6 - 4 - Cost Function (11 min).srt

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In this video we'll talk about
在这段视频中 我们要讲
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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how to fit the parameters theta
如何拟合逻辑回归

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for logistic regression.
模型的参数θ

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In particular, I'd like to
具体来说 我要定义

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define the optimization objective or the
用来拟合参数的

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cost function that we'll use to fit the parameters.
优化目标或者叫代价函数

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Here's to supervised learning problem
这便是监督学习问题中的

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of fitting a logistic regression model.
逻辑回归模型的拟合问题

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We have a training set
我们有一个训练集

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of M training examples.
里面有m个训练样本

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And as usual each of
像以前一样

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our examples is represented by
我们的每个样本

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feature vector that's N plus 1 dimensional.
用n+1维的特征向量表示

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And as usual we have
同样和以前一样

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X 0 equals 1.
x0 = 1

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Our first feature, or our 0
第一个特征变量

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feature is always equal to 1,
或者说第0个特征变量 一直是1

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and because this is a
而且因为这是一个分类问题

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classification problem, our training
我们的训练集

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set has the property that
具有这样的特征

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every label Y, is either 0 or 1.
所有的y 不是0就是1

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This is a hypothesis
这是一个假设函数

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and the parameters of the
它的参数

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hypothesis is this theta over here.
是这里的这个θ

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And the question I want
我要说的问题是

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to talk about is given this
对于这个给定的训练集

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training set how do
我们如何选择

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we choose, or how do we fit the parameters theta?
或者说如何拟合参数θ

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Back when we were developing the
以前我们推导线性回归时

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linear regression model, we use the following cost function.
使用了这个代价函数

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I've written this slightly differently, where
我把这个写成稍微有点儿不同的形式

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instead of 1/2m, I've
不写原先的1/2m

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taken the 1/2 and put it inside the summation instead.
我把1/2放到求和符号里面了

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Now, I want to use
现在我想用

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an alternative way of writing
另一种方法

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out this cost function which is
来写代价函数

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that instead of writing out
去掉这个平方项

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this squared and return here,
把这里写成

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let's write here, cost of
这样的形式

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H of X comma
（具体公式请看屏幕）

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Y, and I'm going
（具体公式请看屏幕）

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to define that term cost
定义这个代价函数Cost函数

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of H of X comma Y to be equal to this.
等于这个

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It's just equal to just one half of the square root error.
等于这个1/2的平方根误差

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So now, we can see more
因此现在

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clearly that the cost
我们能更清楚的看到

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function is a sum
代价函数是这个Cost函数

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over my training set, or
在训练集范围上的求和

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is 1/m times the sum
或者说是1/m倍的

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over my training set of this cost term here.
这个代价项在训练集范围上的求和

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And to simplify this
然后稍微简化一下这个式子

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equation a little bit more, it's gonna
去掉这些上标

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be convenient to get rid of those superscripts.
会显得方便一些

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So just define cost of
所以直接定义

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H of X comma Y to
代价值(h(X), Y)

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be equal to 1/2 of
等于1/2倍的

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this square root error  and the
这个平方根误差

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interpretation of this cost function
对这个代价项的理解是这样的

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is that this is the
这是我所期望的

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cost I want my learning
我的学习算法

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algorithm to, you know,
如果想要达到这个值

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have to pay, if it
也就是这个假设h(x)

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outputs that value it
所需要付出的代价

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this prediction is H of
这个希望的预测值是h(x)

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X, and the actual
而实际值则是y

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label was Y. So just
干脆

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cross off those superscripts. All right.
全部去掉那些上标好了

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And no surprise for linear
显然 在线性回归中

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regression the cost for you to define is that.
代价值会被定义为这个

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Well the cost for this
这个代价值是

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is, that is 1/2
1/2乘以

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times the square difference
预测值h和

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between what are predicted and the
实际值观测的结果y

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actual value that we observe
的差的平方

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for Y. Now, this cost
这个代价值可以

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function worked fine for linear
很好地用在线性回归里

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regression, but here we're interested in logistic regression.
但是我们现在要用在逻辑回归里

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If we could minimize this cost
如果我们可以最小化

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function that is plugged into J here.
代价函数J里面的这个代价值

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That will work okay.
它会工作得很好

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But it turns out that if
但实际上

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we use this particular cost function
如果我们使用这个代价值

