# Hyperparameter optimization

In machine learning, hyperparameter optimization or tuning is the problem of choosing a set of optimal hyperparameters for a learning algorithm.

The same kind of machine learning model can require different constraints, weights or learning rates to generalize different data patterns. These measures are called hyperparameters, and have to be tuned so that the model can optimally solve the machine learning problem. Hyperparameter optimization finds a tuple of hyperparameters that yields an optimal model which minimizes a predefined loss function on given independent data.[1] The objective function takes a tuple of hyperparameters and returns the associated loss.[1] Cross-validation is often used to estimate this generalization performance.[2]

## Approaches

### Grid search

The traditional way of performing hyperparameter optimization has been grid search, or a parameter sweep, which is simply an exhaustive searching through a manually specified subset of the hyperparameter space of a learning algorithm. A grid search algorithm must be guided by some performance metric, typically measured by cross-validation on the training set[3] or evaluation on a held-out validation set.[4]

Since the parameter space of a machine learner may include real-valued or unbounded value spaces for certain parameters, manually set bounds and discretization may be necessary before applying grid search.

For example, a typical soft-margin SVM classifier equipped with an RBF kernel has at least two hyperparameters that need to be tuned for good performance on unseen data: a regularization constant C and a kernel hyperparameter γ. Both parameters are continuous, so to perform grid search, one selects a finite set of "reasonable" values for each, say

${\displaystyle C\in \{10,100,1000\}}$
${\displaystyle \gamma \in \{0.1,0.2,0.5,1.0\}}$

Grid search then trains an SVM with each pair (C, γ) in the Cartesian product of these two sets and evaluates their performance on a held-out validation set (or by internal cross-validation on the training set, in which case multiple SVMs are trained per pair). Finally, the grid search algorithm outputs the settings that achieved the highest score in the validation procedure.

Grid search suffers from the curse of dimensionality, but is often embarrassingly parallel because typically the hyperparameter settings it evaluates are independent of each other.[2]

### Random search

Random Search replaces the exhaustive enumeration of all combinations by selecting them randomly. This can be simply applied to the discrete setting described above, but also generalizes to continuous and mixed spaces. It can outperform Grid search, especially when only a small number of hyperparameters affects the final performance of the machine learning algorithm[2]. In this case, the optimization problem is said to have a low intrinsic dimensionality[5]. Random Search is also embarrassingly parallel, and additionally allows to include prior knowledge by specifying the distribution from which to sample.

### Bayesian optimization

Bayesian optimization is a global optimization method for noisy black-box functions. Applied to hyperparameter optimization, Bayesian optimization builds a probabilistic model of the function mapping from hyperparameter values to the objective evaluated on a validation set. By iteratively evaluating a promising hyperparameter configuration based on the current model, and then updating it, Bayesian optimization, aims to gather observations revealing as much information as possible about this function and, in particular, the location of the optimum. It tries to balance exploration (hyperparameters for which the outcome is most uncertain) and exploitation (hyperparameters expected close to the optimum). In practice, Bayesian optimization has been shown[6][7][8][9] to obtain better results in fewer evaluations compared to grid search and random search, due to the ability to reason about the quality of experiments before they are run.

For specific learning algorithms, it is possible to compute the gradient with respect to hyperparameters and then optimize the hyperparameters using gradient descent. The first usage of these techniques was focused on neural networks.[10] Since then, these methods have been extended to other models such as support vector machines[11] or logistic regression.[12]

A different approach in order to obtain a gradient with respect to hyperparameters consists in differentiating the steps of an iterative optimization algorithm using automatic differentiation.[13][14]

### Evolutionary optimization

Evolutionary optimization is a methodology for the global optimization of noisy black-box functions. In hyperparameter optimization, evolutionary optimization uses evolutionary algorithms to search the space of hyperparameters for a given algorithm.[7] Evolutionary hyperparameter optimization follows a process inspired by the biological concept of evolution:

1. Create an initial population of random solutions (i.e., randomly generate tuples of hyperparameters, typically 100+)
2. Evaluate the hyperparameters tuples and acquire their fitness function (e.g., 10-fold cross-validation accuracy of the machine learning algorithm with those hyperparameters)
3. Rank the hyperparameter tuples by their relative fitness
4. Replace the worst-performing hyperparameter tuples with new hyperparameter tuples generated through crossover and mutation
5. Repeat steps 2-4 until satisfactory algorithm performance is reached or algorithm performance is no longer improving

Evolutionary optimization has been used in hyperparameter optimization for statistical machine learning algorithms[7], automated machine learning[15][16], deep neural network architecture search[17][18], as well as training of the weights in deep neural networks[19].

