Multiclass Classification Strategies (OvR, OvO)
Binary classifiers, multiclass problems
Many foundational classifiers — SVMs, logistic regression in its basic form, kernel methods — are inherently binary: they draw one boundary separating two classes. Yet most real classification tasks have three or more categories: routing a support ticket to one of ten teams, classifying an image into one of a thousand object types. Two general strategies let binary classifiers handle this: One-vs-Rest (OvR) and One-vs-One (OvO). Some models (decision trees, neural networks with softmax, naive Bayes) are natively multiclass and don't need either — but understanding OvR/OvO matters because they're the default reduction strategy wrapping many binary learners.
One-vs-Rest (OvR / OvA)
OvR trains $K$ binary classifiers for $K$ classes, each one separating a single class from all the others combined. Classifier $k$ learns to answer: 'is this example class $k$, or not?' At prediction time, all $K$ classifiers score the input, and the class with the highest confidence score wins. OvR is efficient — only $K$ models to train — and is the default strategy for logistic regression and linear SVMs in most libraries. Its main weakness: each binary sub-problem can be imbalanced (one class vs. everything else), and the raw scores from different classifiers aren't always on a comparable scale, which can bias the argmax decision.
One-vs-One (OvO)
OvO trains a separate binary classifier for every pair of classes, giving $\binom{K}{2} = \frac{K(K-1)}{2}$ classifiers total. For $K=10$ classes that's 45 models instead of 10. At prediction time, every classifier votes for one of its two classes, and the class with the most votes across all pairwise contests wins (majority voting). OvO avoids the imbalance problem of OvR — each sub-problem only ever sees two classes — and tends to work better for kernel SVMs, which is why scikit-learn's SVC defaults to OvO internally. The cost is quadratic growth in the number of models as $K$ increases, which matters for large label spaces.
from sklearn.svm import SVC
from sklearn.multiclass import OneVsRestClassifier, OneVsOneClassifier
from sklearn.datasets import load_iris
from sklearn.model_selection import cross_val_score
X, y = load_iris(return_X_y=True) # 3 classes
ovr = OneVsRestClassifier(SVC(kernel="linear"))
ovo = OneVsOneClassifier(SVC(kernel="linear"))
print("OvR accuracy:", cross_val_score(ovr, X, y, cv=5).mean())
print("OvO accuracy:", cross_val_score(ovo, X, y, cv=5).mean())
# Number of binary classifiers trained:
K = 3
print(f"OvR trains {K} classifiers; OvO trains {K*(K-1)//2} classifiers")
| Strategy | # classifiers (K classes) | Class imbalance per sub-problem | Scales to large K |
|---|---|---|---|
| One-vs-Rest | K | Can be severe (1 vs rest) | Good — linear growth |
| One-vs-One | K(K-1)/2 | Minimal — always balanced pair | Poor — quadratic growth |
Softmax regression sidesteps the reduction entirely
What's next?
OvR and OvO both treat classes as a flat, unordered set with no relationships between them. Many real label spaces aren't flat at all — product categories, biological taxonomies, and organizational ticket types often have a natural tree structure, which calls for a classification strategy that exploits that structure rather than ignoring it.
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