# A snapshot on nonstandard supervised learning problems

###### Abstract

Machine learning is a field which studies how machines can alter and adapt their behavior, improving their actions according to the information they are given. This field is subdivided into multiple areas, among which the best known are supervised learning (e.g. classification and regression) and unsupervised learning (e.g. clustering and association rules).

Within supervised learning, most studies and research are focused on well known standard tasks, such as binary classification, multiclass classification and regression with one dependent variable. However, there are many other less known problems. These are what we generically call nonstandard supervised learning problems. The literature about them is much more sparse, and each study is directed to a specific task. Therefore, the definitions, relations and applications of this kind of learners are hard to find.

The goal of this paper is to provide the reader with a broad view on the distinct variations of nonstandard supervised problems. A comprehensive taxonomy summarizing their traits is proposed. A review of the common approaches followed to accomplish them and their main applications is provided as well.

###### Keywords:

Machine learning Supervised learning Nonstandard learning###### Msc:

MSC 68T05 MSC 68T10## 1 Introduction

According to Mitchell learning-mitchell (), a machine is said to learn from experience related to a class of tasks and performance metric , when its performance at tasks in improves according to after experience .

Supervised learning is one of the fundamental areas of machine learning learning-marsland (). From object detection to ecological modeling to emotion recognition, it covers all kinds of applications. It essentially consists in learning a function by training with a set of input-output pairs. The training stage can be seen as in the previous definition, and the specific task may vary, but usually involves predicting an appropriate output given a new input.

Traditionally, supervised learning problems have been spread into two categories: classification and regression classification (); pattern-rec (). In the first, information is divided into discrete categories, while the latter involves patterns associated to a value in a continuous spectrum.

These problems can be processed by learning from a training dataset, which is composed of instances. Typically, these instances or samples take the form where is a vector of values in the space of input variables and is a value in the target variable. Each problem can be described by the type of its instances: inputs will usually belong to a subset of , and outputs will take values in a specific one-dimensional set, finite or continuous. Once trained, the obtained model can be used to predict the target variable on unseen instances.

Standard classification problems are those where labels are either binary or multiclass classification-duda (); multiclass (). In the binary case, an instance can only be associated with one of two values: positive or negative, which is equivalent to 0 or 1. For example, email messages may be classified into spam or legit, and tumours can be categorized as either benign or malign. Multiclass problems, on the other hand, involve any finite number of classes. That is, any given instance will belong to one of possibly many categories, which is equivalent to it being assigned a natural number below a convenient threshold. As an example, a photograph of a plant or a sound recording from an animal could correspond to one of a variety of species.

A standard regression problem learning-tibshirani (); regression () consists in finding a function which is able to predict, for a given example, a real value among a continuous range, usually an interval or the set of real numbers . For example, the height of a person may be estimated out of several characteristics such as age or country of origin.

Even though these standard problems are applicable in a multitude of cases, there are situations whose correct modeling requires modifications of their structure. For example, a newspaper article can be categorized according to its contents, but it could be desirable to assign several categories simultaneously. Similarly, a social media post could be described by not one but two input vectors, an image and a piece of text. These special circumstances cannot be covered by the traditional one-vector input and one-dimensional output schema. As a consequence, since performance metrics which measure improvements in standard tasks assume the common structure, they lose applicability or sense in these cases. Thus, not only new techniques are needed to tackle the problems, but also new ways of measuring and comparing their success.

This work studies variations on classic supervised problems where the traditional structure is not obeyed, which we call nonstandard variations. These emerge when the structure of the classical components of the problems does not suffice to describe complex situations, such as multiplicity of inputs or outputs, or order restrictions. As a consequence, this manuscript does not cover other singular supervised problems, such as high dimensionality of the feature space highdim () or unbalanced training sets imbalanced (); imbalanced-krawczyk (), nor time-dependent problems, such as data streams streams (); streams2 () or time series timeseries ().