### Others

RBF[20] and spectral[21] approaches have also been developed.

## Software

### Grid search

• LIBSVM comes with scripts for performing grid search.
• scikit-learn is a Python package which includes grid search.

### Random search

• hyperopt, also via hyperas and hyperopt-sklearn, are Python packages which include random search.
• scikit-learn is a Python package which includes random search.
• H2O AutoML provides automated data preparation, hyperparameter tuning via random search, and stacked ensembles in a distributed machine learning platform.

### Bayesian

• spearmint Spearmint is a package to perform Bayesian optimization of machine learning algorithms.
• Bayesopt,[22] an efficient implementation of Bayesian optimization in C/C++ with support for Python, Matlab and Octave.
• MOE MOE is a Python/C++/CUDA library implementing Bayesian Global Optimization using Gaussian Processes.
• Auto-WEKA[23] is a Bayesian hyperparameter optimization layer on top of WEKA.
• Auto-sklearn[24] is a Bayesian hyperparameter optimization layer on top of scikit-learn.
• mlrMBO, also with mlr, is an R package for model-based/Bayesian optimization of black-box functions.
• tuneRanger is an R package for tuning random forests using model-based optimization.
• BOCS is a Matlab package which uses semidefinite programming for minimizing a black-box function over discrete inputs.[25] A Python 3 implementation is also included.

• hypergrad is a Python package for differentiation with respect to hyperparameters.[14]

