 # One-vs-Rest and One-vs-One for Multi-Class Classification

Last Updated on April 27, 2021

Not all classification predictive models support multi-class classification.

Algorithms such as the Perceptron, Logistic Regression, and Support Vector Machines were designed for binary classification and do not natively support classification tasks with more than two classes.

One approach for using binary classification algorithms for multi-classification problems is to split the multi-class classification dataset into multiple binary classification datasets and fit a binary classification model on each. Two different examples of this approach are the One-vs-Rest and One-vs-One strategies.

In this tutorial, you will discover One-vs-Rest and One-vs-One strategies for multi-class classification.

After completing this tutorial, you will know:

• Binary classification models like logistic regression and SVM do not support multi-class classification natively and require meta-strategies.
• The One-vs-Rest strategy splits a multi-class classification into one binary classification problem per class.
• The One-vs-One strategy splits a multi-class classification into one binary classification problem per each pair of classes.

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Let’s get started. How to Use One-vs-Rest and One-vs-One for Multi-Class Classification
Photo by Espen Sundve, some rights reserved.

## Tutorial Overview

This tutorial is divided into three parts; they are:

1. Binary Classifiers for Multi-Class Classification
2. One-Vs-Rest for Multi-Class Classification
3. One-Vs-One for Multi-Class Classification

## Binary Classifiers for Multi-Class Classification

Classification is a predictive modeling problem that involves assigning a class label to an example.

Binary classification are those tasks where examples are assigned exactly one of two classes. Multi-class classification is those tasks where examples are assigned exactly one of more than two classes.

• Binary Classification: Classification tasks with two classes.
• Multi-class Classification: Classification tasks with more than two classes.

Some algorithms are designed for binary classification problems. Examples include:

• Logistic Regression
• Perceptron
• Support Vector Machines

As such, they cannot be used for multi-class classification tasks, at least not directly.

Instead, heuristic methods can be used to split a multi-class classification problem into multiple binary classification datasets and train a binary classification model each.

Two examples of these heuristic methods include:

• One-vs-Rest (OvR)
• One-vs-One (OvO)

Let’s take a closer look at each.

## One-Vs-Rest for Multi-Class Classification

One-vs-rest (OvR for short, also referred to as One-vs-All or OvA) is a heuristic method for using binary classification algorithms for multi-class classification.

It involves splitting the multi-class dataset into multiple binary classification problems. A binary classifier is then trained on each binary classification problem and predictions are made using the model that is the most confident.

For example, given a multi-class classification problem with examples for each class ‘red,’ ‘blue,’ and ‘green‘. This could be divided into three binary classification datasets as follows:

• Binary Classification Problem 1: red vs [blue, green]
• Binary Classification Problem 2: blue vs [red, green]
• Binary Classification Problem 3: green vs [red, blue]

A possible downside of this approach is that it requires one model to be created for each class. For example, three classes requires three models. This could be an issue for large datasets (e.g. millions of rows), slow models (e.g. neural networks), or very large numbers of classes (e.g. hundreds of classes).

The obvious approach is to use a one-versus-the-rest approach (also called one-vs-all), in which we train C binary classifiers, fc(x), where the data from class c is treated as positive, and the data from all the other classes is treated as negative.

— Page 503, Machine Learning: A Probabilistic Perspective, 2012.

This approach requires that each model predicts a class membership probability or a probability-like score. The argmax of these scores (class index with the largest score) is then used to predict a class.

This approach is commonly used for algorithms that naturally predict numerical class membership probability or score, such as:

• Logistic Regression
• Perceptron

As such, the implementation of these algorithms in the scikit-learn library implements the OvR strategy by default when using these algorithms for multi-class classification.

We can demonstrate this with an example on a 3-class classification problem using the LogisticRegression algorithm. The strategy for handling multi-class classification can be set via the “multi_class” argument and can be set to “ovr” for the one-vs-rest strategy.

The complete example of fitting a logistic regression model for multi-class classification using the built-in one-vs-rest strategy is listed below.

The scikit-learn library also provides a separate OneVsRestClassifier class that allows the one-vs-rest strategy to be used with any classifier.

