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{{Short description|Process of finding the optimal set of variables
for a machine learning algorithm}}
In [[machine learning]], '''hyperparameter optimization'''<ref>Matthias Feurer and Frank Hutter. [https://link.springer.com/content/pdf/10.1007%2F978-3-030-05318-5_1.pdf Hyperparameter optimization]. In: ''AutoML: Methods, Systems, Challenges'', pages 3–38.</ref> or tuning is the problem of choosing a set of optimal [[Hyperparameter (machine learning)|hyperparameters]] for a learning algorithm. A hyperparameter is a [[parameter]] whose value is used to control the learning process, which must be configured before the process starts.<ref>{{cite journal |last1=Yang|first1=Li|title=On hyperparameter optimization of machine learning algorithms: Theory and practice|journal=Neurocomputing|year=2020|volume=415|pages=295–316|doi=10.1016/j.neucom.2020.07.061|arxiv=2007.15745 }}</ref><ref>{{cite arXiv |vauthors=Franceschi L, Donini M, Perrone V, Klein A, Archambeau C, Seeger M, Pontil M, Frasconi P |title=Hyperparameter Optimization in Machine Learning |year=2024 |class=stat.ML |eprint=2410.22854 }}</ref>
== Approaches ==
[[File:Hyperparameter Optimization using Grid Search.svg|thumb|Grid search across different values of two hyperparameters. For each hyperparameter, 10 different values are considered, so a total of 100 different combinations are evaluated and compared. Blue contours indicate regions with strong results, whereas red ones show regions with poor results.]]
=== Grid search ===
The traditional
or evaluation on a
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| title = Ten quick tips for machine learning in computational biology
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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.
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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
[[File:Hyperparameter Optimization using Random Search.svg|thumb|Random search across different combinations of values for two hyperparameters. In this example, 100 different random choices are evaluated. The green bars show that more individual values for each hyperparameter are considered compared to a grid search.]]
=== 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. A benefit over grid search is that random search can explore many more values than grid search could for continuous hyperparameters. It can outperform Grid search, especially when only a small number of hyperparameters affects the final performance of the machine learning algorithm.<ref name="bergstra" /> In this case, the optimization problem is said to have a low intrinsic dimensionality.<ref>{{Cite journal|last1=Ziyu|first1=Wang|last2=Frank|first2=Hutter|last3=Masrour|first3=Zoghi|last4=David|first4=Matheson|last5=Nando|first5=de Feitas|date=2016|title=Bayesian Optimization in a Billion Dimensions via Random Embeddings|journal=Journal of Artificial Intelligence Research|language=en|volume=55|pages=361–387|doi=10.1613/jair.4806|arxiv=1301.1942|s2cid=279236}}</ref> Random Search is also [[embarrassingly parallel]], and additionally allows the inclusion of prior knowledge by specifying the distribution from which to sample. Despite its simplicity, random search remains one of the important base-lines against which to compare the performance of new hyperparameter optimization methods.
[[File:Hyperparameter Optimization using Tree-Structured Parzen Estimators.svg|thumb|Methods such as Bayesian optimization smartly explore the space of potential choices of hyperparameters by deciding which combination to explore next based on previous observations.]]
