10 - 6 - Learning Curves (12 min).srt

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In this video, I'd like to tell you about learning curves.
本节课我们介绍学习曲线
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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Learning curves is often a very useful thing to plot.
绘制学习曲线非常有用

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If either you wanted to sanity check
也许你想检查你的学习算法

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that your algorithm is working correctly,
运行是否一切正常

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or if you want to improve the performance of the algorithm.
或者你希望改进算法的表现或效果

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And learning curves is a
那么学习曲线

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tool that I actually use
就是一种很好的工具

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very often to try to
我经常使用学习曲线

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diagnose if a physical learning algorithm may be
来判断某一个学习算法

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suffering from bias, sort of variance problem or a bit of both.
是否处于偏差 方差问题 或是二者皆有

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Here's what a learning curve is.
下面我们就来介绍学习曲线

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To plot a learning curve, what
为了绘制一条学习曲线

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I usually do is plot
我通常先绘制出Jtrain

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j train which is, say,
也就是训练集数据的

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average squared error on my training
平均误差平方和

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set or Jcv which is
或者Jcv 也即交叉验证集数据的

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the average squared error on my cross validation set.
平均误差平方和

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And I'm going to plot
我要将其绘制成一个

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that as a function
关于参数m的函数

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of m, that is as a function
也就是一个关于训练集

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of the number of training examples I have.
样本总数的函数

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And so m is usually a constant like maybe I just have, you know, a 100
所以m一般都是一个常数 比如m等于100

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training examples but what I'm
表示100组训练样本

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going to do is artificially with
但我要自己取一些m的值

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use my training set exercise. So, I
也就是说我要自行对m的取值

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deliberately limit myself to using only,
做一点限制

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say, 10 or 20 or
比如说我取10 20或者

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30 or 40 training examples and
30 40组训练集

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plot what the training error is and
然后绘出训练集误差

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what the cross validation is for this
以及交叉验证集误差

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smallest training set exercises. So
好的 那么我们来看看

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let's see what these plots may look
这条曲线绘制出来是什么样子

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like. Suppose I have only
假设我只有一组训练样本

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one training example like that
也即m=1

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shown in this this first example
正如第一幅图中所示

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here and let's say I'm fitting a quadratic function. Well, I
并且假设使用二次函数来拟合模型

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have only one training example. I'm
那么由于我只有一个训练样本

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going to be able to fit it perfectly
拟合的结果很明显会很好

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right? You know, just fit the quadratic function. I'm
是吧 用二次函数来拟合

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going to have 0
对这一个训练样本拟合

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error on the one training example. If I
其误差一定为0

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have two training examples. Well the quadratic function can also fit that very well. So,
如果有两组训练样本 二次函数也能很好地拟合

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even if I am using regularization,
即使是使用正则化

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I can probably fit this quite well.
拟合的结果也会很好

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And if I am using no neural regularization,
而如果不使用正则化的话

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I'm going to fit this perfectly and
那么拟合效果绝对棒极了

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if I have three training examples
如果我用三组训练样本的话

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again. Yeah, I can fit a quadratic
好吧 看起来依然能很好地

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function perfectly so if
用二次函数拟合

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m equals 1 or m equals 2 or m equals 3,
也就是说 当m等于1 m=2 或m=3时

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my training error
对训练集数据进行预测

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on my training set is
得到的训练集误差

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going to be 0 assuming I'm
都将等于0

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not using regularization or it may
这里假设我不使用正则化

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slightly large in 0 if
当然如果使用正则化

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I'm using regularization and
那么误差就稍大于0

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by the way if I have
顺便提醒一下

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a large training set and I'm artificially
如果我的训练集样本很大

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restricting the size of my
而我要人为地限制训练集

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training set in order to J train.
样本的容量

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Here if I set
比如说这里

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M equals 3, say, and I
我将m值设为3

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train on only three examples,
然后我仅用这三组样本进行训练

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then, for this figure I
然后对应到这个图中

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am going to measure my training error
我只看对这三组训练样本

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only on the three examples that
进行预测得到的训练误差

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actually fit my data too
也是和我模型拟合的三组样本

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and so even I have to say
所以即使我有100组训练样本

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a 100 training examples but if I want to plot what my
而我还是想绘制

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training error is the m equals 3. What I'm going to do
当m等于3时的训练误差

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is to measure the
那么我要关注的仍然是

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training error on the
对这三组训练样本进行预测的误差

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three examples that I've actually fit to my hypothesis 2.
同样 这三组样本也是我们用来拟合模型的三组样本

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And not all the other examples that I have
所有其他的样本