This class can be used to use a binary classifier like Logistic Regression or Perceptron for multi-class classification, or even other classifiers that natively support multi-class classification.

It is very easy to use and requires that a classifier that is to be used for binary classification be provided to the OneVsRestClassifier as an argument.

The example below demonstrates how to use the OneVsRestClassifier class with a LogisticRegression class used as the binary classification model.

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## One-Vs-One for Multi-Class Classification

One-vs-One (OvO for short) is another heuristic method for using binary classification algorithms for multi-class classification.

Like one-vs-rest, one-vs-one splits a multi-class classification dataset into binary classification problems. Unlike one-vs-rest that splits it into one binary dataset for each class, the one-vs-one approach splits the dataset into one dataset for each class versus every other class.

For example, consider a multi-class classification problem with four classes: ‘red,’ ‘blue,’ and ‘green,’ ‘yellow.’ This could be divided into six binary classification datasets as follows:

• Binary Classification Problem 1: red vs. blue
• Binary Classification Problem 2: red vs. green
• Binary Classification Problem 3: red vs. yellow
• Binary Classification Problem 4: blue vs. green
• Binary Classification Problem 5: blue vs. yellow
• Binary Classification Problem 6: green vs. yellow

This is significantly more datasets, and in turn, models than the one-vs-rest strategy described in the previous section.

The formula for calculating the number of binary datasets, and in turn, models, is as follows:

• (NumClasses * (NumClasses – 1)) / 2

We can see that for four classes, this gives us the expected value of six binary classification problems:

• (NumClasses * (NumClasses – 1)) / 2
• (4 * (4 – 1)) / 2
• (4 * 3) / 2
• 12 / 2
• 6

Each binary classification model may predict one class label and the model with the most predictions or votes is predicted by the one-vs-one strategy.

An alternative is to introduce K(K − 1)/2 binary discriminant functions, one for every possible pair of classes. This is known as a one-versus-one classifier. Each point is then classified according to a majority vote amongst the discriminant functions.

— Page 183, Pattern Recognition and Machine Learning, 2006.

Similarly, if the binary classification models predict a numerical class membership, such as a probability, then the argmax of the sum of the scores (class with the largest sum score) is predicted as the class label.

Classically, this approach is suggested for support vector machines (SVM) and related kernel-based algorithms. This is believed because the performance of kernel methods does not scale in proportion to the size of the training dataset and using subsets of the training data may counter this effect.

The support vector machine implementation in the scikit-learn is provided by the SVC class and supports the one-vs-one method for multi-class classification problems. This can be achieved by setting the “decision_function_shape” argument to ‘ovo‘.

The example below demonstrates SVM for multi-class classification using the one-vs-one method.

The scikit-learn library also provides a separate OneVsOneClassifier class that allows the one-vs-one strategy to be used with any classifier.

This class can be used with a binary classifier like SVM, Logistic Regression or Perceptron for multi-class classification, or even other classifiers that natively support multi-class classification.

It is very easy to use and requires that a classifier that is to be used for binary classification be provided to the OneVsOneClassifier as an argument.

The example below demonstrates how to use the OneVsOneClassifier class with an SVC class used as the binary classification model.

This section provides more resources on the topic if you are looking to go deeper.

## Summary

In this tutorial, you discovered One-vs-Rest and One-vs-One strategies for multi-class classification.

Specifically, you learned:

• Binary classification models like logistic regression and SVM do not support multi-class classification natively and require meta-strategies.
• The One-vs-Rest strategy splits a multi-class classification into one binary classification problem per class.
• The One-vs-One strategy splits a multi-class classification into one binary classification problem per each pair of classes.

Do you have any questions?

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### 58 Responses to One-vs-Rest and One-vs-One for Multi-Class Classification

1. Tazar April 13, 2020 at 6:45 am #

Is it possible to access each individual classifier that has been created for each case? In some cases it is required to understand and see result individually

• Jason Brownlee April 13, 2020 at 1:50 pm #

Yes, I believe they are a property on the fit model.

2. Shashi Rekha.H April 13, 2020 at 1:31 pm #

Knowledgeful Information

• Jason Brownlee April 13, 2020 at 1:52 pm #

Thanks!