=== Bayesian optimization ===
{{main
Bayesian optimization is a
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| volume = 6683
| pages = 507–523
| year = 2011
| url = http://www.cs.ubc.ca/labs/beta/Projects/SMAC/papers/11-LION5-SMAC.pdf
| citeseerx = 10.1.1.307.8813
| series = Lecture Notes in Computer Science
| isbn = 978-3-642-25565-6
| s2cid = 6944647
}}</ref><ref name="bergstra11">{{Citation
| last1 = Bergstra
| first1 = James
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| journal = Advances in Neural Information Processing Systems
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| url = http://papers.nips.cc/paper/4443-algorithms-for-hyper-parameter-optimization.pdf }}</ref><ref name="snoek">{{
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| title = Practical Bayesian Optimization of Machine Learning Algorithms
| journal = Advances in Neural Information Processing Systems
| volume =<!-- -->
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| year = 2012
| url = http://papers.nips.cc/paper/4522-practical-bayesian-optimization-of-machine-learning-algorithms.pdf
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}}</ref><ref name="thornton">{{cite journal
| last1 = Thornton
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| title = Auto-WEKA: Combined selection and hyperparameter optimization of classification algorithms
| journal = Knowledge
| volume = <!-- -->
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| year = 2013
| url = http://www.cs.ubc.ca/labs/beta/Projects/autoweka/papers/autoweka.pdf
| bibcode = 2012arXiv1208.3719T
| arxiv = 1208.3719
}}</ref><ref name="krnc">{{Citation
|last=Kernc
|title=SAMBO: Sequential And Model-Based Optimization: Efficient global optimization in Python
|date=2024
|url=https://zenodo.org/records/14461363
|access-date=2025-01-30
|doi=10.5281/zenodo.14461363
}}</ref> 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.
=== Gradient-based optimization ===
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.<ref>{{cite
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]].<ref>{{cite journal|last1=Domke|first1=Justin|title=Generic Methods for Optimization-Based Modeling|journal=
In a different approach,<ref>{{cite arXiv | eprint=1802.09419 | last1=Lorraine | first1=Jonathan | last2=Duvenaud | first2=David | title=Stochastic Hyperparameter Optimization through Hypernetworks | date=2018 | class=cs.LG }}</ref> a hypernetwork is trained to approximate the best response function. One of the advantages of this method is that it can handle discrete hyperparameters as well. Self-tuning networks<ref>{{cite arXiv | eprint=1903.03088 | last1=MacKay | first1=Matthew | last2=Vicol | first2=Paul | last3=Lorraine | first3=Jon | last4=Duvenaud | first4=David | last5=Grosse | first5=Roger | title=Self-Tuning Networks: Bilevel Optimization of Hyperparameters using Structured Best-Response Functions | date=2019 | class=cs.LG }}</ref> offer a memory efficient version of this approach by choosing a compact representation for the hypernetwork. More recently, Δ-STN<ref>{{cite arXiv | eprint=2010.13514 | last1=Bae | first1=Juhan | last2=Grosse | first2=Roger | title=Delta-STN: Efficient Bilevel Optimization for Neural Networks using Structured Response Jacobians | date=2020 | class=cs.LG }}</ref> has improved this method further by a slight reparameterization of the hypernetwork which speeds up training. Δ-STN also yields a better approximation of the best-response Jacobian by linearizing the network in the weights, hence removing unnecessary nonlinear effects of large changes in the weights.
Apart from hypernetwork approaches, gradient-based methods can be used to optimize discrete hyperparameters also by adopting a continuous relaxation of the parameters.<ref>{{cite arXiv | eprint=1806.09055 | last1=Liu | first1=Hanxiao | last2=Simonyan | first2=Karen | last3=Yang | first3=Yiming | title=DARTS: Differentiable Architecture Search | date=2018 | class=cs.LG }}</ref> Such methods have been extensively used for the optimization of architecture hyperparameters in [[neural architecture search]].
=== Evolutionary optimization ===
{{main|Evolutionary algorithm}}
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.<ref name="bergstra11" /> Evolutionary hyperparameter optimization follows a [[Evolutionary algorithm#Implementation|process]] inspired by the biological concept of [[evolution]]:
# Create an initial population of random solutions (i.e., randomly generate tuples of hyperparameters, typically 100+)
# Evaluate the hyperparameter tuples and acquire their [[fitness function]] (e.g., 10-fold [[Cross-validation (statistics)|cross-validation]] accuracy of the machine learning algorithm with those hyperparameters)
# Rank the hyperparameter tuples by their relative fitness
# Replace the worst-performing hyperparameter tuples with new ones generated via [[crossover (genetic algorithm)|crossover]] and [[mutation (genetic algorithm)|mutation]]
