Supervised Learning without Neural Networks

Supervised learning is the term for a machine learning task, where we are given a dataset consisting of input-output pairs \(\lbrace(\mathbf{x}_{1}, y_{1}), \dots, (\mathbf{x}_{m}, y_{m})\rbrace\) and our task is to “learn” a function which maps input to output \(f: \mathbf{x} \mapsto y\). Here we chose a vector-valued input \(\mathbf{x}\) and only a single real number as output \(y\), but in principle also the output can be vector valued. The output data that we have is called the ground truth and sometimes also referred to as “labels” of the input. In contrast to supervised learning, all algorithms presented so far were unsupervised, because they just relied on input-data, without any ground truth or output data.

Within the scope of supervised learning, there are two main types of tasks: Classification and Regression. In a classification task, our output \(y\) is a discrete variable corresponding to a classification category. An example of such a task would be to distinguish stars with a planetary system (exoplanets) from those without given time series of images of such objects. On the other hand, in a regression problem, the output \(y\) is a continuous number or vector. For example predicting the quantity of rainfall based on meteorological data from the previous days.

In this section, we first familiarize ourselves with linear methods for achieving these tasks. Neural networks, in contrast, are a non-linear method for supervised classification and regression tasks.