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this would be a non-convex function of the parameters theta.
它会变成参数θ的非凸函数

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Here's what I mean by non-convex.
我说的非凸函数是这个意思

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We have some cost function J
对于这样一个代价函数J(θ)

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of theta, and for logistic
对于逻辑回归来说

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regression this function H here
这里的h函数

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has a non linearity, right?
是非线性的 对吧？

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It says, you know, 1 over
它是等于 1 除以

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1 plus E to the negative theta transfers
1+e的-θ转置乘以X次方

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X. So it's a pretty complicated nonlinear function.
所以它是一个很复杂的非线性函数

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And if you take the sigmoid
如果对它取Sigmoid函数

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function and plug it in
然后把它放到这里

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here and then take
然后求它的代价值

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this cost function and plug
再把它放到这里

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it in there, and then plot
然后再画出

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what J of theta looks
J(θ)长什么模样

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like, you find that
你会发现

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J of theta can look like a function just like this.
J(θ)可能是一个这样的函数

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You know with many local optima and
有很多局部最优值

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the formal term for this
称呼它的正式术语是

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is that this a non convex function.
这是一个非凸函数

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And you can kind of tell.
你大概可以发现

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If you were to run gradient
如果你把梯度下降法

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descent on this sort of
用在一个这样的函数上

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function, it is not guaranteed
不能保证它会

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to converge to the global minimum.
收敛到全局最小值

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Whereas in contrast, what
相应地

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we would like is to have
我们希望

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a cost function J of theta
我们的代价函数J(θ)

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that is convex, that is
是一个凸函数

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a single bow-shaped function that
是一个单弓形函数

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looks like this, so that
大概是这样

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if you run gradient descent, we
所以如果对它使用梯度下降法

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would be guaranteed that gradient descent, you know,
我们可以保证梯度下降法

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would converge to the global minimum.
会收敛到该函数的全局最小值

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And the problem of using the
但使用这个

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the square cost function is that
平方代价函数的问题是

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because of this very
因为中间的这个

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non linear sigmoid function that
非常非线性的

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appears in the middle here, J of
sigmoid函数的出现

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theta ends up being
导致J(θ)成为

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a non convex function if you
一个非凸函数

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were to define it as the square cost function.
如果你要用平方函数定义它的话

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So what we'd would like to do
所以我们想做的是

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is to instead come up with
另外找一个

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a different cost function that
不同的代价函数

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is convex and so
它是凸函数

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that we can apply a great
使得我们可以使用很好的算法

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algorithm like gradient descent
如梯度下降法

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and be guaranteed to find a global minimum.
而且能保证找到全局最小值

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Here's a cost function that we're going to use for logistic regression.
这个代价函数便是我们要用在逻辑回归上的

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We're going to say the cost
我们认为

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or the penalty that the algorithm
这个算法要付的代价或者惩罚

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pays if it outputs
如果输出值是h(x)

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a value H of X.
或者换句话说

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So, this is some number like 0.7
假如说预测值h(x)

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where it predicts a value H
是一个数 比如0.7

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of X. And the actual
而实际上

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cost label turns out to
真实的标签值是y

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be Y. The cost is
那么代价值将等于

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going to be minus log
-log(h(X))

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H of X if Y is equal 1.
当y=1时

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And minus log, 1 minus
以及-log(1-h(X))

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H of X if Y is equal to 0.
当y=0时

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This looks like a pretty complicated function.
这看起来是个非常复杂的函数

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But let's plot function to
但是让我们画出这个函数

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gain some intuition about what it's doing.
可以直观地感受一下它在做什么

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Let's start up with the case of Y equals 1.
我们从y=1这个情况开始

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If Y is equal equal
如果y等于1

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to 1, then the cost function
那么这个代价函数

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is -log H of X, and
是-log(h(X))

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if we plot that, so let's
如果我们画出它

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say that the horizontal
我们将h(X)

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axis is H of X.
画在横坐标上

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So we know that a hypothesis
我们知道假设函数

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is going to output a value between
的输出值

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0 and 1.
是在0和1之间的

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Right?
对吧？

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So H of X that varies
所以h(X)的值

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between 0 and 1.
在0和1之间变化

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If you plot what this cost function looks like.
如果你画出这个代价函数的样子

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You find that it looks like this.
你会发现它看起来是这样的