The rest of the paper is structured as follows. Section 2 formally defines and describes each nonstandard variation. This is followed by Section 3 establishing relations among the introduced problems and proposing a taxonomy of them. Section 4 describes the most common techniques used to solve them. After that, Section 5 enumerates popular applications of each problem. Section 6 covers other variations further from the ones previously detailed. Lastly, Section 7 draws some conclusions.

## 2 Definitions of nonstandard variations

The problems introduced in this section are generalizations over the traditional versions of classification and regression. The focus is on fully supervised problems, where inputs are always paired with outputs during training. An alternative taxonomy based on different supervision models is introduced in weak-nonstandard ().

### 2.1 Notation

In this work we will establish a notation which intends to be as simple to understand as possible, while being able to encompass every nonstandard variation. First, any supervised learning problem consists in finding a function which will classify, rank or perform regression. It will be noted as

(1) |

where is an input set, or domain, and is an output set, or codomain. It will be assumed that a training dataset is provided, including a finite number of input-output pairs:

(2) |

This way, a learning algorithm will be able to generate the desired function . An additional notation will be the set of labels where convenient.

For example, in standard binary classification and . Similarly, standard regression problems can be defined with the same kind of set and . Thus, we can define very distinct supervised problems by particularizing sets or in different ways.

Other usual notations are based in probability theory, thus involving random variables and probability distributions gaussianproc (); learning-murphy (). In that case, and would be the sample spaces of the input and output variables and , respectively. Predictors would usually attempt to infer a discriminant model from the training dataset.

### 2.2 Multi-instance

The multi-instance (MI) framework mil () assumes a single feature space for all instances, but each training pattern may consist of more than one instance. In this case, a training pattern is composed of a finite multiset or bag of instances and a label. Formally, assuming instances are drawn from a set , the domain can be described as follows:

(3) |

In this case, the learning algorithm will not know labels associated to each instance but to a bag of them. In addition to this, not all instances may share the same relevance or are equally related to the label.

Some MI problems assume that hidden labels are present for each instance in a bag: for example, a training set of drug tests where, for each test, several drug types are analyzed. Additionally, a typical MI assumption in the binary scenario states that a bag is positive when at least one of its instances is positive, and it is negative otherwise mil-assumptions ().

Other MI problems differ in that a per-instance labeling may not be possible or may not make sense: for example, if each bag represents an image and instances are image segments, class beach can only apply to bags with water and sand segments, but it cannot apply to an individual instance.

### 2.3 Multi-view

A learning problem is considered to be multi-view (MV) mviewl () when inputs are composed of several components of very different nature.

For example, if a learning pattern consists of an image as well as a piece of text representing the same instance, they can be seen as two views on it. In that case, images and texts would belong to distinct feature spaces and respectively, an input pattern being . More generally, we can describe the input space as:

(4) |

where is the number of views offered by the problem and is the dimension of the feature space of the -th view.

### 2.4 Multi-label

The multi-label (ML) learning field mlc (); mltutorial () studies problems related to simultaneously assigning multiple labels to a single instance. That is, if the codomain consists of all possible selections of these labels, also known as labelsets:

(5) |

As shown by this formulation, it is equivalent to think of a selection of labels as a subset of and as a binary vector. For example, the labelset composed of the first and third labels can be represented either by or .

The difference that arises when comparing ML problems to binary or multiclass ones is that labels may interact with each other. For example, a news piece classified in economy is more likely to be labeled politics than sports. Similarly, a photograph labeled ocean is less likely to have the mountains label rather than beach. Methods may take advantage of label co-ocurrence scumble () in order to reduce the search space when predicting a labelset.

### 2.5 Multi-dimensional

Multi-dimensional (MD) learning mdc () is a generalized classification problem where categorization is performed simultaneously along several dimensions. Each instance can belong to one of many classes in each dimension, thus the output space can be formally described as:

(6) |

where is the label space for the -th dimension.

As with ML learning, label dimensions may be related in some way and treating them independently would only be a naive solution to the problem.