3. Christos April 13, 2020 at 3:33 pm #

Excellent article. My questions are how to decide which one to use when we have to build a predictive model with specific input dataset. Should we try both methods and Keep the one with the best results in terms of i.e. kappa, F1 etc of the classification table? Does the literature propose a specific choice if we have an unbalanced training dataset?

4. Prasad April 13, 2020 at 6:22 pm #

I believed multinomial logistic regression vould solve multi-label classification.

If we employ one to rest or one to one; wouldnt it takr more time in midel building

• Jason Brownlee April 14, 2020 at 6:06 am #

Agreed on both points.

• Amal April 18, 2020 at 12:02 am #

Thank you very much
Please which type of machine learning you recommend to use for classification faults for PV solar

5. Dave April 17, 2020 at 10:35 pm #

Great article as always.

1. How do you choose between setting parameters in the model or using the seperate ovr/ovo class?

2. Do you have any links to examples with output using these methods? Would you suggest a real-world dataset to work with with to show the value here?

Thanks!

• Jason Brownlee April 18, 2020 at 5:56 am #

Thanks!

What do you mean model vs separate class? You mean as in logistic regression? In that case, probably better to use the built-in case to avoid introducing bugs.

I’m sure I have examples of LR on multi-class classification. It is not about value, it about being able to use algorithms on problems were they could not be used before. That is valuable if those algorithms are as/more skillful than other methods tried on the same problem.

6. Bonelwa April 18, 2020 at 6:28 pm #

Excellent article.

Do you perhaps have a similar article where you use R.

Thanks.

• Jason Brownlee April 19, 2020 at 5:51 am #

Sorry, I do not.

7. Bastia May 6, 2020 at 7:22 am #

thanks for the explanation. I do have a question:

for xgboost, it is internally for binary classification. Therefore, in xgboost in we choose to use “multi:softprob” as the objective function, it will create n forests for each class, right? So we don’t need to build a xgboost classifier with “binary:logistic” and wrap it with OneVsRestClassifier, right?

• Jason Brownlee May 6, 2020 at 1:36 pm #

Not quite, each tree makes a multi-class prediction directly in xgboost.

• Bastia May 7, 2020 at 5:44 am #

Thanks for your answer. Just want to confirm my understanding with you. The author of xbgoost says in:
https://github.com/dmlc/xgboost/issues/3655
so, each class has a binary forest; however, they are trained based on multi-class cross-entropy loss. So it seems that xgboost does build binary classifiers, but not trains them on binary loss.

Then my question is how about for logstic regression in sklearn, if we choose the multi_class parameter to be ovr, then what loss function it should use? If it also use multiclass crossentropy loss function, then what is the meaning here. We can train N 1-by-m vectors for N binary classifier, and we can also train a N-by-m matrix to directily get the softmax result.

• Jason Brownlee May 7, 2020 at 7:02 am #

Fascinating. Thanks.

You can train a multinomial logistic regression, e.g. true multi-class. Or you can use binary cross entropy and fit multiple OvR models, which is what sklearn does I believe.

8. Mehmet May 13, 2020 at 9:36 pm #

Very nice article.
1) Is it possible to use with MultiLayerPerceptron ?

• Jason Brownlee May 14, 2020 at 5:49 am #

Thanks.

Perhaps. I don’t have an example, sorry.

9. Isak Imanuel Leong May 22, 2020 at 8:49 pm #

Hello Jason, thank you for your article.

My name is Isak and I’m from Indonesia. So, sorry for my bad English. I want to ask about the code on this line:

# define model
model = SVC()
# define ovo strategy
ovo = OneVsOneClassifier(model)

I have read sklearn documentation that explains two things below:

=== For SVC() ===
The multiclass support is handled according to a one-vs-one scheme.

=== For LinearSVC() ===
This class supports both dense and sparse input and the multiclass support is handled according to a one-vs-the-rest scheme.

My Question:
1. SVC already supports multi-class classification with OVO approach. Why use “OneVsOneClassifier” module for SVC ()?

2. I am currently working on my final project on multiclass classification using the SVM method with the OnevsAll approach. Is it possible to change the “OnevsOneClassifier” with “OnevsRestClassifier” on SVC() ? Or should I use the “OnevsRestClassifier” module on LinearSVC() ?