## References

1. ^ a b Claesen, Marc; Bart De Moor (2015). "Hyperparameter Search in Machine Learning". arXiv: [cs.LG].
2. ^ a b c Bergstra, James; Bengio, Yoshua (2012). "Random Search for Hyper-Parameter Optimization" (PDF). J. Machine Learning Research. 13: 281–305.
3. ^ Chin-Wei Hsu, Chih-Chung Chang and Chih-Jen Lin (2010). A practical guide to support vector classification. Technical Report, National Taiwan University.
4. ^ Chicco D (December 2017). "Ten quick tips for machine learning in computational biology". BioData Mining. 10 (35): 1–17. doi:10.1186/s13040-017-0155-3. PMC . PMID 29234465.
5. ^ Ziyu, Wang,; Frank, Hutter,; Masrour, Zoghi,; David, Matheson,; Nando, de Feitas, (2016). "Bayesian Optimization in a Billion Dimensions via Random Embeddings". Journal of Artificial Intelligence Research. 55. doi:10.1613/jair.4806 (inactive 2018-06-26).
6. ^ Hutter, Frank; Hoos, Holger; Leyton-Brown, Kevin (2011), "Sequential model-based optimization for general algorithm configuration" (PDF), Learning and Intelligent Optimization
7. ^ a b c Bergstra, James; Bardenet, Remi; Bengio, Yoshua; Kegl, Balazs (2011), "Algorithms for hyper-parameter optimization" (PDF), Advances in Neural Information Processing Systems
8. ^ Snoek, Jasper; Larochelle, Hugo; Adams, Ryan (2012). "Practical Bayesian Optimization of Machine Learning Algorithms" (PDF). Advances in Neural Information Processing Systems. arXiv: [stat.ML]. Bibcode:2012arXiv1206.2944S.
9. ^ Thornton, Chris; Hutter, Frank; Hoos, Holger; Leyton-Brown, Kevin (2013). "Auto-WEKA: Combined selection and hyperparameter optimization of classification algorithms" (PDF). Knowledge Discovery and Data Mining. arXiv: [cs.LG]. Bibcode:2012arXiv1208.3719T.
10. ^ Larsen, Jan; Hansen, Lars Kai; Svarer, Claus; Ohlsson, M (1996). "Design and regularization of neural networks: the optimal use of a validation set". Proceedings of the 1996 IEEE Signal Processing Society Workshop.
11. ^ Olivier Chapelle; Vladimir Vapnik; Olivier Bousquet; Sayan Mukherjee (2002). "Choosing multiple parameters for support vector machines" (PDF). Machine Learning. 46: 131–159. doi:10.1023/a:1012450327387.
12. ^ Chuong B; Chuan-Sheng Foo; Andrew Y Ng (2008). "Efficient multiple hyperparameter learning for log-linear models". Advances in Neural Information Processing Systems 20.
13. ^ Domke, Justin (2012). "Generic Methods for Optimization-Based Modeling" (PDF). Aistats. 22.
14. ^ a b Maclaurin, Douglas; Duvenaud, David; Adams, Ryan P. (2015). "Gradient-based Hyperparameter Optimization through Reversible Learning". arXiv: [stat.ML].
15. ^ a b Olson RS, Urbanowicz RJ, Andrews PC, Lavender NA, Kidd L, Moore JH (2016). "Automating biomedical data science through tree-based pipeline optimization". Proceedings of EvoStar 2016. Lecture Notes in Computer Science. 9597: 123–137. arXiv:. doi:10.1007/978-3-319-31204-0_9. ISBN 978-3-319-31203-3.
16. ^ a b Olson RS, Bartley N, Urbanowicz RJ, Moore JH (2016). "Evaluation of a Tree-based Pipeline Optimization Tool for Automating Data Science". Proceedings of EvoBIO 2016: 485–492. doi:10.1145/2908812.2908918. ISBN 9781450342063.
17. ^ Miikkulainen R, Liang J, Meyerson E, Rawal A, Fink D, Francon O, Raju B, Shahrzad H, Navruzyan A, Duffy N, Hodjat B (2017). "Evolving Deep Neural Networks". arXiv: [cs.NE].
18. ^ Jaderberg M, Dalibard V, Osindero S, Czarnecki WM, Donahue J, Razavi A, Vinyals O, Green T, Dunning I, Simonyan K, Fernando C, Kavukcuoglu K (2017). "Population Based Training of Neural Networks". arXiv: [cs.LG].
19. ^ Such FP, Madhavan V, Conti E, Lehman J, Stanley KO, Clune J (2017). "Deep Neuroevolution: Genetic Algorithms Are a Competitive Alternative for Training Deep Neural Networks for Reinforcement Learning". arXiv: [cs.NE].
20. ^ a b Diaz, Gonzalo; Fokoue, Achille; Nannicini, Giacomo; Samulowitz, Horst (2017). "An effective algorithm for hyperparameter optimization of neural networks". arXiv: [cs.AI].
21. ^ a b Hazan, Elad; Klivans, Adam; Yuan, Yang (2017). "Hyperparameter Optimization: A Spectral Approach". arXiv: [cs.LG].
22. ^ Martinez-Cantin, Ruben (2014). "BayesOpt: A Bayesian Optimization Library for Nonlinear Optimization, Experimental Design and Bandits" (PDF). Journal of Machine Learning Research. 15: 3915−3919. arXiv: [cs.LG]. Bibcode:2014arXiv1405.7430M.
23. ^ Kotthoff L, Thornton C, Hoos HH, Hutter F, Leyton-Brown K (2017). "Auto-WEKA 2.0: Automatic model selection and hyperparameter optimization in WEKA". Journal of Machine Learning Research: 1–5.
24. ^ Feurer M, Klein A, Eggensperger K, Springenberg J, Blum M, Hutter F (2015). "Efficient and Robust Automated Machine Learning". Advances in Neural Information Processing Systems 28 (NIPS 2015): 2962–2970.
25. ^ Baptista, Ricardo; Poloczek, Matthias (2018). "Bayesian Optimization of Combinatorial Structures". arXiv: [stat.ML].
26. ^ Gorissen, Dirk; Crombecq, Karel; Couckuyt, Ivo; Demeester, Piet; Dhaene, Tom (2010). "A Surrogate Modeling and Adaptive Sampling Toolbox for Computer Based Design" (PDF). J. Machine Learning Research. 11: 2051–2055.