ICLR 2020：杨立昆与基于能量的模型

确实，我觉得思考GANs很有意思，我认为StyleGAN2模型的一个关键点在于它们如何引入随机噪声。

Definitely like it like thinking about GANs and I think that's a big thing in like the style GAN two model is how they put that that random noise.

这些噪声直接注入到中间特征中，帮助模型实现更丰富的生成效果——它不再是确定性生成器，即每次对z采样都产生完全相同的面孔。

It just gets injected in the intermediate features and that helps it do, you know, more like it's not like a deterministic generator where for every sampling of the z, it produces the exact same face.

你可以通过将这个潜在向量z添加到预测过程中来实现这一点。

And you can do that by adding this sample latent vector z into the forecast.

因此，这个函数f(x, y)在x和y相互兼容时取值较低，在y与x不兼容时取值较高。

So this function f of x y will take low values is if x and y are compatible with each other and higher value if y is incompatible with x.

例如，如果它不是视频的合理延续。

If it's not a good continuation for the video, for example.

我在这里使用的符号与图模型中的因子图非常相似，只是额外加入了确定性函数的符号。

The symbolism I'm using here is very similar to factor graphs in graphical models except for this extra symbol of deterministic function.

现在，我将提倡使用基于能量的模型，它本质上通过这个能量函数来衡量x和y之间的兼容性。

Now I'm going to advocate to use energy based models, which, you know, basically measure the compatibility between x and y through this energy function.

同样，当x和y兼容时能量值较低，不兼容时则较高。

Again, that takes low value effects and y are compatible and higher value if if they're not.

推理过程是，对于给定的x，找到能使能量最小化的y。

Inference is performed by for a given x finding y's that minimizes energy.

可能对应多个y。

It could be multiple y's.

这是一种无需借助概率就能处理不确定性的方法。

And this is a way of handling uncertainty without resorting to probabilities.

所以，无需借助概率？

So without resorting to probabilities?

是的。

Yeah.

所以，这里的区别在于，你只是想要一个函数，它能告诉你何时对某个输入感到满意，这种满意会用一个较低的数字来表示，而不满意时则用较高的数字。

So so the again, the the difference is here, you you simply want a function that tells you when when it is happy with some input, and And that indicates by a lower number than when it is not happy.

较高的数字就表示不满意。

That would be indicated by a higher number.

然后他基本上说的是推理。

And then he basically says inference.

现在，如果我得到一个能量函数，对吧，那么我可以——假设我有一个视频，我想预测下一帧，并且我得到了一个能量函数，我能做的就是简单地找到在给定输入下最小化该能量函数的帧。

Now if I'm given an energy function, right, then I can so if I have, let's say, a video and I want to predict the next frame and I am given an energy function, what I can do is simply I can find the frame that minimizes that energy function given my input.

所以这基本上是对他所做工作的重新表述。

So that's kind of a reformulation of what he's doing.

尽管他接下来要讨论的模型并非都符合这个特定标准。

Though not all the models that he is gonna talk about fit that particular criteria.

是的。

Yeah.

起初，我觉得他提出可能存在多个可能的下一帧视频画面这种说法很傻，但他可能是在谈论电脑游戏之类的东西，如果你以不同的方式与之互动，那么下一帧就会不同。

At first, I I thought that it was silly suggesting that there were possible multiple possible video frames that come next, but he might be talking about a computer game or something where if you interact with it differently, then the next frame will will be different.

但正如你所说，这是一个非常通用的框架。

But but this is a very general framework as you say.

所以他的意思是，看看这个能量曲面，然后直接最小化。

So he's saying, look at this energy surface and just minimize.

也就是说，在所有那些点中，找到最小的那个，并将其作为我预测的下一帧。

So across all of those points, find the one which is the smallest and represent use that as my predicted next frame.

但即使对于视频来说，我也明白你的意思。

But even even if even for the video, I I get what you mean.

你的意思是，在已录制的视频中，只有一个下一帧。

You mean that in the video that was recorded, there is only one next frame.

是的。

Yeah.

但如果你只是剪掉最后部分，只看这个序列，你就无法知道摄像师是要往左转还是往右转。

But if if you simply cut away the last part and just look at the sequence, you don't know whether the camera person is going to gear left or right.

所以在真实世界中，存在多种可能的后续情况。

So there are multiple, in the true world, there are multiple continuations that are possible.

你的能量函数就是要捕捉这一点。

Your energy function is supposed to capture that.

当然，你会通过提供样本来训练它，但你希望它能泛化，告诉你真实世界中存在这么多可能的后续，而其他那些纯粹是胡言乱语的后续则不好。

Of course, you're going to train it by giving it the samples, but you hope that it is going to generalize into telling you there are all of these continuations that are possible in the real world, and all of these other continuations that are just gibberish are not good.

当y的取值变得如此高维，可以对应大量不同的y值与我们的x配对时，我们该如何采样？如何用我们的能量函数来搜索这些y值？

So once y becomes like this high cardinality, you know, can take on tons of different y's to pair with our x, how do we sample like, how are we gonna search for the y's with our energy function?

是的

Yep.

我认为这就是z发挥作用的地方。

That's I think that that's where the z comes into play.

而且我认为，他几乎把机器学习中的所有内容都重新表述为能量函数的形式。

Well, and that I I think that is the if so he's he's gone on to rephrase pretty much everything in machine learning in terms of an energy function.

我认为，如何实现这一确切目标，正是每个模型一直以来试图解决的问题。

And I think the question of how do we do this exact thing is that's what every model has ever attempted to do.

对吧？

Right?

而且他将讨论一些具体的例子。

So and and he's going to talk about specific ones.