### 2.6 Label distribution learning

In label distribution learning (LDL) problems ldl (), otherwise known as probabilistic class label problems ldl-prob (), any instance can be described in different degrees by each label. This can be modeled as a discrete distribution over the labels, where the probability of a label given a specific instance is called its degree of description. Analitically, the objective is, for each instance, to predict a real-valued vector which sums exactly 1:

(7) |

In this case, we would say that the -th label in describes an instance with degree .

### 2.7 Label ranking

In a label ranking (LR) problem lrankpairwise (); lranksurvey () the objective is not to find a function able to choose one or several labels from the label space. Instead, it must evaluate their relevance for each unseen instance. The most general version of the problem involves a training set where is the set of all partial orders of , and the obtained function also maps individual instances to partial orders. This way, for each test instance the function will output a sequence of preferences where some labels will be seen as more relevant than others.

However, the typical situation in label ranking problems is that the orders are total, which means any two labels can always be compared. This is called a ranking and does not exclude the possibility of ties. When ties are not allowed it is said to be a sorting or permutation, and can be formulated as follows:

(8) |

where is the amount of labels. can also be seen as the set of all permutations of the labels in , usually known as the symmetric group of order , and noted as .

### 2.8 Multi-target regression

A regression problem where the output space has more than just one dimension is usually called multi-target regression (MTR) and is also known as multi-output, multi-variate or multi-response moutr (). In this case, a formal description is simply that the codomain is a continuous multi-dimensional real set:

(9) |

and is the number of target variables.

As with other multiple target extensions, the key difference with single-target regression in this case is the possible interactions among output variables.

### 2.9 Ordinal regression

A problem where the target space is discrete but ordered is called ordinal regression (OR) or, alternatively, ordinal classification ord-survey (). It can be located midway between classification and regression. More specifically, it consists in labeling instances with a finite number of choices where these are ordered

(10) |

In OR, the training phase consists in learning from a set of feature vectors which have a specific label associated to them, and testing can be performed over individual instances. This means that, although labels are ordered, the main objective is not to rank or sort instances as in learning to rank ltr (), but to simply classify them. The labels themselves do not provide any metric information either, they only carry qualitative information about the order among themselves.

### 2.10 Monotonicity constraints

Order relations can exist not only in the label space but in the feature space as well. Partial orders among real-valued feature vectors are always possible, and there may be cases where the order among instances is determined by just one or a few of their attributes.

When inputs as well as outputs are at least partially ordered, it is common to look for predictions which respect their order relations. In that case, the objective is to obtain a classifier or regression function which enforces the following constraint:

(11) |

When is discrete the problem is usually called monotone classification (MC), monotonic classification or ordinal classification with monotonicity constraints mc-salva (). If, on the contrary, is continuous, it is known as isotonic regression (IR) ir-book ().

### 2.11 Absence or partiality of information

Some problems do not directly alter the structure of and from the standard supervised problem. Instead, they restrict which data can belong to a training set, or remove labelings from training examples. In this case, training information is presented partially or with some exclusions.

According to which kind of information is missing from the training set, a learning task can usually be categorized as semi-supervised semi-sup (), one-class learning oneclass (), PU-learning pu-learn (), zero-shot learning zeroshot () or one-shot learning oneshot (). These are described further in Section 6.1.

### 2.12 Variation combinations

Some of the components described above can be combined to compose a more complex problem overall. Usually, one of these combinations will take components from different variation types, for example, simultaneous multiplicity of inputs and outputs.

More specifically, there exist several studies involving MI ML scenarios miml (); miml2 (). In this case, examples from the input space are composed of several feature vectors and are associated to various labels. As a consequence, this model can represent many complicated problems where inputs and outputs have more structure than usual.

Other more uncommon situations are MV MI ML problems mvmiml (), where patterns have several instances which may or may not belong to the same space, a multi-output version of OR named graded ML classification graded-ml () and more complex input structures such as multi-layer MI MV mlmimv (), where a hierarchy of instances is present in each example.