Thank you…

• Jason Brownlee May 23, 2020 at 6:19 am #

Correct. I do it explicitly here as a demonstration. You can adapt it to use any method you like.

Yes, try it!

10. Dennis June 6, 2020 at 12:29 am #

Nice Article! My two cents:

“decision_function_shape{‘ovo’, ‘ovr’}, default=’ovr’
Whether to return a one-vs-rest (‘ovr’) decision function of shape (n_samples, n_classes) as all other classifiers, or the original one-vs-one (‘ovo’) decision function of libsvm which has shape (n_samples, n_classes * (n_classes – 1) / 2). However, one-vs-one (‘ovo’) is always used as multi-class strategy. The parameter is ignored for binary classification.”

Thus, for SVM ovo is always used for multi class classfication, and this parameter just decides how the decision function looks like, don’t you agree?

• Jason Brownlee June 6, 2020 at 7:53 am #

Thanks for the note.

11. Yatin August 12, 2020 at 2:56 am #

Thanks for this wonderful article, Very well explained different OvO and OvR methods

• Jason Brownlee August 12, 2020 at 6:13 am #

Thanks!

12. Mark Littlewood August 13, 2020 at 2:57 am #

I presume that this one v rest creates, in your case 3 models. The first where its either a 1 for red and 0 for anything else, then 1 for blue and 0 for anything else etc. It then predicts by running all three models on a line and taking the highest prob’. What if you have a problem involving ‘races’. Here the one is a runner and the rest are the other runners in the race. One could make all races have equal runners by padding out but my question is whether there is a neat library routine or parameter to handle this ?. Can ti be made to perform one v rest id the data has a raceId and a runnerId field

• Jason Brownlee August 13, 2020 at 6:20 am #

Yes, padding, but it does not sound like a good fit for that type of problem.

Any tracking of row to a subject (e.g. an id) must happen in the application (code around the model), not in the predictive model.

13. Teerath August 22, 2020 at 2:28 pm #

Can we use binary classification for mutli class classifications?

• Jason Brownlee August 22, 2020 at 2:34 pm #

Yes, the above techniques show how how.

14. Łukasz Wojtów September 14, 2020 at 10:33 pm #

This caught my attention:
“This approach requires that each model predicts a class membership probability or a probability-like score. The argmax of these scores (class index with the largest score) is then used to predict a class.”

I’ve implemented some toy AI tool myself self and what I do instead is that I compare scores for each model (OvR) that says “yes, that’s the class”. So if:
model “RED” has training score 0.6 and says YES
model “BLUE” has training score 0.7 and says NO
model “GREEN” has training score 0.65 says YES
then I assign GREEN label.
So it’s not directly “class membership probability” but more like “probability that model is correct”.

• Jason Brownlee September 15, 2020 at 5:24 am #

Nice. If it works well/better than other approaches then use it.

15. Alex October 16, 2020 at 2:59 pm #

Hey Jason. Interesting concepts. Does each methodology you described computes a probability score value for each class variable? So In other words , each class variable in the test dataset will get a probability value assigned based on training data. What I am getting at is the difference between say a random forest where only the highest value is assigned to one of the three class variables but others are 0 and your description seems like suggest that not only does the class with highest probability identified but also classes with lower probability.

• Jason Brownlee October 17, 2020 at 5:55 am #

Some models will natively predict a probability, e.g. logistic regression.

This is not related to using binary classification models for multi-class classification problems as described above. The topics are orthogonal.

16. Paul October 21, 2020 at 3:41 am #

I think i don’t really understand OvO, if i have three classes A, B, C then i have pairs :

A vs B, A vs C, B vs C,

let’s say we want to use model for class C, then we need to choose between A vs C, B vs C right? but both of them did flas C as negative observation, then we are choosing one with higher number of negatives not postivies(not sure if this vote is just higher num of positives)?

• Jason Brownlee October 21, 2020 at 6:42 am #

You predict with all 3 models and the label with the most support wins/is predicted.

