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Linear Regression

Our goal in linear regression is to predict a target value yy starting from a vector of input values x∈Rnx∈ℜn. For example, we might want to make predictions about the price of a house so that yy represents the price of the house in dollars and the elements xjxj of xx represent “features” that describe the house (such as its size and the number of bedrooms). Suppose that we are given many examples of houses where the features for the i’th house are denoted x(i)x(i) and the price is y(i)y(i). For short, we will denote the

Our goal is to find a function y=h(x)y=h(x) so that we have y(i)≈h(x(i))y(i)≈h(x(i)) for each training example. If we succeed in finding a function h(x)h(x) like this, and we have seen enough examples of houses and their prices, we hope that the function h(x)h(x) will also be a good predictor of the house price even when we are given the features for a new house where the price is not known.

To find a function h(x)h(x) where y(i)≈h(x(i))y(i)≈h(x(i)) we must first decide how to represent the function h(x)h(x). To start out we will use linear functions: hθ(x)=∑jθjxj=θ⊤xhθ(x)=∑jθjxj=θ⊤x. Here, hθ(x)hθ(x) represents a large family of functions parametrized by the choice of θθ. (We call this space of functions a “hypothesis class”.) With this representation for hh, our task is to find a choice of θθ so that hθ(x(i))hθ(x(i)) is as close as possible to y(i)y(i). In particular, we will search for a choice of θθ that minimizes:

J(θ)=12∑i(hθ(x(i))−y(i))2=12∑i(θ⊤x(i)−y(i))2