梯度下降 （Gradient Descent）

前面我们了解到代价函数（Cost Function）是一种衡量我们的假设函数h(x)与数据集的拟合程度的函数，而要使得假设函数与数据集拟合程度最大，必须找到使得代价函数J值最小的参数θ的值。（这里使用单变量线性回归作为例子）

而梯度下降算法是一种能帮助我们找到合适的参数θi从而使得代价函数J(θ0, θ1)值最小的一种算法，它的基本思想是：

• Start with some θ0, θ1, ... , θn (we can initialize θ0 = 0, θ1 = 0, ... , θn = 0)
• Keep changing θ0, θ1, ... , θn to reduce J(θ0, θ1, ... , θn) until we hopefully end up at minimum (or maybe a local minimum)

这里的参数的组合(θ0, θ1, ... , θn)一般是初始化为0，当然也可以自己选择，但不同的参数组合可能得到不同的局部最小值，如下图：

梯度下降算法的公式为：

在这个公式里面，参数α是学习率（Learning Rate），α右边的则是代价函数对于参数θi的偏导数，表达式中 “:=” 这个符号表示赋值。α越大则代价函数值下降得越快，但过大的α值可能会导致我们找不到最小值；α越小时，代价函数值下降得越慢。

并且我们在更新参数θ的值时要做到同步更新，即同时更新θ0，θ1的值：

基于高数中的多元微分学的知识，我们能很容易得到：

当我们接近局部最小值的时候，梯度下降会自动地减小它迈的步子（因为此时偏导会逐渐趋于0），当参数θi不再变化时，说明我们已经找到了局部最小值（此时偏导应为0）。

阅读材料

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, ... , θn. 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.