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deliberately omitted from the training
我都在训练过程中选择性忽略了

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process. So just to summarize what we've
好的 总结一下

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seen is that if the training set
我们现在已经看到

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size is small then the
当训练样本容量m很小的时候

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training error is going to be small as well.
训练误差也会很小

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Because you know, we have a
因为很显然

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small training set is
如果我们训练集很小

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going to be very easy to
那么很容易就能把

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fit your training set
训练集拟合到很好

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very well may be even
甚至拟合得天衣无缝

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perfectly now say
现在我们来看

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we have m equals 4 for example. Well then
当m等于4的时候

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a quadratic function can be
好吧 二次函数似乎也能

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a longer fit this data set
对数据拟合得很好

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perfectly and if I
那我们再看

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have m equals 5 then you
当m等于5的情况

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know, maybe quadratic function will fit to stay there so
这时候再用二次函数来拟合

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so, then as my training set gets larger.
好像效果有下降但还是差强人意

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It becomes harder and harder to
而当我的训练集越来越大的时候

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ensure that I can
你不难发现 要保证使用二次函数

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find the quadratic function that process through
的拟合效果依然很好

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all my examples perfectly. So
就显得越来越困难了

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in fact as the training set size
因此 事实上随着训练集容量的增大

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grows what you find
我们不难发现

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is that my average training error
我们的平均训练误差

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actually increases and so if you plot
是逐渐增大的

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this figure what you find
因此如果你画出这条曲线

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is that the training set
你就会发现

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error that is the average
训练集误差 也就是

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error on your hypothesis grows
对假设进行预测的误差平均值

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as m grows and just to repeat when the intuition is that when
随着m的增大而增大

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m is small when you have very
再重复一遍对这一问题的理解

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few training examples. It's pretty
当训练样本很少的时候

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easy to fit every single
对每一个训练样本

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one of your training examples perfectly and
都能很容易地拟合到很好

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so your error is going
所以训练误差将会很小

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to be small whereas
而反过来

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when m is larger then gets
当m的值逐渐增大

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harder all the training
那么想对每一个训练样本都拟合到很好

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examples perfectly and so
就显得愈发的困难了

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your training set error becomes
因此训练集误差就会越来越大

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more larger now, how about the cross validation error.
那么交叉验证集误差的情况如何呢

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Well, the cross validation is
好的 交叉验证集误差

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my error on this cross
是对完全陌生的交叉验证集数据

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validation set that I haven't seen and
进行预测得到的误差

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so, you know, when I have
那么我们知道

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a very small training set, I'm
当训练集很小的时候

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not going to generalize well, just
泛化程度不会很好

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not going to do well on that.
意思是不能很好地适应新样本

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So, right, this hypothesis here doesn't
因此这个假设

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look like a good one, and
就不是一个理想的假设

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it's only when I get
只有当我使用

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a larger training set that,
一个更大的训练集时

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you know, I'm starting to get
我才有可能

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hypotheses that maybe fit
得到一个能够更好拟合数据的

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the data somewhat better.
可能的假设

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So your cross validation error and
因此 你的验证集误差和

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your test set error will tend
测试集误差

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to decrease as your training
都会随着训练集样本容量m的增加

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set size increases because the
而减小 因为你使用的数据越多

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more data you have, the better
你越能获得更好地泛化表现

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you do at generalizing to new examples.
或者说对新样本的适应能力更强

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So, just the more data you have, the better the hypothesis you fit.
因此 数据越多 越能拟合出合适的假设

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So if you plot j train,
所以 如果你把Jtrain和Jcv绘制出来

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and Jcv this is the sort of thing that you get.
就应该得到这样的曲线

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Now let's look at what
现在我们来看看

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the learning curves may look like
当处于高偏差或者高方差的情况时

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if we have either high
这些学习曲线

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bias or high variance problems.
又会变成什么样子

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Suppose your hypothesis has high
假如你的假设处于高偏差问题

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bias and to explain this
为了更清楚地解释这个问题

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I'm going to use a, set an
我要用一个简单的例子来说明

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example, of fitting a straight
也就是用一条直线

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line to data that, you
来拟合数据的例子

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know, can't really be fit well by a straight line.
很显然一条直线不能很好地拟合数据

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So we end up with a hypotheses that maybe looks like that.
所以最后得到的假设很有可能是这样的

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Now let's think what would
现在我们来想一想

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happen if we were to increase
如果我们增大训练集样本容量

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the training set size. So if
会发生什么情况呢

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instead of five examples like
所以现在不像画出的这样

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what I've drawn there, imagine that
只有这五组样本了

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we have a lot more training examples.
我们有了更多的训练样本