# Repeat steps 2-4 until satisfactory algorithm performance is reached or is no longer improving.
Evolutionary optimization has been used in hyperparameter optimization for statistical machine learning algorithms,<ref name="bergstra11" /> [[automated machine learning]], typical neural network <ref name="kousiouris1">{{cite journal |vauthors=Kousiouris G, Cuccinotta T, Varvarigou T | year = 2011 | title= The effects of scheduling, workload type and consolidation scenarios on virtual machine performance and their prediction through optimized artificial neural networks | url= https://www.sciencedirect.com/science/article/abs/pii/S0164121211000951 | journal = Journal of Systems and Software | volume = 84 | issue = 8 | pages = 1270–1291| doi = 10.1016/j.jss.2011.04.013 | hdl = 11382/361472 | hdl-access = free }}</ref> and [[Deep learning#Deep neural networks|deep neural network]] architecture search,<ref name="miikkulainen1">{{cite arXiv | vauthors = Miikkulainen R, Liang J, Meyerson E, Rawal A, Fink D, Francon O, Raju B, Shahrzad H, Navruzyan A, Duffy N, Hodjat B | year = 2017 | title = Evolving Deep Neural Networks |eprint=1703.00548| class = cs.NE }}</ref><ref name="jaderberg1">{{cite arXiv | vauthors = Jaderberg M, Dalibard V, Osindero S, Czarnecki WM, Donahue J, Razavi A, Vinyals O, Green T, Dunning I, Simonyan K, Fernando C, Kavukcuoglu K | year = 2017 | title = Population Based Training of Neural Networks |eprint=1711.09846| class = cs.LG }}</ref> as well as training of the weights in deep neural networks.<ref name="such1">{{cite arXiv | vauthors = Such FP, Madhavan V, Conti E, Lehman J, Stanley KO, Clune J | year = 2017 | title = Deep Neuroevolution: Genetic Algorithms Are a Competitive Alternative for Training Deep Neural Networks for Reinforcement Learning |eprint=1712.06567| class = cs.NE }}</ref>
===
Population Based Training (PBT) learns both hyperparameter values and network weights. Multiple learning processes operate independently, using different hyperparameters. As with evolutionary methods, poorly performing models are iteratively replaced with models that adopt modified hyperparameter values and weights based on the better performers. This replacement model warm starting is the primary differentiator between PBT and other evolutionary methods. PBT thus allows the hyperparameters to evolve and eliminates the need for manual hypertuning. The process makes no assumptions regarding model architecture, loss functions or training procedures.
PBT and its variants are adaptive methods: they update hyperparameters during the training of the models. On the contrary, non-adaptive methods have the sub-optimal strategy to assign a constant set of hyperparameters for the whole training.<ref>{{cite arXiv|last1=Li|first1=Ang|last2=Spyra|first2=Ola|last3=Perel|first3=Sagi|last4=Dalibard|first4=Valentin|last5=Jaderberg|first5=Max|last6=Gu|first6=Chenjie|last7=Budden|first7=David|last8=Harley|first8=Tim|last9=Gupta|first9=Pramod|date=2019-02-05|title=A Generalized Framework for Population Based Training|eprint=1902.01894|class=cs.AI}}</ref>
=== Early stopping-based ===
[[File:Successive-halving-for-eight-arbitrary-hyperparameter-configurations.png|thumb|Successive halving for eight arbitrary hyperparameter configurations. The approach starts with eight models with different configurations and consecutively applies successive halving until only one model remains.]]
