Bayesian optimization is as follows:

- Compute the value of your black-box function at a point
- Store this point and function value in your history of points previously sampled
- Use this history to decide what point to inspect next
- Repeat

Bayesian optimization is a 'sequential' strategy: you compute function values at points one at a time. In the case of hyperparameter tuning, this is often referred to as Sequential Model Based Optimization (SMBO).

We can restate this general strategy more precisely: start by placing a prior distribution over your function (the prior distribution can be uniform). Use the prior distribution to choose a point to sample. There are a few ways to choose what point to sample - informally, the goal is to sample a point with a high probability of maximizing (or minimizing) your function. Compute the function value at this point, and incorporate this data to create a posterior distribution. Begin again: your posterior is your new prior.

There are many algorithms for how to create distributions, and how to choose what point to sample (see references).

## Bayesian optimization for hyperparameter tuning

In the case of hyperparameter tuning, the 'black-box function' generally consists of two steps:

1. train a model
2. compute a performance metric

This black-box function takes values of hyperparameters as inputs, and returns a performance metric. The performance metric can be anything (f1-score, AUC-ROC, accuracy, etc.), and it can take into account penalties for undesirable features (training time, evaluation time, memory use, etc).

## Code Examples
- `Examples/Hyperopt_Predict_Affairs.ipynb` shows how to use hyperopt with a tree of parzen estimators for hyperparameter tuning, and for model selection. 
- `Examples/Scikit-optimize_Predict_Affairs.ipynb` shows how to use scikit-optimize with Gaussian processes and trees for hyperparameter tuning (using expected improvement).

## References
### Papers

- [Algorithms for Hyper-Parameter Optimization](https://papers.nips.cc/paper/4443-algorithms-for-hyper-parameter-optimization.pdf)
  - Authors: Bergstra, Bardenet, Bengio, Kégl
- [Automatic Model Construction with Gaussian Processes](http://www.cs.toronto.edu/~duvenaud/thesis.pdf)
  - Author: Duvenaud
- [Auto-WEKA: Combined Selection and Hyperparameter Optimization of Classification Algorithms](https://arxiv.org/pdf/1208.3719.pdf)
  - Authors: Thornton, Hutter, Hoos, Brown
- [Bayesian Hyperparameter Optimization for Ensemble Learning](https://www.arxiv.org/abs/1605.06394)
  - Authors: Lévesque, Gagné, Sabourin
- [Practical Bayesian Optimization of Machine Learning Algorithms](https://papers.nips.cc/paper/4522-practical-bayesian-optimization-of-machine-learning-algorithms.pdf)
  - Authors: Snoek, Larochelle, Adams
- [Sequential Model-Based Optimization for General Algorithm Configuration](http://www.cs.ubc.ca/~hutter/papers/11-LION5-SMAC.pdf)
  - Authors: Hutter, Hoos, Leyton-Brown
- [Towards an Empirical Foundation for Assessing Bayesian Optimization of Hyperparameters](https://www.cs.ubc.ca/~hoos/Publ/EggEtAl13.pdf)
  - Authors: Eggensperger, Feurer, Hutter, Bergstra, Snoek, Hoos, Leyton-Brown
- [Modular mechanisms for Bayesian optimization](http://mlg.eng.cam.ac.uk/hoffmanm/papers/hoffman:2014b.pdf)
  - Authors: Hoffman, Shahriari

### Books
- [Gaussian Processes for Machine Learning](http://www.gaussianprocess.org/gpml/)
   - Authors: Rasmussen, Williams

### Software (list curated primarily for Python)
- [Hyperopt](https://github.com/hyperopt/hyperopt)
- [Scikit-Optimize](https://scikit-optimize.github.io/)
- [Spearmint](https://github.com/JasperSnoek/spearmint)
- [MOE](https://github.com/Yelp/MOE)
- [SigOpt](https://sigopt.com/)
- [GPyOpt](https://sheffieldml.github.io/GPyOpt/)
- [BayesOpt](http://rmcantin.bitbucket.org/html/)

### Videos
- [Introduction to Gaussian Processes from Neil Lawrence](https://www.youtube.com/watch?v=ewJ3AxKclOg)

### Other 
- [Bayesian Optimization on Wikipedia](https://en.wikipedia.org/wiki/Bayesian_optimization)

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