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Well what happens, if you fit a straight line to this.
那么如果你用一条直线来拟合

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What you find is that, you
不难发现

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end up with you know, pretty much the same straight line.
还是会得到类似的一条直线假设

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I mean a straight line that
我的意思是

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just cannot fit this
刚才的情况用一条直线不能很好地拟合

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data and getting a ton more data, well
而现在把样本容量扩大了

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the straight line isn't going to change that much.
这条直线也基本不会变化太大

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This is the best possible straight-line
因为这条直线是对这组数据

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fit to this data, but the
最可能也是最接近的拟合

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straight line just can't fit this
但一条直线再怎么接近

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data set that well. So,
也不可能对这组数据进行很好的拟合

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if you plot across validation error,
所以 如果你绘出交叉验证集误差

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this is what it will look like.
应该是这样子的

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Option on the left, if you have already a miniscule training set size like you know,
最左端表示训练集样本容量很小 比如说只有一组样本

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maybe just one training example and is not going to do well.
那么表现当然很不好

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But by the time you have
而随着你增大训练集样本数

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reached a certain number of training
当达到某一个容量值的时候

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examples, you have almost
你就会找到那条最有可能

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fit the best possible straight
拟合数据的那条直线

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line, and even if
并且此时即便

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you end up with a much
你继续增大训练集的

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larger training set size, a
样本容量

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much larger value of m,
即使你不断增大m的值

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you know, you're basically getting the same straight line,
你基本上还是会得到的一条差不多的直线

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and so, the cross-validation error
因此 交叉验证集误差

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- let me label that -
我把它标在这里

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or test set error or
或者测试集误差

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plateau out, or flatten out
将会很快变为水平而不再变化

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pretty soon, once you reached
只要训练集样本容量值达到

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beyond a certain the number
或超过了那个特定的数值

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of training examples, unless you
交叉验证集误差和测试集误差就趋于不变

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pretty much fit the best possible straight line.
这样你会得到最能拟合数据的那条直线

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And how about training error?
那么训练误差又如何呢

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Well, the training error will again be small.
同样 训练误差一开始也是很小的

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And what you find
而在高偏差的情形中

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in the high bias case is
你会发现训练集误差

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that the training error will end
会逐渐增大

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up close to the cross
一直趋于接近

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validation error, because you
交叉验证集误差

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have so few parameters and so
这是因为你的参数很少

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much data, at least when m is large.
但当m很大的时候 数据太多

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The performance on the training
此时训练集和交叉验证集的

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set and the cross validation set will be very similar.
预测效果将会非常接近

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And so, this is what your
这就是当你的学习算法处于

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learning curves will look like,
高偏差情形时

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if you have an algorithm that has high bias.
学习曲线的大致走向

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And finally, the problem with
最后补充一点

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high bias is reflected in
高偏差的情形

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the fact that both the
反映出的问题是

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cross validation error and the
交叉验证集和训练集

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training error are high,
误差都很大

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and so you end up with
也就是说 你最终会得到一个

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a relatively high value of
值比较大Jcv

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both Jcv and the j train.
和Jtrain

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This also implies something very
这也得出一个很有意思的结论

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interesting, which is that,
那就是

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if a learning algorithm has high
如果一个学习算法

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bias, as we
有很大的偏差

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get more and more training examples,
那么当我们选用更多的训练样本时

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that is, as we move to
也就是在这幅图中

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the right of this figure, we'll
随着我们增大横坐标

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notice that the cross
我们发现交叉验证集误差的值

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validation error isn't going
不会表现出明显的下降

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down much, it's basically fattened
实际上是变为水平了

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up, and so if
所以如果学习算法

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learning algorithms are really suffering from high bias.
正处于高偏差的情形

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Getting more training data by
那么选用更多的训练集数据

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itself will actually not help
对于改善算法表现无益

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that much,and as our figure
正如我们右边的

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example in the figure
这两幅图所体现的

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on the right, here we had only five training.
这里我们只有五组训练样本

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examples, and we fill certain straight line.
然后我们找到这条直线来拟合

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And when we had a ton
然后我们增加了更多的训练样本

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more training data, we still
但我们仍然得到几乎一样的

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end up with roughly the same straight line.
一条直线

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And so if the learning algorithm
因此如果学习算法

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has high bias give me a lot more training data.
处于高偏差时

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That doesn't actually help you
给我再多的训练数据也于事无补

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get a much lower cross validation
交叉验证集误差或测试集误差

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error or test set error.
也不会降低多少

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So knowing if your learning
所以 能够看清你的算法正处于

