单变量线性回归（二）

梯度下降（Gradient Descent）

我们将使用梯度下降算法来求出使得代价函数J(θ₀, θ₁)值最小的参数θ的值。

梯度下降算法的基本思想：首先我们随机选择一个参数的组合(θ₀, θ₁, ... , θ_n)，计算代价函数，然后我们寻找下一个能让代价函数值最小的参数组合，且一直这样寻找下去直至找到一个局部最小值。为什么将这个最小值称为局部最小值？因为，我们在寻找过程中并没有尝试寻找所有的参数组合，所以我们不能确定这个最小值为全局最小值。

梯度下降算法的公式为：

其中：

因此，我们可将表达式改写为：

其中α是学习率（Learning Rate），α越大则代价函数值下降得越快；反之，代价函数值下降得越慢。表达式中 “:=” 这个符号表示赋值。

注：我们在更新参数θ的值时要做到同步更新，即同时更新θ₀，θ₁的值。

补充笔记

Gradient Descent

So we have our hypothesis function and we have a way of measuring how well it fits into the data. Now we need to estimate the parameters in the hypothesis function. That's where gradient descent comes in.

Imagine that we graph our hypothesis function based on its fields θ₀ and θ₁ (actually we are graphing the cost function of the parameter estimates). We are not graphing x and y itself, but the parameter range of our hypothesis function and the cost resulting from selecting a particular set of parameters.

We put θ₀ on the x axis and θ₁ on the y axis, with the cost function on the vertical z axis. The points on our graph will be the result of the cost function using our hypothesis with those specific theta parameters. The graph below depicts such a setup.

We will know that we have succeeded when our cost function is at the very bottom of the pits in our graph, i.e. when its value is the minimum. The red arrows show the minimum points in the graph.

The way we do this is by taking the derivative (the tangential line to a function) of our cost function. The slope of the tangent is the derivative at that point and it will give us a direction to move towards. We make steps down the cost function in the direction with the steepest descent. The size of each step is determined by the parameter α, which is called the learning rate.

The gradient descent algorithm is:

repeat until convergence:

where
j = 0, 1 represents the feature index number.

At each iteration j, one should simultaneously update the parameters θ₁, θ₂, ... , θ_n. Updating a specific parameter prior to calculating another one on the j^(th) iteration would yield to a wrong implementation.

Gradient Descent Intuition

We explored the scenario where we used one parameter θ₁ and its cost function to implement a gradient. Our formula for a single parameter was:

repeat until convergence:

On a side note, we should adjust our parameter α to ensure that the gradient descent algorithm converges in a reasonable time. Failure to converge or too much time to obtain the minimum value imply that our step size is wrong.

How does gradient descent converge with a fixed step size α?

Gradient Descent For Linear Regression

When specifically applied to the case of linear regression, a new form of the gradient descent equation can be derived. We can substitute our actual cost function and our actual hypothesis function and modify equation to:

where m is the size of the training set θ₀ a constant that will be changing simultaneously with θ₁ and x_i, y_i are values of the given training set (data).

The point of all this is that if we start with a guess for our hypothesis and then repeatedly apply these gradient descent equations, our hypothesis will become more and more accurate.

So, this is simply gradient descent on the original cost function J. This method looks at every example in the entire training set on every step, and is called batch gradient descent. Not that, while gradient descent can be susceptible to local minimum in general, the optimization problem we have posed here for linear regression has only one global, and no other local, optima; thus gradient descent always converges (assuming the learning rate α is not too large) to the global minimum. Indeed, J is a convex quadratic function. Here is an example of gradient descent as it is run to minimize a quadratic function.

The ellipses shown above are the contours of a quadratic function. Also shown is the trajectory taken by gradient descent, which was initialized at (48, 30). The x's in the figure (joined by straight lines) mark the successive values of θ that gradient descent went through as it converged to its minimum.

注：国外与国内关于凹凸函数的定义是反的。