# label *Entry of the [Dictionary of Applied Machine Learning](https://dictionaryofml.org) (course edition), by Alexander Jung, Aalto University. CC BY 4.0. Updated 2026-10-28. [HTML](https://dictionaryofml.org/terms/label.html) · [typeset PDF](https://dictionaryofml.org/terms/label.pdf) · [LaTeX](https://dictionaryofml.org/terms/label.tex) · [Python demo](https://dictionaryofml.org/terms/label.py) · [notebook](https://dictionaryofml.org/terms/label.ipynb)* Math is LaTeX with the book's macros expanded, so no notation package is needed to read it. ## Abstract A label is an attribute of a data point that represents a quantity of interest and that typically requires human expertise to determine. Predicting the label of a data point from its features is the goal of supervised learning. Determining the label of a data point is error-prone and often involves human annotation. The choice of label determines the learning task: whether the task is classification or regression depends on the label space together with the loss function. In self-supervised learning, the label is constructed from the features themselves, so that no manual annotation is required. ## Definition For a data point that represents a patient, a feature could be the body weight, while the label could be the presence of a disease. A label of a data point ${\mathbf z}$ is one of its attributes that represents a higher-level fact, or quantity of interest, and that typically requires human expertise or domain knowledge to determine (Anon., 2003; Everitt and Skrondal, 2010; Gujarati and Porter, 2009). Unlike a feature, which can be computed or measured easily, a label is usually harder to obtain. Whether a given attribute serves as a feature or a label is often a design choice that depends on the resources available in a given machine learning application. If an oncologist is readily available, a cancer prognosis can serve as a feature. On the other hand, if no oncologist is available, the cancer prognosis is a label that needs to be predicted from patient features. The notion of a label is made precise by introducing a labeling function $\bar{h}: \mathcal{Z} \rightarrow \mathcal{Y}$. The domain of this function is the data point space $\mathcal{Z}$, the set of all possible data points that can occur in a given machine learning application. Its values lie in the label space $\mathcal{Y}$, the set of all possible values the label of a data point can take on. The function assigns each data point ${\mathbf z}$ its label $y = \bar{h}({\mathbf z})$ (Shalev-Shwartz and Ben-David, 2014, Sect. 2.1). The labeling function $\bar{h}$ acts on the data point ${\mathbf z}$ itself, that is, on the whole patient. A trained model (or learned hypothesis) $\widehat{h}$ instead acts on the feature vector ${\mathbf x}$, a fixed list of recorded features such as the patient's body weight and age. The feature vector usually records only some attributes of a data point. The data point space $\mathcal{Z}$ contains patients, whereas the feature space $\mathcal{X}$ contains only their recorded feature vectors. Two patients with the same feature vector may therefore carry different labels, since they can differ in information that was never recorded as a feature. No hypothesis $\widehat{h}$ that depends on ${\mathbf x}$ alone can then deliver perfect predictions. In practice, the true label $\bar{h}({\mathbf z})$ of a data point is rarely known exactly. Even with full access to the data point, its label must be determined by human judgment or measurement, which is imperfect. The available label is then a noisy version of $\bar{h}({\mathbf z})$, as in the additive error model of Hastie et al. (2009, Sect. 2.6). For instance, an object category assigned by a human annotator or a diagnosis obtained from a clinician may be wrong, even though the annotator or clinician examines the data point directly. The discrepancy between such an observed label and the true label is called label noise and can degrade training and raise the risk of the learned hypothesis. This source of error differs from the incompleteness of the recorded features discussed above: here the data point is accessible, but its label is determined imperfectly. One way to reduce the effect of label noise is label smoothing, which slightly perturbs the labels of data points used for model training (Goodfellow et al., 2016, Sect. 7.5.1). It