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algorithm is suffering from high
高偏差的情形

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bias seems like a useful
是一件很有意义的事情

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thing to know because this can
因为这样可以让你避免

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prevent you from wasting a
把时间浪费在

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lot of time collecting more training
收集更多的训练集数据上

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data where it might just not end up being helpful.
因为再多的数据也是无意义的

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Next let us look at the
接下来我们再来看看

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setting of a learning algorithm
当学习算法正处于高方差的时候

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that may have high variance.
学习曲线应该是什么样子的

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Let us just look at the
首先我们来看

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training error in a around if
训练集误差

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you have very smart training
如果你的训练集样本容量很小

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set like five training examples shown on
比如像图中所示情形

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the figure on the right and
只有五组训练样本

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if we're fitting say a
如果我们用很高阶次的

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very high order polynomial,
多项式来拟合

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and I've written a hundredth degree polynomial which
比如这里我用了100次的多项式函数

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really no one uses, but just an illustration.
当然不会有人这么用的 这里只是演示

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And if we're using a
并且假设我们使用

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fairly small value of lambda,
一个很小的lambda值

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maybe not zero, but a fairly
可能不等于0

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small value of lambda, then
但足够小的lambda

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we'll end up, you know,
那么很显然 我们会对这组数据

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fitting this data very well that with
拟合得非常非常好

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a function that overfits this.
因此这个假设函数对数据过拟合

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So, if the training
所以 如果训练集

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set size is small, our training
样本容量很小时

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error, that is, j train
训练集误差Jtrain

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of theta will be small.
将会很小

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And as this training set size increases
随着训练集样本容量的增加

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a bit, you know, we may
可能这个假设函数仍然会

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still be overfitting this
对数据或多或少

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data a little bit but
有一点过拟合

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it also becomes slightly harder to
但很明显此时要对数据很好地拟合

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fit this data set perfectly,
显得更加困难和吃力了

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and so, as the training set size
所以 随着训练集样本容量的增大

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increases, we'll find that
我们会发现Jtrain的值

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j train increases, because
会随之增大

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it is just a little harder to fit
因为当训练样本越多的时候

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the training set perfectly when we have
我们就越难跟训练集数据拟合得很好

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more examples, but the training set error will still be pretty low.
但总的来说训练集误差还是很小

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Now, how about the cross validation error?
交叉验证集误差又如何呢

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Well, in high variance
好的 在高方差的情形中

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00:09:31,040 --> 00:09:32,760
setting, a hypothesis is
假设函数对数据过拟合

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overfitting and so the
因此交叉验证集误差

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cross validation error will remain
将会一直都很大

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high, even as we
即便我们选择一个

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get you know, a moderate number
比较合适恰当的

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of training examples and, so
训练集样本数

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maybe, the cross validation
因此交叉验证集误差

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00:09:43,730 --> 00:09:45,520
error may look like that.
画出来差不多是这样的

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And the indicative diagnostic that we
所以算法处于高方差情形

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have a high variance problem,
最明显的一个特点是

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is the fact that there's
在训练集误差

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this large gap between
和交叉验证集误差之间

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the training error and the cross validation error.
有一段很大的差距

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And looking at this figure.
而这个曲线图也反映出

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If we think about adding more
如果我们要考虑增大训练集的样本数

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training data, that is, taking
也就是在这幅图中

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00:10:02,110 --> 00:10:03,660
this figure and extrapolating to
向右延伸曲线

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00:10:03,790 --> 00:10:05,220
the right, we can kind
我们大致可以看出

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00:10:05,330 --> 00:10:06,830
of tell that, you know the
这两条学习曲线

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00:10:07,030 --> 00:10:08,120
two curves, the blue curve
蓝色和红色的两条曲线

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00:10:08,480 --> 00:10:10,480
and the magenta curve, are converging to each other.
正在相互靠近

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And so, if we were to
因此 如果我们将曲线

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extrapolate this figure to
向右延伸出去

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00:10:13,980 --> 00:10:21,230
the right, then it
那么似乎

305
00:10:21,360 --> 00:10:23,000
seems it likely that the
训练集误差很可能会

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00:10:23,170 --> 00:10:24,120
training error will keep on
逐渐增大

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00:10:24,270 --> 00:10:25,740
going up and the
而交叉验证集误差

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00:10:27,130 --> 00:10:29,040
cross-validation error would keep on going down.
则会持续下降

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00:10:30,000 --> 00:10:32,340
And the thing we really care about is the cross-validation error
当然我们最关心的还是交叉验证集误差

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00:10:33,010 --> 00:10:35,150
or the test set error, right?
或者测试集误差 对吧