## 3 Taxonomy

A first categorization of the variations analyzed in this work can be made according to how they differ from the standard problem. There can be multiplicity in the input space or the output space, order constraints may exist, or only partial information may be given in some cases. Fig. 1 shows ways in which the traditional problems can be generalized.

Problems introducing multiple inputs are MI and MV, whereas multiple outputs can be found on ML, MD, LR, LDL and MTR. Problems where orders are present are OR, MC and IR. Likewise, tasks with only partial information are, among others, semi-supervised learning, one-shot classification and zero-shot classification.

Finally, a generalized problem can be built out of combining several of these components: for example, a multiple-input multiple-output problem where the inputs and outputs can belong to structures like the ones defined above.

The rest of this section studies variations on the structure of the input space and output space, establishes relations among problems, and describes how they can be particularized or generalized to one another.

### 3.1 Input structure

In a standard supervised problem, the input space consists of single feature vectors and does not impose a specific order.

Problems where learning patterns are composed of multiple instances can usually be categorized into either MI, if the inputs share the same structure, or MV, otherwise. Their combination can also be considered as well, e.g. a problem where an example is composed of one or more photographs and one or more pieces of text. This would be a case of a MV MI problem.

There are also problems where there exists a partial or total order among instances, which is coupled with an order constraint in relation to the outputs. These are MC and IR.

Fig. 2 summarizes these structural traits in a hierarchy and indicates problems where these traits are present.

### 3.2 Output structure

The diversity in output variations is higher than that of the input ones. A first sorting criterion is whether the codomain is discrete or continuous. This way, problems are either classification or regression ones.

Further subdivision of problems allows to separate these traits according to whether outputs remain scalars or become vectors. In the first case we consider order in the discrete scenario a nonstandard variation, which is present in OR and MC. In the second case, classification problems are spread into ML, LR and MD, and regression ones into LDL and MTR.

Fig. 3 organizes these traits in a hierarchy based on the previous criteria. Each leaf of the tree also includes problems where each one is present.

The variations in the structure of target spaces in supervised problems can be seen as generalizations of the standard problems. Furthermore, some of them are also more general than others. For example, ML problems can be seen as LR ones where, for a given instance, labels over a threshold are active and those below are not. Thus, LR is a generalization of the ML scenario. More relations of this kind are displayed in Fig. 4.

As shown in the graph, an inclusion of more target variables of the same type transforms a binary problem into ML, a multiclass problem into MD and a single-target regression one into MTR. Similarly, inclusion of more values into each variable allows to generalize binary problems to multiclass, and ordinal to single-target regression, as well as ML ones to MD and these to MTR. LDL can be seen as a generalization of ML where real numbers between 0 and 1 are also allowed as values for a label. LR is a generalization of ML by the argument discussed before.

### 3.3 Summary

In this section input and output variations of standard supervised problems have been categorized and related. Table 1 allows to identify specific problems according to which input and output traits are present.

Unordered outputs | Ordered outputs | |||||

Scalar | Multiple | Scalar | Multiple | |||

Discrete | Continuous | Discrete | Continuous | |||

Unordered inputs | standard classification classification () | ML/MD classification mlc (); mdc () | OR ord-survey () | standard regression regression () | Graded ML graded-ml () | MTR moutr () |

Ordered inputs | - | - | MC mc-salva () | IR ir-book () | - | - |

Multiple instances | MI classification mil () | MIML/MIMD classification miml () | - | MI regression mil () | - | - |

Multiple views | MV classification mviewl () | MVML/MVMD classification mvmiml () | - | MV regression mviewl () | - | - |

## 4 Common approaches to tackle nonstandard problems

When tackling a nonstandard problem, most techniques follow one of two main approaches: problem transformation or algorithm adaptation. The first one relies on appropriate transformations of the data which result in one or more simpler, standard problems. The latter implies an extension or development of previously existing algorithms, in order to adapt them to the complexities induced by the structure of the data.