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One way to see why the
理解这个函数为什么是这样的

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plot like this it is because
一个方式是

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if you were to plot log Z
如果你画出log(z)

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with Z on the horizontal axis.
z在横轴上

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Then that looks like that.
它看起来会是这样

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And it's approach is minus infinity.
它趋于负无穷

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So this is what the log function looks like.
这是对数函数的样子

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And so this is 0, this is 1.
所以这里是0 这里是1

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Here Z is of
显然 这里的Z

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course playing the role  of
就是代表h(x)的角色

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H of X.  And so
因此

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minus log Z will look like this.
-log(Z)看起来这样

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Right just flipping the sign.
就是翻转一下符号

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Minus log Z. And we're
-log(Z)

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interested only in the
我们所感兴趣的是

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range of when this function
函数在0到1

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goes between 0 and 1.
之间的这个区间

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So, get rid of that.
所以 忽略那些

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And so, we're just left with,
所以只剩下

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you know, this part of the curve.
曲线的这部分

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And that's what this curve on the left looks like.
这就是左边这条曲线的样子

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Now this cost function
现在这个代价函数

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has a few interesting and desirable properties.
有一些有趣而且很好的性质

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First you notice that if
首先 你注意到

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Y is equal to 1 and H of X is equal 1, in
如果y=1而且h(X)=1

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other words, if the hypothesis
也就是说

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exactly, you know, predicts
如果假设函数

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H equals 1, and Y
刚好预测值是1

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is exactly equal to what I predicted.
而且y刚好等于我预测的

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Then the cost is equal 0.
那么这个代价值等于0

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Right?
对吧？

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That corresponds to, the curve doesn't actually flatten out.
这对应于… 这个曲线并不是平的

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The curve is still going. First, notice
曲线还在继续走

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that if H of X
首先 注意到如果h(x)=1

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equals 1, if the hypothesis
如果假设函数

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predicts that Y is equal to 1.
预测Y=1

200
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And if indeed Y is
并且如果y确实等于1

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equal to 1 then the cost is equal to 0.
那么代价值等于0

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That corresponds to this point down here.
这对应于下面这个点

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Right?
对吧？

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If H of X is equal
如果h(X)=1

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to 1, and we're only
这里我们只需要考虑

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concerned the case that Y equals 1 here.
y=1的情况

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But if H of X is equal to 1.
如果h(x)等于1

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Then the cost is down here is equal to 0.
那么代价值等于0

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And that is what we like it to be.
这是我们所希望的

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Because, you know, if we
因为如果我们

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correctly predict the output Y then the cost is 0.
正确预测了输出值y 那么代价值是0

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But now, notice also
但是现在 同样注意到

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that H of X approaches 0.
h(x)趋于0时

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So, that's H. As the
所以 那是h

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output of the hypothesis approaches 0
当假设函数的输出趋于0时

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the cost blows up, and it goes to infinity.
代价值激增 并且趋于无穷

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And what this does is
我们这样描述

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it captures the intuition that if
体现出了这样一种直观的感觉

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a hypothesis, you know, outputs 0.
那就是如果假设函数输出0

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That's like saying, our hypothesis is
相当于说

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saying, the chance of Y
我们的假设函数说

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equals 1 is equal to 0.
Y=1的概率等于0

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It's kind of like our going
这类似于

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to our medical patient and saying,
我们对病人说

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"The probability that you
你有一个恶性肿瘤的概率

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00:07:45,610 --> 00:07:47,337
have a malignant tumor, the
也就是说

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probability that Y equals 1 is zero."
y=1的概率是0

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So, it's like absolutely impossible that your
就是说你的肿瘤

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tumor is malignant.
完全不可能是恶性的

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00:07:55,150 --> 00:07:56,776
But if it turns out that
然而结果是

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the tumor, the patient's tumor, actually is malignant.
病人的肿瘤确实是恶性的

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So if Y is equal to
所以如果y=1

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1 even after we told them
即使我们告诉他们

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you know, the probability of it happening is 0.
它发生的概率是0

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It's absolutely impossible for it to be malignant.
它完全不可能是恶性的

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But if we told them
如果我们告诉他们这个

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this with that level of certainty,
和我们的确信程度

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and we turn out to be wrong,
并且最后我们是错的

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then we penalize the learning algorithm
那么我们用非常非常大的代价值

