2 - 7 - Gradient Descent For Linear Regression (10 min).srt

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在以前的视频中我们谈到
In previous videos, we talked

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关于梯度下降算法
about the gradient descent algorithm

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梯度下降是很常用的算法 它不仅被用在线性回归上
and talked about the linear

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和线性回归模型、平方误差代价函数
regression model and the squared error cost function.

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在这段视频中  我们要
In this video, we're going to

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将梯度下降
put together gradient descent with

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和代价函数结合
our cost function, and that

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在后面的视频中 我们将用到此算法 并将其应用于
will give us an algorithm for

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具体的拟合直线的线性回归算法里
linear regression for fitting a straight line to our data.

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这就是
So, this is

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我们在之前的课程里所做的工作
what we worked out in the previous videos.

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这是梯度下降法
That's our gradient descent algorithm, which

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这个算法你应该很熟悉
should be familiar, and you

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这是线性回归模型
see the linear linear regression model

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还有线性假设和平方误差代价函数
with our linear hypothesis and our squared error cost function.

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我们将要做的就是
What we're going to do is apply

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用梯度下降的方法
gradient descent to minimize

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来最小化平方误差代价函数
our squared error cost function.

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为了
Now, in order to apply

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使梯度下降 为了
gradient descent, in order

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写这段代码
to write this piece of

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我们需要的关键项
code, the key term

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是这里这个微分项
we need is this derivative term over here.

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所以.我们需要弄清楚
So, we need to figure out

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这个偏导数项是什么
what is this partial derivative term,

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并结合这里的
and plug in the

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代价函数J
definition of the cost

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的定义
function J, this turns

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就是这样
out to be this "inaudible"

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一个求和项
equals sum 1 through M of

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代价函数就是
this squared error

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这个误差平方项
cost function term, and all

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我这样做 只是
I did here was I just

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把定义好的代价函数
you know plugged in the definition of

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插入了这个微分式
the cost function there, and simplifying

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再简化一下
little bit more, this turns

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这等于是
out to be equal to, this

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这一个求和项
"inaudible" equals sum 1 through M

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θ0 + θ1x(1) - y(i)
of tetha zero plus tetha one, XI

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θ0 + θ1x(1) - y(i)
minus YI squared.

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这一项其实就是
And all I did there was took

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我的假设的定义
the definition for my hypothesis

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然后把这一项放进去
and plug that in there.

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实际上我们需要
And it turns out we need

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弄清楚这两个
to figure out what is

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偏导数项是什么
the partial derivative of two

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这两项分别是 j=0
cases for J equals

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和j=1的情况
0 and for J equals 1 want

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因此我们要弄清楚
to figure out what is this

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θ0 和 θ1 对应的
partial derivative for both the

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偏导数项是什么
theta(0) case and the theta(1) case.

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我只把答案写出来
And I'm just going to write out the answers.

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事实上 第一项可简化为
It turns out this first term simplifies

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1 / m 乘以求和式
to 1/M, sum over

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对所有训练样本求和
my training set of just

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求和项是 h(x(i))-y(i)
that, X(i)-  Y(i).

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而这一项
And for this term, partial derivative

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对θ(1)的微分项
with respect to theta(1), it turns

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得到的是这样一项
out I get this term: -Y(i)<i>X(i).</i>

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对吧
Okay.

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计算出这些
And computing these partial

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偏导数项
derivatives, so going from

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从这个等式
this equation to either

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到下面的等式
of these equations down there, computing

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计算这些偏导数项需要一些多元微积分
those partial derivative terms requires some multivariate calculus.

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如果你掌握了微积分
If you know calculus, feel free

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你可以随便自己推导这些
to work through the derivations yourself

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然后你检查你的微分
and check take the derivatives

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你实际上会得到我给出的答案
you actually get the answers that I got.

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但如果你
But if you are less

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不太熟悉微积分
familiar with calculus you don't

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别担心
worry about it, and it

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你可以直接用这些
is fine to just take these equations

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已经算出来的结果
worked out, and you

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你不需要掌握微积分
won't need to know calculus or

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或者别的东西
anything like that in order to

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来完成作业 你只需要会用梯度下降就可以
do the homework, so to implement gradient descent you'd get that to work.

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在定义这些以后
But so, after these definitions,

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在我们算出
or after what we've worked

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这些微分项以后
out to be the derivatives, which

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这些微分项
is really just the slope of

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实际上就是代价函数J的斜率
the cost function J.  We

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现在可以将它们放回
can now plug them back into

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我们的梯度下降算法
our gradient descent algorithm.

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所以这就是专用于
So here's gradient descent, or

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线性回归的梯度下降
the regression, which is going

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反复执行括号中的式子直到收敛
to repeat until convergence, theta 0

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θ0和θ1不断被更新
and theta one get updated as,

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都是加上一个-α/m
you know, the same minus alpha

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乘上后面的求和项
times the derivative term.