In the following subsections several methods based on both approaches are enumerated for each analysed problem.

### 4.1 Problem transformation

Problem transformation methods assume that a solution can be achieved by extracting one or more simpler problems out of the original one. For example, a problem with multi-dimensional targets could be transformed into many problems with scalar outputs. Then, these problems could be solved independently by a classical algorithm. A solution for the original problem would be the concatenation of those extracted from the simpler ones.

Next, the most common transformation techniques are described for each nonstandard supervised learning task previously introduced.

#### – Mi.

The taxonomy proposed in mic-taxonomy () describes an Embedded Space paradigm, where each bag is transformed into a single feature vector representing the relevant information about the whole bag. This transformation brings the MI problem into a single-instance one. Most of these methods are vocabulary-based, which means that the embedding uses a set of concepts to classify each bag according to its instances, resulting in a single vector with one component per concept.

#### – Mv.

Some naive transformations consist in ignoring every view except one, or concatenating feature vectors from all views, thus training a single-view model in both cases mv-spectral (). A preprocessing based on Canonical Correlation Analysis mv-cca () is able to project data from multiple views onto a lower-dimensional, single-view space.

#### – Ml.

Transformation methods for ML classification mlmethods () are diverse: Binary Relevance trains separate binary classifiers for each label. Label Powerset reduces the problem to a multiclass one by treating each individual labelset as an independent class label, and Random k-Labelsets ml-rakel () extracts an ensemble of multiclass problems similarly. Classifier chains ml-chains () trains subsequent binary classifiers accumulating previous predictions as inputs. ML problems can also be transformed to LR ml-clr ().

#### – Md.

In some cases, independent classifiers can be trained for several dimensions mdc (); mdc-indep () but this method ignores possible correlations among dimensions. An alternative transformation, building a different label from each combination of classes, would produce a much larger label space and thus is not typically applied.

#### – Ldl.

A LDL problem can be reduced to multiclass classification by extracting as many single-label examples as labels for each one of the training instances ldl (). These new examples are assigned a class corresponding to each label and weighted according to its degree of description. During the prediction process, the classifier must be able to output the score/confidence for each label, which can be used as its description degree.

#### – Lr.

A reduction of this problem to several binary problems can be achieved by learning pairwise preferences lrankpairwise (). This transforms a -label problem into binary problems describing a comparison among two labels. An alternative reduction by means of constraint classification lr-constraint () builds a single binary classification dataset by expanding each label preference into a new positive instance and a new negative instance. The feature space of the new binary problem has dimension , where is the original dimension and the number of labels, due to the constraints embedded in it by Kesler’s construction nilsson ().

#### – Mtr.

There are several ways to transform a MTR problem into several single-target regression ones. Some of them are inspired by the ML field, such as a one-vs-all single-target reduction, multi-target stacking and regressor chains mtrviaml (). All of them train single-target regressors for several extracted problems, and then combine the obtained predictions. A different approach based on support vectors mtr-lssvr () extends the feature space which expresses the multi-output problem as a single-target one that can be solved using least squares support vector regression machines.

#### – Or.

An ordinal problem with classes can be transformed into binary classification problems by using each class from the second to the last one as a threshold for the positive class ord-simple (). This decomposition can be called ordered partitions and is not the only possible one: others are one-vs-next, one-vs-followers and one-vs-previous ord-survey (). Several 3-class problems can also be obtained by using, for the -th problem, classes “”, “” and “”.

#### – Mc.

The authors in monotonicity () describe a procedure to tackle binary MC problems by means of IR. Multiclass MC cases can be reduced to several binary MC ones, which in turn are solved as IR problems.

### 4.2 Algorithm adaptation

Existing methods for classical problems can be extended in order to introduce the necessary complexities of nonstandard variations. As an example, nearest neighbor methods could be coupled with new distance metrics in order to be able to measure similarity among multiple inputs.