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by a very, very large cost,
惩罚这个学习算法

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and that's captured by having this
它是被这样体现出来

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cost goes infinity if Y
这个代价值趋于无穷

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equals 1 and H
如果y=1

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of X approaches 0.
而h(x)趋于0

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00:08:24,334 --> 00:08:26,725
This might consider of
这是y=1时的情况

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00:08:26,725 --> 00:08:28,875
Y1, let's look at what
我们再来看看

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the cost function looks like for Y0.
y=0时 代价值函数是什么样

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If Y is equal to 0, then the cost
如果y=0

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00:08:35,720 --> 00:08:39,121
looks like this expression over here.
那么代价值是这个表达式

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00:08:39,121 --> 00:08:40,403
And if you plot
如果画出函数

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00:08:40,403 --> 00:08:42,751
the function minus log 1
-log(1-z)

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00:08:42,780 --> 00:08:45,839
minus Z what you
那么你得到的

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00:08:45,839 --> 00:08:49,245
get is the cost function actually looks like this.
代价函数实际上是这样

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So, it goes from 0 to 1.
它从0到1

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00:08:50,270 --> 00:08:53,263
Something like that.
差不多这样

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And so if you plot
如果你画出

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00:08:54,611 --> 00:08:55,872
the cost function for the case
y=0情况下的

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00:08:55,872 --> 00:08:57,823
of y equals zero, you find that it looks
代价函数

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00:08:57,823 --> 00:09:00,763
like this and what
你会发现大概是这样

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00:09:00,763 --> 00:09:02,404
this curve does is it
它现在所做的是

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00:09:02,404 --> 00:09:04,937
now blows up,
在h(X)趋于1时激增

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00:09:04,937 --> 00:09:08,273
and it goes to plus infinity as H of X goes to 1.
趋于正无穷

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00:09:08,290 --> 00:09:09,880
Because it's saying that
因为它是说

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if Y turns out to be
如果最后发现

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00:09:11,200 --> 00:09:12,168
equal to 0, but we
y等于0

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00:09:12,168 --> 00:09:13,966
predicted that you know, Y is
而我们却几乎

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00:09:13,966 --> 00:09:15,286
equal to 1 with almost
非常肯定地预测

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00:09:15,320 --> 00:09:17,281
certainty with probability 1, then
y=1的概率是1

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00:09:17,281 --> 00:09:21,569
we end up paying a very large cost.
那么我们最后就要付出非常大的代价值

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00:09:21,569 --> 00:09:23,143
Let's plot the cost function for
（以下这一段和前面重复了）让我们画出y=0时的

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00:09:23,143 --> 00:09:25,063
the case of Y equals 0.
代价函数

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00:09:25,063 --> 00:09:29,702
So if Y equals 0 that's going to be our cost function.
所以如果y=0 这就是我们的代价值函数

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00:09:29,702 --> 00:09:31,914
If you look at this expression,
如果你看着这个表达式

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00:09:31,914 --> 00:09:33,726
and if you plot, you know, minus
然后你画出

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00:09:33,726 --> 00:09:36,221
log 1 minus Z, if
-log(1-Z)

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00:09:36,221 --> 00:09:37,428
you figure out what that looks like,
如果你清楚它是什么样的

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you get a figure that looks like this.
你会得到这样一个图形

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00:09:40,071 --> 00:09:41,669
Where, which goes from 0
这样随着

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00:09:41,680 --> 00:09:43,610
to 1 with the Z
横轴上的z

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00:09:43,610 --> 00:09:45,850
axis on the horizontal axis.
从0到1

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00:09:45,850 --> 00:09:47,221
So If you take this cost
如果你画出

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00:09:47,221 --> 00:09:48,397
function and plot it for
y=0时的

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00:09:48,397 --> 00:09:49,614
the case of Y equals 0,
代价函数

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00:09:49,614 --> 00:09:51,186
what you get is
你会发现

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00:09:51,186 --> 00:09:55,109
that the cost function looks like this.
代价函数是这样的

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00:09:55,109 --> 00:09:56,743
And what this cost function
它所做的是

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00:09:56,743 --> 00:09:58,650
does is that it blows
代价函数会在这里激增

288
00:09:58,660 --> 00:09:59,530
up or it goes to a
趋于正无穷

289
00:09:59,560 --> 00:10:01,448
positive infinity as each
随着h(X)的增大

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00:10:01,448 --> 00:10:03,707
H of X goes to one
而趋近于1