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所以这里这一项
So, this term here.

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所以这就是我们的线性回归算法
So, here's our linear regression algorithm.

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这里的第一项
This first term here that

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当然
term is, of course, just

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这一项就是关于θ0的偏导数
a partial derivative of respective

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在上一张幻灯片中推出的
theta zero, that we worked on in the previous slide.

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而第二项
And this second term here,

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这一项是刚刚的推导出的
that term is just

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关于θ1的
a partial derivative with respect to

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偏导数项
theta one that we worked out on the previous line.

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提醒一下
And just as a quick reminder,

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执行梯度下降时
you must, when implementing gradient descent,

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有一个细节要注意
there's actually there's detail that, you

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就是必须要
know, you should be implementing

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同时更新θ0和θ1
it so the update theta zero and theta one simultaneously.

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所以 让我们来看看梯度下降是如何工作的
So, let's see how gradient descent works.

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我们用梯度下降解决问题的
One of the issues we solved

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一个原因是 它更容易得到局部最优值
gradient descent is that it can be susceptible to local optima.

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当我第一次解释梯度下降时
So, when I first explained gradient

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我展示过这幅图
descent, I showed you this picture

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在表面上
of it, you know, going downhill

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不断下降
on the surface and we

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并且我们知道了
saw how, depending on where

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根据你的初始化 你会得到不同的局部最优解
you're initializing, you can end up with different local optima.

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你知道.你可以结束了.在这里或这里。
You know, you can end up here or here.

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但是 事实证明
But, it turns out that

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用于线性回归的
the cost function for gradient

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代价函数
of cost function for linear regression

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总是这样一个
is always going to be

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弓形的样子
a bow-shaped function like this.

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这个函数的专业术语是
The technical term for this

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这是一个凸函数
is that this is called a convex function.

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我不打算在这门课中
And I'm not going

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给出凸函数的定义
to give the formal definition for what

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凸函数(convex function)
is a convex function, c-o-n-v-e-x, but

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但不正式的说法是
informally a convex function

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它就是一个弓形的函数
means a bow-shaped function, you know, kind of like a bow shaped.

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因此 这个函数
And so, this function doesn't

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没有任何局部最优解
have any local optima, except

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只有一个全局最优解
for the one global optimum.

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并且无论什么时候
And does gradient descent on

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你对这种代价函数
this type of cost function which

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使用线性回归
you get whenever you're using linear

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梯度下降法得到的结果
regression, it will always convert

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总是收敛到全局最优值 因为没有全局最优以外的其他局部最优点
to the global optimum, because there are no other local optima other than global optimum.

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现在 让我们来看看这个算法的执行过程
So now, let's see this algorithm in action.

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像往常一样
As usual, here are plots of

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这是假设函数的图
the hypothesis function and of

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还有代价函数J的图
my cost function J.

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让我们来看看如何
And so, let's see how

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初始化参数的值
to initialize my parameters at this value.

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通常来说
You know, let's say, usually you

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初始化参数为零
initialize your parameters at zero

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θ0和θ1都在零
for zero, theta zero and zero.

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但为了展示需要
For illustration in this

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在这个梯度下降的实现中
specific presentation, I have

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我把θ0初始化为-900
initialised theta zero at

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θ1初始化为-0.1
about 900, and theta one at about minus 0.1, okay?

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这对应的假设
And so, this corresponds to H

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就应该是这样
over X, equals, you know,

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h(x)是等于-900减0.1x
minus 900 minus 0.1 x

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这对应我们的代价函数
is this line, so out here on the cost function.

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现在 如果我们进行
Now if we take one

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一次梯度下降
step of gradient descent, we end

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从这个点开始
up going from this point

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在这里.一点点
out here, a little

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向左下方移动了一小步
bit to the down left

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这就得到了第二个点
to that second point over there.

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而且你注意到这条线改变了一点点
And, you notice that my line changed a little bit.

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然后我再进行
And, as I take another step

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一步梯度下降 左边这条线又变一点
at gradient descent, my line on the left will change.

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对吧
Right.

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同样地
And I have also

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我又移到代价函数上的另一个点
moved to a new point on my cost function.

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再进行一步梯度下降
And as I think further step

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我觉得我的代价项
is gradient descent, I'm going

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应该开始下降了
down in cost, right, so

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所以我的参数是
my parameter is following

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跟随着这个轨迹
this trajectory, and if

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再看左边这个图
you look on the left, this corresponds

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这个表示的是假设函数h(x)
to hypotheses that seem

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它变得好像
to be getting to be

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越来越拟合数据
better and better fits for the

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直到它渐渐地
data until eventually,

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收敛到全局最小值
I have now wound up at the global minimum.

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这个全局最小值
And this global minimum corresponds to

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对应的假设函数 给出了最拟合数据的解
this hypothesis, which gives me a good fit to the data.