The rest of this section presents some algorithm adaptations which can be used to tackle nonstandard supervised tasks.

#### – Mi.

Methods that work on instance level are adaptations of algorithms from single-instance classification whose responses are then aggregated to build the bag-level classification mic-taxonomy (). They typically assume that one positive instance implies a positive bag. Adaptations of common algorithms have been proposed with support vector machines (SVM) mi-svm () and neural networks mi-nn (), whereas some original methods in this area are Axis-Parallel Rectangles mi-apr () and Diverse Density mi-framework (). In the bag-space paradigm, methods treat bags as a whole and use specific distance metrics with distance as well as kernel-based classifiers, such as k-nearest neighbor (k-NN) mi-knn () or SVM mi-kernel ().

#### – Mv.

Supervised methods for MV are comparatively less developed than semi-supervised ones. Nonetheless, there is an extension of SVM mv-svm () which simultaneously looks for two SVMs, one in each of the feature spaces of a two-view problem. There is an extension of Fisher discriminant analysis as well mv-fda ().

#### – Ml.

The most relevant algorithm adaptations mlmethods () are based on standard classification algorithms with added support for choosing more than one class at a time: adaptations exist for k-NN ml-knn (), decision trees ml-dt (), SVMs ml-svm (), association rules ml-rules () and ensembles mlensembles ().

#### – Md.

Specific Bayesian networks have been proposed for the MD scenario md-bayes (); md-bayes2 (), as well as Maximum Entropy-based algorithms mdc (); mdc-indep ().

#### – Ldl.

Proposals in ldl () are adaptations of k-NN, with a special derivation of the label distribution of an unseen instance given its neighbors, and backpropagated neural networks, where the output layer indicates the label distribution of an instance. Other proposed methods are based on the optimization algorithms BFGS and Improved Iterative Scaling.

#### – Lr.

Boosting methods have been adapted to LR lr-boost (), as well as the SVM proposed in ml-svm () for ML which can be naturally extended to LR lranksurvey (). An adaptation of online learning algorithms such as the perceptron has also been developed lr-online ().

#### – Mtr.

First methods able to treat MTR problems were actually generalizations of statistical methods for single-target regression mtr-rank (); mtr-canon (). Other common methods which have been extended to predict multiple regression variables are support vector regression mtr-svr1 (); mtr-svr2 (), kernel-based methods mtr-kern1 (); mtr-kern2 (), and regression trees mtr-trees () as well as random forests mtr-rf ().

#### – Or.

Neural networks can be used to tackle OR with slight changes in the loss function or the output layer or-nn (); or-nn2 (). Similarly, extreme learning machines have also been applied to this problem or-elm (); or-elm2 (). Common techniques such as k-NN or decision trees have been coupled with global constraints for OR or-knn-dt (), and extensions of other well known algorithms such as Gaussian processes or-gp () and AdaBoost or-ada () have been proposed as well.

#### – Mc.

Algorithm adaptations generally take a well known technique and add monotonicity constraints. For example, there exist in the literature adaptations of k-NN mc-knn (), decision trees mc-trees (), decision rules mc-rules (); mc-rules2 () and artificial neural networks mc-monnets ().

Table 2 gathers all the methods described previously to tackle nonstandard supervised tasks.

Task | Problem transformation | Algorithm adaptation |

MI | Embedded-space mic-taxonomy () | SVM mi-svm (); mi-kernel () |

Neural networks mi-nn () | ||

k-NN mi-knn () | ||

MV | Canonical correlation analysis mv-cca () | SVM mv-svm () |

Fisher discriminant analysis mv-fda () | ||

ML | Binary Relevance mlmethods () | |

Label Powerset mlmethods () | ||

Classifier chains ml-chains () | k-NN ml-knn () | |

Decision trees ml-dt () | ||

SVM ml-svm () | ||

Association rules ml-rules () | ||

Ensembles mlensembles () | ||

MD | Independent classifiers mdc (); mdc-indep () | Bayesian networks md-bayes (); md-bayes2 () |

