can a perceptron be used for regression?

In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers. A binary classifier is a function which can decide whether or not an input, represented by a vector of numbers, belongs to some specific class.

on wiki page, the problems section includes regression without more explanation. can a perceptron used for regression?

The deep feedforward neural networks used for regression are nothing but multilayer perceptron architectures. Originally, perceptrons were used as binary classifiers i.e to classify binary labels ( 0 or 1 ). But, if no non-linear activation function is applied to the dot product of the features and weights, then it is simply a linear regressor.

If the linear function is $$f(x) = x$$ and $$N$$ is the number of features then,

$$\Large y = \sum_{i=0}^N x_i w_i$$

or using vector notation,

$$\Large y = \vec{x}.\vec{w} ...(1)$$

For multiple regression, we use the below equation with mean-squared error loss function to optimize the $$\beta$$ parameters,

$$\Large y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 ... \beta_n X_n$$

Let ,

$$\Large \vec{W} = [ \beta_1 \ \beta_2 \ \beta_3 \ ... \ \beta_n ] \\ \vec{X} = [ X_1 \ X_2 \ X_3 \ ... \ X_n ]$$

Where $$N$$ is the number of features, $$\vec{X}$$ is the feature vector and $$\vec{W}$$ is the weights vector. Then,

$$\Large y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 ... \beta_n X_n = \beta_0 + \vec{W}.\vec{X} ...(2)$$

Where, $$\beta_0$$ is also called as the bias coefficient in Artificial Neural Networks. You can relate equations 1 and 2 and understand the concept.

Hence, $$\Large y = \vec{x}.\vec{w} + bias$$ generally represents a hyperplane which is used in linear regression.

Yes a perceptron (one fully connected unit) can be used for regression. It will just be a linear regressor. If you use no activation function you get a regressor and if you put a sigmoid activation you get a classifier.

Actually, with neural networks, classification is a special case of regression where we "regress" the probability of belonging to a given class. That's why the loss function for classification is called "logistic regression".