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这就是梯度下降法
And so that's gradient

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我们刚刚运行了一遍
descent, and we've just run

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并且最终得到了
it and gotten a good

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房价数据的最好拟合结果
fit to my data set of housing prices.

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现在你可以用它来预测
And you can now use it to predict.

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比如说 假如你有个朋友
You know, if your friend has a

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他有一套房子
house with a

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面积1250平方英尺(约116平米)
size 1250 square feet, you

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现在你可以通过这个数据
can now read off the value

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然后告诉他们
and tell them that, I don't

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也许他的房子
know, maybe they can get

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可以卖到35万美元
$350,000 for their house.

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最后 我想再给出
Finally, just to give

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另一个名字
this another name, it turns out

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实际上 我们
that the algorithm that we

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刚刚使用的算法
just went over is sometimes

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有时也称为批量梯度下降
called batch gradient descent.

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实际上 在机器学习中
And it turns out in machine

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我们这些搞机器学习的人
learning, I feel like us machine

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通常不太会
learning people, we're not always

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给算法起名字
created has given me some algorithms.

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但这个名字"批量梯度下降"
But the term batch gradient descent

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指的是
means that refers to the

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在梯度下降的
fact that, in every step

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每一步中
of gradient descent we're looking

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我们都用到了所有的训​​练样本
at all of the training examples.

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在梯度下降中
So, in gradient descent, you

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在计算微分求导项时
know, when computing derivatives, we're computing

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我们需要进行求和运算
these sums, this sum of.

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所以 在每一个单独的
So, in every separate

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梯度下降中 我们最终
gradient descent, we end up

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都要计算这样一个东西
computing something like this, that

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这个项需要对所有m个训练样本求和
sums over our M training examples.

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因此 批量梯度下降法
And so the term batch gradient

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这个名字 说明了
descent refers to the fact

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我们需要考虑
when looking at the entire batch

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所有这一"批"训练样本
of training examples, and again,

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这确实不是一个
this is really, really not

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很好听的名字
a great name, but this is

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但是搞机器学习的人就是这么称呼的
what Machine Learning people call it.

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而事实上
And it turns out there are

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有时也有其他类型的
sometimes other versions of

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梯度下降法
gradient descent that are not

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不是这种"批量"型的
batch versions but instead do

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不考虑整个的训练集
not look at the entire traning set

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而是每次只关注
but look at small subsets

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训练集中的一些小的子集
of the training sets at the time,

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在后面的课程中 我们也将介绍这些方法
and we'll talk about those versions later in this course as well.

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但就目前而言
But for now, using the algorithm

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应用刚刚学到的算法
you just learned, now we're

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你应该已经掌握了批量梯度算法
using batch gradient descent, you

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并且能把它应用到
now know how to implement

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线性回归中了
gradient descent, or linear regression.

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这就是用于线性回归的梯度下降法
So that's linear regression with gradient descent.

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如果你之前学过线性代数
If you've seen advanced linear algebra

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有些同学之前
before so some you may

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可能已经学过
have taken a class with advanced

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高等线性代数
linear algebra, you might

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你应该知道
know that there exists a solution

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有一种计算
for numerically solving for the

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代价函数J最小值的
minimum of the cost function

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数值解法
J, without needing to

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不需要梯度下降这种迭代算法
use and iterative algorithm like gradient descent.

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在后面的课程中
Later in this course we will

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我们也会谈到这个方法
talk about that method as

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它可以在不需要
well that just solves for the

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多步梯度下降的情况下
minimum cost function J without

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也能解出代价函数J的最小值
needing this multiple steps of gradient descent.

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这是另一种称为正规方程(normal equations)的方法
That other method is called normal equations methods.

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可能你之前已经
And, but in case you

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听说过这种方法
have heard of that method, it turns

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实际上在数据量较大的情况下
out gradient descent will

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梯度下降法比正规方程
scale better to larger data

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要更适用一些
sets than that normal equals

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现在我们已经
method and, now that

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掌握了梯度下降
we know about gradient descent, we'll

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我们可以在不同的环境中
be able to use it in

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使用梯度下降法
lots of different contexts, and we'll

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我们还将在不同的机器学习问题中大量地使用它
use it in lots of different Machine Learning problems as well.

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所以 祝贺大家成功
So, congrats on learning

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学会你的第一个机器学习算法
about your first Machine Learning algorithm.

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我们稍后将有一些练习
We'll later have exercises in

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这些练习会要求你
which we'll ask you to

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实现梯度下降
implement gradient descent and

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希望大家能让这些算法真正地为你工作
hopefully see these algorithms work for yourselves.

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但在此之前
But before that I first

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我还想先
want to tell you in

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在下一组视频中
the next set of videos, the

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告诉你
first want to tell you about

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泛化的梯度下降
a generalization of the gradient descent

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算法 这将使
algorithm that will make

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梯度下降更加强大
it much more powerful and I

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在下一段视频中我将介绍这一问题
guess I will tell you about that in the next video.