17. Dennis October 21, 2020 at 4:41 am #

Hey Jason, great article can you describe how OvO works for testing dataset? How voting works?

18. Jimmy January 6, 2021 at 3:17 am #

Hi Jason,

Why do we even use One vs Rest and why don’t we train our model on k classes simply and what trouble does it cause if we use k-classes instead of creating a binary class out of every class?

And how does One Vs Rest solve the problem?

Thank You
Jimmy

• Jason Brownlee January 6, 2021 at 6:32 am #

It may work well or best for a specific binary classification model and dataset.

The way it solves the problem is described in the above tutorial.

19. Monlu January 25, 2021 at 8:49 pm #

Hi Jason, very interesting article.
I wanted to ask you a question.
In a multiclass classification problem (3 classes), how do I determine which classifier works best between multiclass classifiers (such as naive bayes or mln) and svm classifier (one vs one) which is binary??

• Jason Brownlee January 26, 2021 at 5:52 am #

Generally, you would select a metric and evaluate a suite of models and model configurations until you find one that performs best.

• M. Gafer April 20, 2021 at 5:53 am #

Hi Dr Jason, Thank you very much for the interesting topic.

If I need to compare three classifiers performance in the one-vs-one method. I select to use sensitivity and accuracy as metrics. For each pair of classes, I will produce my metrics (sensitivity and accuracy). How to estimate the overall metric for the classifier.

for example, I have a three-class problem, assuming that I have balanced-class data, for each classifier I will have 3 values for sensitivity and three values for accuracy one for each pair of classes. how to estimate the overall accuracy and overall sensitivity for the classifier.
can I just average them, or there is another way?

Thanks

• Jason Brownlee April 20, 2021 at 6:03 am #

Perhaps you can use repeated k-fold cv to evaluate each method and compare the mean score.

20. syeda January 28, 2021 at 5:04 am #

Hi jason, very interesting article. I need your help. I have a dataset which have 11 classes and I am using SVM classifier for multiclass classification but my accuracy is not good. but when I perform binary classification, it gives me good accuracy. There is something wrong with multiclass classifier. Please help me.

• Jason Brownlee January 28, 2021 at 6:08 am #

Thanks.

Perhaps SVM is no good for your data.

Perhaps try a different model?
Perhaps try a different data preparation?

21. Mustafa February 9, 2021 at 9:12 pm #

Hi dr. Brownlee, I was wondering how does the “.predict_proba” work with ovr? since ovr uses multiple independent binary classifiers… I actually tried the example above and got 3 probabilities with a sum equals to 1 just like the multi class model.

22. Bex June 2, 2021 at 10:49 pm #

Does RandomForestClassifier work in the same way? I mean, does Sklearn create multiple RandomForestClassifiers for multi-class classification too?

• Jason Brownlee June 3, 2021 at 5:36 am #

No, I believe random forest directly supports multi-class classification.

23. Mary June 17, 2021 at 3:17 am #

Hi,

Since we may encounter the challenge of imbalanced classification in One vs All, how should we handle it?

Thank you.

Regards,
Mary

• Jason Brownlee June 17, 2021 at 6:19 am #

Perhaps try rebalancing the training dataset.

24. rafia June 18, 2021 at 3:25 am #

Hi Jason Brownlee great work,

Does stacking ensemble support multilabel classification. or let me rephrase my question can i use multilabel classifiers(such as MLKNN, BRKNN, or transformation classifier : Labelpowerst, binaryrelevance) at level 0 and 1 for

• Jason Brownlee June 18, 2021 at 5:47 am #

I’m not sure off the cuff, perhaps try it and see.

25. Irzan September 21, 2021 at 10:54 pm #

Let’s say we have 3 classes: red, green, and blue.
So, in one-vs-one (OVO) we create 3 pairs of classifier:
– red vs blue
– red vs green
– blue vs green

How if the result for each classifier is like this?
– red vs blue , resulting red
– red vs green , resulting green
– blue vs green , resulting blue

Which class should be the winner?
Or maybe I misunderstood how this OVO concept works?

• Adrian Tam September 23, 2021 at 3:08 am #

No, you didn’t misunderstand. That’s correct and we have no concrete answer in this case. I would rather say, “undetermined”.