311
00:10:35,300 --> 00:10:36,460
So in this sort
所以从这幅图中

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00:10:36,730 --> 00:10:37,850
of figure, we can tell that
我们基本可以预测

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if we keep on adding training
如果继续增大训练样本的数量

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00:10:39,820 --> 00:10:40,930
examples and extrapolate to the
将曲线向右延伸

315
00:10:41,050 --> 00:10:42,650
right, well our cross validation
交叉验证集误差将会

316
00:10:43,290 --> 00:10:44,610
error will keep on coming down.
逐渐下降

317
00:10:45,120 --> 00:10:46,090
And, so, in the high
所以 在高方差的情形中

318
00:10:46,330 --> 00:10:47,980
variance setting, getting more
使用更多的训练集数据

319
00:10:48,180 --> 00:10:49,550
training data is, indeed,
对改进算法的表现

320
00:10:50,170 --> 00:10:51,240
likely to help.
事实上是有效果的

321
00:10:51,520 --> 00:10:52,810
And so again, this seems like a
这同样也体现出

322
00:10:53,060 --> 00:10:54,180
useful thing to know if your
知道你的算法正处于

323
00:10:54,330 --> 00:10:55,830
learning algorithm is suffering
高方差的情形

324
00:10:56,150 --> 00:10:57,460
from a high variance problem, because
也是非常有意义的

325
00:10:57,810 --> 00:10:59,150
that tells you, for example that it
因为它能告诉你

326
00:10:59,220 --> 00:11:00,100
may be be worth your while
是否有必要花时间

327
00:11:00,680 --> 00:11:02,430
to see if you can go and get some more training data.
来增加更多的训练集数据

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00:11:03,700 --> 00:11:04,920
Now, on the previous slide
好的 在前一页和这一页幻灯片中

329
00:11:05,330 --> 00:11:06,450
and this slide, I've drawn fairly
我画出的学习曲线

330
00:11:06,970 --> 00:11:08,510
clean fairly idealized curves.
都是相当理想化的曲线

331
00:11:08,900 --> 00:11:10,050
If you plot these curves for
针对一个实际的学习算法

332
00:11:10,170 --> 00:11:11,970
an actual learning algorithm, sometimes
如果你画出学习曲线的话

333
00:11:12,500 --> 00:11:13,910
you will actually see, you know, pretty
你会看到基本类似的结果

334
00:11:14,560 --> 00:11:15,900
much curves, like what I've drawn here.
就像我在这里画的一样

335
00:11:16,600 --> 00:11:17,730
Although, sometimes you see curves
虽然如此

336
00:11:18,150 --> 00:11:19,160
that are a little bit noisier and
有时候你也会看到

337
00:11:19,230 --> 00:11:20,820
a little bit messier than this.
带有一点噪声或干扰的曲线

338
00:11:21,090 --> 00:11:22,440
But plotting learning curves like
但总的来说

339
00:11:22,620 --> 00:11:23,850
these can often tell
像这样画出学习曲线

340
00:11:24,120 --> 00:11:25,460
you, can often help you
确实能帮助你

341
00:11:25,570 --> 00:11:26,650
figure out if your learning algorithm is
看清你的学习算法

342
00:11:26,950 --> 00:11:29,080
suffering from bias, or variance or even a little bit of both.
是否处于高偏差 高方差 或二者皆有的情形

343
00:11:29,170 --> 00:11:31,030
So when I'm
所以当我打算

344
00:11:31,200 --> 00:11:32,700
trying to improve the performance of
改进一个学习算法

345
00:11:32,760 --> 00:11:34,060
a learning algorithm, one thing
的表现时

346
00:11:34,260 --> 00:11:35,720
that I'll almost always do
我通常会进行的一项工作

347
00:11:35,960 --> 00:11:37,440
is plot these learning
就是画出这些学习曲线

348
00:11:37,970 --> 00:11:39,460
curves, and usually this will
一般来讲 这项工作会让你

349
00:11:39,490 --> 00:11:41,710
give you a better sense of whether there is a bias or variance problem.
更轻松地看出偏差或方差的问题

350
00:11:44,280 --> 00:11:45,180
And in the next video
在下一节视频中

351
00:11:45,420 --> 00:11:46,440
we'll see how this can
我们将介绍如何判断

352
00:11:46,650 --> 00:11:48,370
help suggest specific actions is
是否应采取具体的某个行为

353
00:11:48,450 --> 00:11:49,580
to take, or to not take,
来改进学习算法的表现

354
00:11:50,260 --> 00:11:53,250
in order to try to improve the performance of your learning algorithm.