A class of early stopping-based hyperparameter optimization algorithms is purpose built for large search spaces of continuous and discrete hyperparameters, particularly when the computational cost to evaluate the performance of a set of hyperparameters is high. Irace implements the iterated racing algorithm, that focuses the search around the most promising configurations, using statistical tests to discard the ones that perform poorly.<ref name="irace">{{cite journal |last1=López-Ibáñez |first1=Manuel |last2=Dubois-Lacoste |first2=Jérémie |last3=Pérez Cáceres |first3=Leslie |last4=Stützle |first4=Thomas |last5=Birattari |first5=Mauro |date=2016 |title=The irace package: Iterated Racing for Automatic Algorithm Configuration |journal=Operations Research Perspective |volume=3 |issue=3 |pages=43–58 |doi=10.1016/j.orp.2016.09.002|doi-access=free |hdl=10419/178265 |hdl-access=free }}</ref><ref name="race">{{cite journal |last1=Birattari |first1=Mauro |last2=Stützle |first2=Thomas |last3=Paquete |first3=Luis |last4=Varrentrapp |first4=Klaus |date=2002 |title=A Racing Algorithm for Configuring Metaheuristics |journal=Gecco 2002 |pages=11–18}}</ref>
Another early stopping hyperparameter optimization algorithm is successive halving (SHA),<ref>{{cite arXiv|last1=Jamieson|first1=Kevin|last2=Talwalkar|first2=Ameet|date=2015-02-27|title=Non-stochastic Best Arm Identification and Hyperparameter Optimization|eprint=1502.07943|class=cs.LG}}</ref> which begins as a random search but periodically prunes low-performing models, thereby focusing computational resources on more promising models. Asynchronous successive halving (ASHA)<ref>{{cite arXiv|last1=Li|first1=Liam|last2=Jamieson|first2=Kevin|last3=Rostamizadeh|first3=Afshin|last4=Gonina|first4=Ekaterina|last5=Hardt|first5=Moritz|last6=Recht|first6=Benjamin|last7=Talwalkar|first7=Ameet|date=2020-03-16|title=A System for Massively Parallel Hyperparameter Tuning|class=cs.LG|eprint=1810.05934v5}}</ref> further improves upon SHA's resource utilization profile by removing the need to synchronously evaluate and prune low-performing models. Hyperband<ref>{{cite journal|last1=Li|first1=Lisha|last2=Jamieson|first2=Kevin|last3=DeSalvo|first3=Giulia|last4=Rostamizadeh|first4=Afshin|last5=Talwalkar|first5=Ameet|date=2020-03-16|title=Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization|journal=Journal of Machine Learning Research|volume=18|pages=1–52|arxiv=1603.06560}}</ref> is a higher level early stopping-based algorithm that invokes SHA or ASHA multiple times with varying levels of pruning aggressiveness, in order to be more widely applicable and with fewer required inputs.
=== Others ===
[[Radial basis function|RBF]]<ref name=abs1705.08520>{{cite arXiv |eprint=1705.08520|last1=Diaz|first1=Gonzalo|title=An effective algorithm for hyperparameter optimization of neural networks|last2=Fokoue|first2=Achille|last3=Nannicini|first3=Giacomo|last4=Samulowitz|first4=Horst|class=cs.AI|year=2017}}</ref> and [[spectral method|spectral]]<ref name=abs1706.00764>{{cite arXiv |eprint=1706.00764|last1=Hazan|first1=Elad|title=Hyperparameter Optimization: A Spectral Approach|last2=Klivans|first2=Adam|last3=Yuan|first3=Yang|class=cs.LG|year=2017}}</ref> approaches have also been developed.
== Issues with hyperparameter optimization ==
When hyperparameter optimization is done, the set of hyperparameters are often fitted on a training set and selected based on the generalization performance, or score, of a validation set. However, this procedure is at risk of overfitting the hyperparameters to the validation set. Therefore, the generalization performance score of the validation set (which can be several sets in the case of a cross-validation procedure) cannot be used to simultaneously estimate the generalization performance of the final model. In order to do so, the generalization performance has to be evaluated on a set independent (which has no intersection) of the set (or sets) used for the optimization of the hyperparameters, otherwise the performance might give a value which is too optimistic (too large). This can be done on a second test set, or through an outer [[Cross-validation (statistics)|cross-validation]] procedure called nested cross-validation, which allows an unbiased estimation of the generalization performance of the model, taking into account the bias due to the hyperparameter optimization.
== See also ==
* [[Automated machine learning]]
* [[Neural architecture search]]
* [[Meta-optimization]]
* [[Model selection]]
* [[Self-tuning]]
* [[XGBoost]]
* [[Optuna]]
== References ==
{{Reflist|30em}}
{{Differentiable computing}}
[[Category:Machine learning]]
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