is a form of data augmentation that acts as regularization. Label noise also matters for regulation. The EU AI Act requires the training set of a high-risk artificial intelligence system to be, to the best extent possible, free of errors (European Parliament and Council of the European Union, 2024, Art. 10(3)) and the system to achieve an appropriate level of accuracy (European Parliament and Council of the European Union, 2024, Art. 15(1)); erroneous labels are one such error. When labels are personal data, the general data protection regulation requires them to be accurate (European Parliament and Council of the European Union, 2016, Art. 5(1)(d)). Beyond how reliably a label is determined, which attribute is taken as the label, i.e., the choice of the labeling function $\bar{h}$, determines the learning task. For the image of a cow herd in Fig. 1, the label could be a binary indicator of whether the image contains cows (binary classification). Other choices are the number of cows or the average green level (regression), and a masked center pixel predicted from the surrounding pixels (self-supervised learning). ![Figure 1 of the entry label](https://dictionaryofml.org/terms/fig/label-fig1.svg) *Figure 1: The same data point yields different learning tasks depending on the choice of label* Whether a learning task is a classification problem or a regression problem depends not only on the label space $\mathcal{Y}$ but also on the loss function. For example, the label space $\mathcal{Y} = \{0,1,2,3,\ldots\}$ can be used for a classification method that uses the 0/1 loss as the loss function. The same label space can also be used for a regression method that uses the squared error loss as the loss function. In self-supervised learning, the label is constructed from the features themselves, so no manual annotation is required. A prominent example is large language model training: consider the sentence ``All human beings are born free and equal,'' tokenized as the sequence (``All'', ``human'', ``beings'', ``are'', ``born'', ``free'', ``and'', ``equal''). The label for the subsequence (``All'', ``human'', ``beings'') is the next token ``are,'' the label for (``All'', ``human'', ``beings'', ``are'') is ``born,'' and so on. Taking every prefix of the sequence in turn, a single sentence produces many labeled data points from plain text alone. **Synonyms:** target, response, ground truth, quantity of interest. **See also:** [data point](https://dictionaryofml.org/terms/datapoint.html), [feature](https://dictionaryofml.org/terms/feature.html), label space, learning task, [classification](https://dictionaryofml.org/terms/classification.html), regression, [self-supervised learning](https://dictionaryofml.org/terms/selfsupervisedlearning.html), [data augmentation](https://dictionaryofml.org/terms/dataaug.html). ## References - Anon. (2003). The Oxford Dictionary of Statistical Terms. *Oxford Univ. Press*. - Everitt and Skrondal (2010). The Cambridge Dictionary of Statistics. *Cambridge Univ. Press*. - Gujarati and Porter (2009). Basic Econometrics. *McGraw-Hill/Irwin*. - Shalev-Shwartz and Ben-David (2014). Understanding Machine Learning: From Theory to Algorithms. *Cambridge Univ. Press*. - Goodfellow et al. (2016). Deep Learning. *MIT Press*. - European Parliament and Council of the European Union (2024). Regulation (EU) 2024/1689 of the European Parliament and of the Council of 13 June 2024 laying down harmonised rules on artificial intelligence and amending Regulations (EC) No 300/2008, (EU) No 167/2013, (EU) No 168/2013, (EU) 2018/858, (EU) 2018/1139 and (EU) 2019/2144 and Directives 2014/90/EU, (EU) 2016/797 and (EU) 2020/1828 (Artificial Intelligence Act) (Text with EEA relevance). - European Parliament and Council of the European Union (2016). Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation) (Text with EEA relevance). - Hastie et al. (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. *Springer Science+Business Media*. ## Cite this entry ```bibtex @incollection{dictml_label, author = {Jung, Alexander}, editor = {Olioumtsevits, Konstantina and Schnoor, Ekkehard}, title = {label}, booktitle = {Dictionary of Applied Machine Learning}, publisher = {Springer Nature Singapore}, year = {2026}, doi = {10.1007/978-981-92-2012-0_12-1}, isbn = {978-981-92-2012-0}, url = {https://doi.org/10.1007/978-981-92-2012-0_12-1} } ```