Maximum Entropy mdc (); mdc-indep () | ||

LDL | Multiclass reduction ldl () | k-NN ldl () |

Neural networks ldl () | ||

LR | Pairwise preferences lrankpairwise () | |

Constraint classification lr-constraint () | Boosting lr-boost () | |

SVM lranksurvey () | ||

Perceptron lr-online () | ||

MTR | ML mtrviaml () | |

Support vectors mtr-lssvr () | Generalizations mtr-rank (); mtr-canon () | |

Support vector regression mtr-svr1 (); mtr-svr2 () | ||

Kernel-based mtr-kern1 (); mtr-kern2 () | ||

Regression trees mtr-trees () | ||

Random forests mtr-rf () | ||

OR | Ordered partitions ord-simple () | |

One-vs-next, One-vs-followers, One-vs-previous ord-survey () | ||

3-class problems ord-survey () | Neural networks or-nn (); or-nn2 () | |

Extreme learning machines or-elm (); or-elm2 () | ||

Decision trees or-knn-dt () | ||

Gaussian processes or-gp () | ||

AdaBoost or-ada () | ||

MC | Reduction to IR monotonicity () | k-NN mc-knn () |

Decision trees mc-trees () | ||

Decision rules mc-rules (); mc-rules2 () | ||

Neural networks mc-monnets () |

## 5 Applications. Original real word scenarios

The problems studied in this work have their origins in real-world scenarios which are related below:

#### – Mi.

Problems modeled under MI learning are drug activity prediction mi-apr (), where each pattern describes a molecule and its different forms are represented by instances; image classification mic-taxonomy (), and bankruptcy ami-bank (). Most of the datasets used in experimentations, however, are usually synthetic.

#### – Mv.

Some situations where data is described in multiple views are multilingual text categorization amv-multilingual (), face detection with several poses amv-face (), user localization in a WiFi network amv-wifi (), advertisements described by their image and surrounding text amv-ads-webkb () and image classification with several color-based views and texture-based views amv-corel ().

#### – Ml.

Problems which fall naturally under the ML definition are text classification under several categories simultaneously aml-text (), image labeling aml-scene (), question tagging in forums where tags can co-exist aml-question (), protein classification aml-protein ().

#### – Md.

Applications of MD classification include classification of biomedical text mdc (), where predicted dimensions for a given document are its focus, evidence type, certainty level, polarity and trend; gene function identification md-bayes (); tumor classification, and illness diagnosis in animals md-bayes2 ().

#### – Lr.

The field known as preference learning has been gaining interest lrankpairwise (), and LR is one of the problem that falls under this term. LR is also frequently applied in ML scenarios lrank4ml (), where a threshold can be applied in order to transform an obtained ranking into a labelset.

#### – Ldl.

Data with relative importance of each label appears in applications such as analysis of gene expression levels in yeast aldl-yeast (), or emotion description from facial expressions aldl-face (), where a face can depict several emotions in different grades.

#### – Mtr.

Applications modeled as MTR problems are diverse, including modeling of vegetation condition in ecosystems assigning several scores which depend on the vegetation type amtr-eco (), prediction of audio spectrums of wind tunnel tests amtr-wind (), and estimation of several biophysical parameters from remote sensing images amtr-remote ().

#### – Or.

The most salient fields where OR can be found are text classification aor-text (), where the predicted variable may be an opinion scale or a degree of satisfaction; image categorization aor-image (); medical research aor-medical (); credit rating aor-credit (), and age estimation aor-age ().

#### – Mc.

Monotonicity constraints are found in problems related to customer satisfaction analysis amc-customer (), in which overall appreciation of a product must increase along with the evaluation of its features; house pricing mc-trees (); bankruptcy risk evaluation amc-bank (), and cancer prediction amc-cancer (), among others.

## 6 Other nonstandard variations

This section covers variations of the standard supervised problem which are further from the central focus of this paper less related to those above.