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00:10:03,710 --> 00:10:05,443
and this captures the
这体现了这样一个直观的感觉

292
00:10:05,443 --> 00:10:07,159
intuition that if a hypothesis
如果假设函数预测

293
00:10:07,180 --> 00:10:08,847
predicted that, you know, H of
h(X)=1

294
00:10:08,850 --> 00:10:10,406
X is equal to 1 with
并且非常确定

295
00:10:10,406 --> 00:10:12,121
certainty, with like probability 1,
比如这样的概率是1

296
00:10:12,121 --> 00:10:14,283
it's absolutely got to be Y equals 1.
认为y肯定是1

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00:10:14,283 --> 00:10:15,563
But if Y turned out to
但是最后发现

298
00:10:15,563 --> 00:10:17,219
be equal to 0 then
y其实等于0

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00:10:17,219 --> 00:10:18,206
it makes sense to make the
这就必须要让假设函数

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00:10:18,206 --> 00:10:21,940
hypothesis, or make the learning algorithm pay a very large cost.
或者学习算法付出一个很大的代价

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00:10:21,940 --> 00:10:24,609
And conversely, if H
反过来

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00:10:24,610 --> 00:10:25,942
of X is equal to
如果h(x)=0

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00:10:25,950 --> 00:10:27,483
0 and Y equals zero,
而且y=0

304
00:10:27,483 --> 00:10:28,983
then the hypothesis nailed it.
那么假设函数预测对了

305
00:10:29,000 --> 00:10:30,626
The predicted Y is equal
预测的是y=0

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00:10:30,630 --> 00:10:32,371
to zero and it turns
并且y就是等于0

307
00:10:32,371 --> 00:10:34,376
out Y is equal to zero
并且Y就是等于0

308
00:10:34,376 --> 00:10:36,701
so at this point the cost
那么代价值函数在这点上

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00:10:36,750 --> 00:10:40,139
function is going to be 0.
应该等于0

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00:10:40,160 --> 00:10:42,163
In this video, we
在这个视频中

311
00:10:42,163 --> 00:10:43,886
have defined the cost function
我们定义了

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00:10:43,886 --> 00:10:46,428
for a single training example.
单训练样本的代价函数

313
00:10:46,428 --> 00:10:50,251
The topic of convexity analysis is beyond the scope of this course.
凸性分析的内容是超出这门课的范围的

314
00:10:50,270 --> 00:10:51,594
But it is possible to show
但是可以证明

315
00:10:51,620 --> 00:10:53,080
that with our particular choice
我们所选的

316
00:10:53,150 --> 00:10:54,774
of cost function this would
代价值函数

317
00:10:54,774 --> 00:10:57,926
give us a convex optimization problem
会给我们一个

318
00:10:57,960 --> 00:11:00,081
as cost function, overall cost function
凸优化问题

319
00:11:00,081 --> 00:11:01,463
J of theta will be
代价函数J(θ)会是一个凸函数

320
00:11:01,463 --> 00:11:04,368
convex and local optima free.
并且没有局部最优值

321
00:11:04,370 --> 00:11:05,691
In the next video we're going
在下一个视频中

322
00:11:05,691 --> 00:11:07,753
to take these ideas of the
我们会把单训练样本的

323
00:11:07,753 --> 00:11:08,923
cost function for a single
代价函数的这些理念

324
00:11:08,923 --> 00:11:10,839
training example and develop that
进一步发展

325
00:11:10,839 --> 00:11:12,522
further and define the
然后给出

326
00:11:12,522 --> 00:11:13,773
cost function for the entire
整个训练集的代价函数的定义

327
00:11:13,780 --> 00:11:16,104
training set, and we'll also
我们还会找到一种

328
00:11:16,104 --> 00:11:17,404
figure out a simpler way to
比我们目前用的

329
00:11:17,404 --> 00:11:19,699
write it than we have been using so far.
更简单的写法

330
00:11:19,699 --> 00:11:21,016
And based on that will
基于这些推导出的结果

331
00:11:21,030 --> 00:11:22,779
work out gradient descent, and
我们将应用梯度下降法

332
00:11:22,779 --> 00:11:25,835
that will give us our logistic regression algorithm.
得到我们的逻辑回归算法