To establish notation for future use, we’ll use
A pair (x(i),y(i)) is called a training example
the dataset that we’ll be using to learn—a list of m training examples (x(i),y(i));i=1,...,m—is called a training set.
the superscript “(i)” in the notation is simply an index
into the training set, and has nothing to do with exponentiation
In this example
X = Y = R
To describe the supervised learning
problem slightly more formally, our goal is,
given a training set, to learn afunction h : X → Y
so that h(x)
is a “good” predictor for the corresponding value of y.
For historical reasons, this function h
is called a hypothesis
. Seen pictorially, the process is therefore like this
continuous
, such as in our housing example代价函数是用来测量实际值和预测值精确度的一个函数模型.
We can measure the accuracy of our hypothesis function by using acost function
.
This takes an average difference (actually a fancier version of an average) of all the results of the hypothesis with inputs from x's and the actual output y's.
首先需要搞清楚假设函数和代价函数的区别 当假设函数为线性时,即线性回归方程,其由两个参数组成:theta0和theta1
我们要做的就是选取两个参数的值,使其代价函数的值达到最小化
J(θ0,θ1)=12m∑i=1m(y^i−yi)2=12m∑i=1m(hθ(xi)−yi)2
To break it apart, it is 1/2 x ̄ where x ̄ is the mean of the squares of hθ(xi)−yi , or the difference
between the predicted value and the actual value.
This function is otherwise called theSquared error function
, or Mean squared error
.
The mean is halved (1/2)as a convenience for the computation of the gradient descent, as the derivative term of the square function will cancel out the 1/2 term.
The following image summarizes what the cost function does: