7 - 2 - Cost Function (10 min).srt

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In this video, I'd like to
在这段视频中 我想要
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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convey to you, the main intuitions
传达给你一个直观的感受

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behind how regularization works.
告诉你正规化是如何进行的

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And, we'll also write down
而且 我们还要写出

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the cost function that we'll use, when we were using regularization.
我们使用正规化时 需要使用的代价函数

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With the hand drawn examples that
根据我们幻灯片上的

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we have on these slides, I
这些例子

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think I'll be able to convey part of the intuition.
我想我可以给你一个直观的感受

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But, an even better
但是 一个更好的

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way to see for yourself, how
让你自己去理解正规化

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regularization works, is if
如何工作的方法是

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you implement it, and, see it work for yourself.
你自己亲自去实现它 并且看看它是如何工作的

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And, if you do the
如果在这节课后

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appropriate exercises after this,
你进行一些适当的练习

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you get the chance
你就有机会亲自体验一下

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to self see regularization in action for yourself.
正规化到底是怎么工作的

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So, here is the intuition.
那么 这里就是一些直观解释

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In the previous video, we saw
在前面的视频中 我们看到了

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that, if we were to fit
如果说我们要

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a quadratic function to this
用一个二次函数来

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data, it gives us a pretty good fit to the data.
拟合这些数据 它给了我们一个对数据很好的拟合

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Whereas, if we were to
然而 如果我们

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fit an overly high order
用一个更高次的

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degree polynomial, we end
多项式去拟合 我们最终

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up with a curve that may fit
可能得到一个曲线

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the training set very well, but,
能非常好地拟合训练集 但是

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really not be a,
这真的不是一个好的结果

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but overfit the data
它过度拟合了数据

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poorly, and, not generalize well.
因此 一般性并不是很好

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Consider the following, suppose we
让我们考虑下面的假设

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were to penalize, and, make
我们想要加上惩罚项 从而使

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the parameters theta 3 and theta 4 really small.
参数 θ3 和 θ4 足够的小

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Here's what I
这里我的意思就是

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mean, here is our optimization
这是我们的优化目标

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objective, or here is our
或者客观的说 这就是我们需要

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optimization problem, where we minimize
优化的问题 我们需要尽量减少

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our usual squared error cause function.
代价函数的均方误差

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Let's say I take this objective
对于这个函数

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and modify it and add
我们对它进行一些 添加一些项

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to it, plus 1000 theta
加上 1000 乘以 θ3 的平方

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3 squared, plus 1000 theta 4 squared.
再加上 1000 乘以 θ4 的平方

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1000 I am just writing down as some huge number.
1000 只是我随便写的某个较大的数字而已

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Now, if we were to
现在 如果我们要

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minimize this function, the
最小化这个函数

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only way to make this
为了使这个

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new cost function small is
新的代价函数最小化

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if theta 3 and data
我们要让 θ3 和 θ4

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4 are small, right?
尽可能小 对吧？

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Because otherwise, if you have
因为 如果你有

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a thousand times theta 3, this
1000 乘以 θ3 这个

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new cost functions gonna be big.
新的代价函数将会是很大的

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So when we minimize this
所以 当我们最小化

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new function we are going
这个新的函数时 我们将使

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to end up with theta 3
θ3 的值

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close to 0 and theta
接近于0

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4 close to 0, and as
θ4 的值也接近于0

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if we're getting rid
就像我们忽略了

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of these two terms over there.
这两个值一样

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And if we do that, well then,
如果我们做到这一点

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if theta 3 and theta 4
如果 θ3 和 θ4

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close to 0 then we are
接近0 那么我们

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being left with a quadratic function,
将得到一个近似的二次函数

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and, so, we end up with
所以 我们最终

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a fit to the data, that's, you know, quadratic
恰当地拟合了数据  你知道

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function plus maybe, tiny
二次函数加上一些项

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contributions from small terms,
这些很小的项 贡献很小

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theta 3, theta 4, that they may be very close to 0.
因为 θ3 θ4 它们是非常接近于0的

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And, so, we end up with
所以 我们最终得到了

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essentially, a quadratic function, which is good.
实际上 很好的一个二次函数

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Because this is a
因为这是一个

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much better hypothesis.
更好的假设

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In this particular example, we looked at the effect
在这个具体的例子中 我们看到了

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of penalizing two of
惩罚这两个

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the parameter values being large.
大的参数值的效果

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More generally, here is the idea behind regularization.
更一般地 这里给出了正规化背后的思路

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The idea is that, if we
这种思路就是 如果我们

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have small values for the
的参数值

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parameters, then, having
对应一个较小值的话

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small values for the parameters,
就是说 参数值比较小

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will somehow, will usually correspond
那么往往我们会得到一个

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to having a simpler hypothesis.
形式更简单的假设

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So, for our last example, we
所以 我们最后一个例子中

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penalize just theta 3 and
我们惩罚的只是 θ3 和

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theta 4 and when both
θ4 使这两个

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of these were close to zero,
值均接近于零

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we wound up with a much simpler
我们得到了一个更简单的假设

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hypothesis that was essentially a quadratic function.
也即这个假设大抵上是一个二次函数

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But more broadly, if we penalize all
但更一般地说 如果我们就像这样

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the parameters usually that, we
惩罚的其它参数 通常我们

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can think of that, as trying
可以把它们都想成是

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to give us a simpler hypothesis
得到一个更简单的假设

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as well because when, you
因为你知道

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know, these parameters are
当这些参数越接近这个例子时

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as close as you in this
假设的结果越接近

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example, that gave us a quadratic function.
一个二次函数

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But more generally, it is
但更一般地

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possible to show that having
可以表明

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smaller values of the parameters
这些参数的值越小

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corresponds to usually smoother
通常对应于越光滑的函数

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functions as well for the simpler.
也就是更加简单的函数

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And which are therefore, also, less prone to overfitting.
因此 就不易发生过拟合的问题

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I realize that the reasoning for
我知道

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why having all the parameters be small.
为什么要所有的部分参数变小的这些原因

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Why that corresponds to a simpler
为什么越小的参数对应于一个简单的假设

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hypothesis; I realize that
我知道这些原因

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reasoning may not be entirely clear to you right now.
对你来说现在不一定完全理解

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And it is kind of hard
但现在解释起来确实比较困难

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to explain unless you implement
除非你自己实现一下

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yourself and see it for yourself.
自己亲自运行了这部分

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But I hope that the example of
但是我希望 这个例子中

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having theta 3 and theta
使 θ3 和 θ4

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4 be small and how
很小 并且这样做

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that gave us a simpler
能给我们一个更加简单的

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hypothesis, I hope that
假设 我希望这个例子

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helps explain why, at least give
有助于解释原因 至少给了

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some intuition as to why this might be true.
我们一些直观感受 为什么这应该是这样的

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Lets look at the specific example.
来让我们看看具体的例子

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For housing price prediction we
对于房屋价格预测我们

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may have our hundred features
可能有上百种特征

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that we talked about where may
我们谈到了一些可能的特征

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be x1 is the size, x2
比如说 x1 是房屋的尺寸

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is the number of bedrooms, x3
x2 是卧室的数目

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is the number of floors and so on.
x3 是房屋的层数等等

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And we may we may have a hundred features.
那么我们可能就有一百个特征

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And unlike the polynomial
跟前面的多项式例子不同

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example, we don't know, right,
我们是不知道的 对吧

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we don't know that theta 3,
我们不知道 θ3

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theta 4, are the high order polynomial terms.
θ4 是高阶多项式的项

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So, if we have just a
所以 如果我们有一个袋子

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bag, if we have just a
如果我们有一百个特征

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set of a hundred features, it's hard
在这个袋子里 我们是很难

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to pick in advance which are
提前选出那些

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the ones that are less likely to be relevant.
关联度更小的特征的

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So we have a hundred or a hundred one parameters.
也就是说如果我们有一百或一百零一个参数

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And we don't know which
我们不知道

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ones to pick, we
挑选哪一个

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don't know which
我们并不知道

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parameters to try to pick, to try to shrink.
如何选择参数 如何缩小参数的数目

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So, in regularization, what we're
因此在正规化里

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going to do, is take our
我们要做的事情 就是把我们的

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cost function, here's my cost function for linear regression.
代价函数 这里就是线性回归的代价函数

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And what I'm going to do
我现在要做的就是

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is, modify this cost
来修改这个代价函数

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function to shrink all
从而缩小

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of my parameters, because, you know,
我所有的参数值 因为你知道

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I don't know which
我不知道是哪个

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one or two to try to shrink.
哪一个或两个要去缩小

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So I am going to modify my
所以我就修改我的

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cost function to add a term at the end.
代价函数 在这后面添加一项

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Like so we have square brackets here as well.
就像我们在方括号里的这项

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When I add an extra
当我添加一个额外的

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regularization term at the
正则化项的时候

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end to shrink every
我们收缩了每个

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single parameter and so this
参数 并且因此

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term we tend to shrink
我们会使

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all of my parameters theta 1,
我们所有的参数 θ1

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theta 2, theta 3 up
θ2 θ3

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to theta 100.
直到 θ100 的值变小

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By the way, by convention the summation
顺便说一下 按照惯例来讲

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here starts from one so I
我们从第一个这里开始

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am not actually going penalize theta
所以我实际上没有去惩罚 θ0

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zero being large.
因此 θ0 的值是大的

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That sort of the convention that,
这就是一个约定

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the sum I equals one through
从1到 n 的求和

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N, rather than I equals zero
而不是从0到 n 的求和

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through N. But in practice,
但其实在实践中

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it makes very little difference, and,
这只会有非常小的差异

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whether you include, you know,
无论你是否包括这项

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theta zero or not, in
就是 θ0 这项

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practice, make very little difference to the results.
实际上 结果只有非常小的差异

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But by convention, usually, we regularize
但是按照惯例 通常情况下我们还是只

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only theta  through theta
从 θ1 到 θ100 进行正规化

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100. Writing down
这里我们写下来

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our regularized optimization objective,
我们的正规化优化目标

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our regularized cost function again.
我们的正规化后的代价函数

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Here it is. Here's J of
就是这样的

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theta where, this term
J(θ) 这个项

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on the right is a regularization
右边的这项就是一个正则化项

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term and lambda
并且 λ

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here is called the regularization parameter and
在这里我们称做正规化参数

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what lambda does, is it
λ 要做的就是控制

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controls a trade off
在两个不同的目标中

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between two different goals.
的一个平衡关系

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The first goal, capture it
第一个目标

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by the first goal objective, is
第一个需要抓住的目标

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that we would like to train,
就是我们想要训练

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is that we would like to fit the training data well.
使假设更好地拟合训练数据

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We would like to fit the training set well.
我们希望假设能够很好的适应训练集

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And the second goal is,
而第二个目标是

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we want to keep the parameters
我们想要保持参数值较小

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small, and that's captured by
这就是第二项的目标

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the second term, by the regularization objective. And by the regularization term.
通过正则化目标函数

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And what lambda, the regularization
这就是λ 这个正则化

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parameter does is the controls the trade of
参数需要控制的

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between these two
它会这两者之间的平衡

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goals, between the goal of fitting the training set well
目标就是平衡拟合训练的目的

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and the
和

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goal of keeping the parameter plan
保持参数值较小的目的

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small and therefore keeping the hypothesis relatively
从而来保持假设的形式相对简单

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simple to avoid overfitting.
来避免过度的拟合

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For our housing price prediction
对于我们的房屋价格预测来说

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example, whereas, previously, if
这个例子 尽管我们之前有

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we had fit a very high
我们已经用非常高的

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order polynomial, we may
高阶多项式来拟合 我们将会

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have wound up with a very,
得到一个

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sort of wiggly or curvy function like
非常弯曲和复杂的曲线函数

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this. If you still fit a high order polynomial
就像这个 如果你还是用高阶多项式拟合

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with all the polynomial
就是用这里所有的多项式特征来拟合的话

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features in there, but instead,
但现在我们不这样了

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you just make sure, to use
你只需要确保使用了

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this sole of regularized objective, then what
正规化目标的方法

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you can get out is in
那么你就可以得到

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fact a curve that isn't
实际上是一个曲线 但这个曲线不是

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quite a quadratic function, but is
一个真正的二次函数

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much smoother and much simpler
而是更加的流畅和简单

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and maybe a curve like the magenta
也许就像这条紫红色的曲线一样

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line that, you know, gives a
那么 你知道的

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much better hypothesis for this data.
这样就得到了对于这个数据更好的假设

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Once again, I realize
再一次说明下

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it can be a bit difficult to see why strengthening the
我了解这部分有点难以明白 为什么加上

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parameters can have
参数的影响可以具有

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this effect, but if you
这种效果 但如果你

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implement yourselves with regularization
亲自实现了正规化

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you will be able to see
你将能够看到

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this effect firsthand.
这种影响的最直观的感受

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In regularized linear regression, if
在正规化线性回归中 如果

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the regularization parameter monitor
正规化参数值

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is set to be very large,
被设定为非常大

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then what will happen is
那么将会发生什么呢？

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we will end up penalizing the
我们将会非常大地惩罚

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parameters theta 1, theta
参数θ1 θ2

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2, theta 3, theta
θ3 θ4

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4 very highly.
也就是说

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That is, if our hypothesis is this is one down at the bottom.
如果我们的假设是底下的这个

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And if we end up penalizing
如果我们最终惩罚

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theta 1, theta 2, theta
θ1 θ2 θ3

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3, theta 4 very heavily, then we
θ4 在一个非常大的程度 那么我们

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end up with all of these parameters close to zero, right?
会使所有这些参数接近于零的 对不对？

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Theta 1 will be close to zero; theta 2 will be close to zero.
θ1 将接近零   θ2 将接近零

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Theta three and theta four
θ3 和 θ4

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will end up being close to zero.
最终也会接近于零

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And if we do that, it's as
如果我们这么做 那么就是

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if we're getting rid of these
我们的假设中

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terms in the hypothesis so that
相当于去掉了这些项 并且使

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we're just left with a hypothesis
我们只是留下了一个简单的假设

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that will say that.
这个假设只能表明

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It says that, well, housing
那就是 房屋价格

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prices are equal to theta zero,
就等于 θ0 的值

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and that is akin to fitting
那就是类似于拟合了

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a flat horizontal straight line to the data.
一条水平直线 对于数据来说

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And this is an
这就是一个

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example of underfitting, and
欠拟合 (underfitting)

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in particular this hypothesis, this
这种情况下这一假设

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straight line it just fails
它是条失败的直线

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to fit the training set
对于训练集来说

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well. It's just a fat straight
这只是一条平滑直线

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line, it doesn't go, you know, go near.
它没有任何趋势

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It doesn't go anywhere near most of the training examples.
它不会去趋向大部分训练样本的任何值

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And another way of saying this
这句话的另??一种方式来表达就是

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is that this hypothesis has
这种假设有

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too strong a preconception or
过于强烈的"偏见" 或者

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too high bias that housing
过高的偏差 (bais)

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prices are just equal
认为预测的价格只是

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to theta zero, and despite
等于 θ0 并且

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the clear data to the contrary,
尽管我们的数据集

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you know chooses to fit a sort
选择去拟合一条

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of, flat line, just a
扁平的直线 仅仅是一条

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flat horizontal line. I didn't draw that very well.
扁平的水平线  我画得不好

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This just a horizontal flat line
对于数据来说

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to the data. So for
这只是一条水平线 因此

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regularization to work well, some
为了使正则化运作良好

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care should be taken,
我们应当注意一些方面

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to choose a good choice for
应该去选择一个不错的

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the regularization parameter lambda as well.
正则化参数 λ

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And when we talk about multi-selection
并且当我们以后讲到多重选择时

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later in this course, we'll talk
在后面的课程中 我们将讨论

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about a way, a variety
一种方法

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of ways for automatically choosing
一系列的方法来自动选择

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the regularization parameter lambda as well. So, that's
正则化参数 λ  所以

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the idea of the high regularization
这就是高度正则化的思路

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and the cost function reviews in
回顾一下代价函数

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order to use regularization In the
为了使用正则化

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next two videos, lets take
在接下来的两段视频中 让我们

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these ideas and apply them
把这些概念 应用到

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to linear regression and to
到线性回归和

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00:10:05,440 --> 00:10:07,111
logistic regression, so that
逻辑回归中去

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we can then get them to
那么我们就可以让他们

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avoid overfitting.
避免过度拟合了