### 6.1 Learning with partial information

In a standard supervised classification setting, it is assumed that every training example is labeled accordingly and that there exist examples for every class that may appear in the testing phase. When only a fraction of the training instances are labeled, the problem is considered semi-supervised semi-sup (), but generally there still exist labeled samples for each class.

In positive-unlabeled learning pu-learn (); pu-text (), however, labeled examples provided within the training set are only positive. This means the learning algorithm only knows about the class of positive instances, and unlabeled ones can have either class.

A different scenario arises when the training set only consists of negative (or only positive) instances, and no unlabeled examples are provided. This is known as one-class classification oneclass (), and data of this nature can be obtained from outlier detection applications, where positive examples are hardly recorded.

A problem which may be seen as a generalization of one-class classification is zero-shot learning zeroshot (), a situation where unseen classes are to be predicted in the testing stage. That is, the label space includes some values which are not present in any training pattern, but the classifier must be able to predict them. For example, if in a speech recognition problem is the set of all words in English, the training set is unlikely to have at least one instance for each word, thus the classifier will only succeed if it is capable of assigning unlearned words to test examples.

A relaxation on the obstacles of zero-shot learning is present in one-shot learning oneshot (), where algorithms attempt to generalize from very few (1 to 5) examples of each class. This is a common circumstance in the field of image classification, where the cost of collecting and labeling data samples is high.

A classification of these problems according to the type of missing information can be found in Table 3.

Trait | Problem types |

Presence of unlabeled instances | Semi-supervised semi-sup () |

Positive-unlabeled pu-learn () | |

No representation of some classes | One-class oneclass () |

Positive-unlabeled pu-learn () | |

Zero-shot zeroshot () | |

Scarce representation of some classes | One-shot oneshot () |

### 6.2 Prediction of structured data

The nonstandard variations described in this work generalize traditional supervised problems where the predicted output is at most a vector whose components take values in either a finite set or . Further generalizations are possible if other kinds of structures are allowed. For example, the target may take the form of an ordered sequence or a tree. In this case, the problem usually enters the scope of structured prediction str-pred (), a generalization of supervised learning where methods must build structured data associated to input instances.

A particular case of supervised problem which can be seen under the umbrella of structured prediction is learning to rank ltr (), which does not involve a label space as such. Instead, training consists in learning from a set of feature vectors with a series of preferences among them, that is, a partial or total order in the training set. During testing a set of feature vectors is provided and the desired output is a ranking (with a predefined number of relevance levels, allowing ties) or a sorting (simply an ordering of the instances). This problem differs from OR in that individual classifications are usually meaningless: only relative distances among ranked instances matter.

## 7 Conclusions

Traditional supervised learning comprises two well known problems in machine learning: classification and regression. However, the multitude of applications which do not strictly fit the structure of the standard versions of those problems have favored the development of alternative versions which are more flexible and allow the analysis of more complex situations.

In this work an overview of nonstandard variations of supervised learning problems has been presented. A novel taxonomy under several criteria has described relationships among these variations, where the main differentiating properties are multiplicity of inputs, multiplicity of outputs, presence of order relations and constraints, and partial information. Afterwards, common methods for tackling these problems have been outlined and their main applications have been mentioned as well. Finally, some additional variants which were left out of the scope of the previous analysis have been introduced as well.

Design of novel algorithms for nonstandard supervised tasks is scarcer than adaptations and transformations, but there exist some approximations and even more open possibilities for tackling these from classical algorithmic perspectives, such as probabilistic and heuristic methods, information theory and linear algebra, among others.

###### Acknowledgements.

D. Charte is supported by the Spanish Ministry of Science, Innovation and Universities under the FPU National Program (Ref. FPU17/04069). This work has been partially supported by projects TIN2017-89517-P (FEDER Founds) of the Spanish Ministry of Economy and Competitiveness and TIN2015-68454-R of the Spanish Ministry of Science, Innovation and Universities.## References

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