但归根结底，这正是该方法本身所要解决的问题。

But ultimately, that is the the problem of the method itself.

为什么会有如此高的基数？

Why is so high cardinality?

你打算怎么做这件事？

How are you going to do this?

可以通过训练一个生成器来实现，对吧，让它直接预测一个能量较低的y。

It could be by training a generator, right, to simply predict a why that has a low energy.

也可以通过梯度下降进行优化，找到能量最低的那个y。

It could be by really optimizing with gradient descent to find the y that has the lowest possible.

还可以采用消息传递的方法。

It could be by a message passing method.

可能是通过很多方式实现的。

It could be, you know, via many things.

F在白色空间中是平滑的，可以通过基于梯度的优化算法或其他一些推理方法来完成。

F is smooth in white space can be done through gradient based optimization algorithms or some other inference methods.

当然，离散的方式要容易得多，我们不必处理那个问题。

Of course, way is discrete is much easier and we don't have to deal with that.

我的意思是，他说如果y是连续的，我们可以通过基于梯度的优化方法找到一个好的y。

I mean, he says if y is continuous, we can find a good y through gradient based optimization method.

这正是我们所说的。

That's what we said.

如果我们有一个能量函数，就可以直接用梯度下降来最小化它。

If we have an energy function, we can just minimize it using gradient descent.

然后他说，如果 y 是离散的，那当然要简单得多。

Then he says, If y is discrete, of course, that's much easier.

我不确定，因为通常离散优化问题确实如此。

I am not sure because usually discrete optimization problems Yeah.

所以一方面，如果你考虑一个监督分类问题，就可以这样来表述。

So on the one hand, if you think of a supervised classification problem, then you can phrase it like this.

这样一来就非常简单了。

And then it's super easy.

你只需尝试每一个类别，哪个类别的能量最低，就意味着哪个类别的可能性最高，会被输出为标签。

You just try every one of the classes and whichever one has the lowest energy, that means whichever one has the highest likelihood that you're gonna output as the label.

但在类似语言模型这样的场景中，这极其困难，因为你必须尝试每一个可能的句子，找出能量最低的那个。

But in exactly something like a language model, it is extremely hard because you're gonna have to try every single sentence that's possible and take find the one with the lowest energy functions.

我对他说的内容感到困惑。

I'm confused by what he said.

但如果y是离散的，这是否意味着能量函数f(x,y)不是平滑的？

But if y is discrete, does that imply that f, the energy function, f of x y is not smooth?

这是个难题，就像是一年级的数学题一样。

It's a difficult quest these are like first year math questions.

我不太确定，这并不意味着那样。

I'm not so sure it it doesn't imply that.

这不可能，这不可能，这绝对不行。

It can't it can't be it cannot.

嗯，这取决于它是如何定义的。

Well, it depends on how it is defined.

如果它定义在连续空间上，但集合y恰好是离散的，那么它可以是平滑的。

If it is defined on the continuous space, but simply the set y happens to be discrete, then it can be smooth.

但如果它定义在离散集合上，那么我非常确定平滑性就没有意义了。

But if it is defined on the discrete set, then I'm pretty sure smoothness makes no sense.

我也是这么想的。

That's what I think too.

如果是离散的，你只是在点与点之间跳跃，没有任何连续性。

If it's discrete, you're just jumping from point to point with no connection at all.

但函数本身可能在学习某种插值，只是可能不光滑。

But the function itself might be learning some kind of interpolation, but it might not be smooth.

推理方法，当然，这就是为什么离散化更容易，我们不必处理那个问题。

Inference methods, of course, is why it's discrete is much easier and we don't have to deal with that.

基于能量的模型分为条件版本和无条件版本。

There are conditional and unconditional versions of energy based models.

在条件版本中，变量x是始终已知的，而y是需要预测的那个。

In conditional version, the variable x is the one that's always known and y is the one that sort of needs to be predicted.

无条件版本的技巧在于训练机器根据y的其他部分来预测y的某一部分，但我们永远不知道哪部分是已知的，哪部分是未知的。

The unconditional version, the the trick here is to train the machine to predict part of y from part from other parts of y, but we never know which one is known and which one is unknown.

所以这大致捕捉了变量之间的相互依赖关系，正如这里左下角的图示所象征的那样，它代表了这种情况下的能量函数，与k均值对齐，其中训练样本绘制在这条紫色小曲线上。

So this is sort of capturing the mutual dependencies between the the variables as symbolized by the the drawing here on the on the left, on the bottom left that that represents energy function in this case here aligned with k means where the the training samples are drawn on this little purple curve.

因为我认为在几乎所有的机器学习应用场景中，它都是条件性能量模型，因为我们学习的是x和y之间的依赖关系。

Because I think in almost all machine learning use cases, it is a conditional EBM because we're learning a dependency between x's and y's.

是的。

Yeah.

我是说，他哈哈，他立刻就把这跟k均值算法联系起来了。

I mean, he he he goes and immediately makes the connection to k means.

对吧？

Right?

这又回到了我们之前所说的，它最终将涵盖几乎所有的机器学习，而这里我们看到它是如何涵盖k均值的。

And that's again, we were saying before, this is going to ultimately encompass pretty much every all of machine learning, and here we see how it encompasses k means.

所以在k均值中，我仅仅是想通过计算这些均值，创建一个对数据分布中任何点都满意的函数。

So in k means, I simply want to create a function that is happy with any point in the data distribution by doing these means.

但这本质上只是在学习数据的分布，因为我在k均值中没有进行归一化，所以它是一种基于能量的方法，而非概率方法。

But ultimately this is just learning the distribution of the data, but because I'm not normalizing in k means, it is an energy based method and not a probabilistic method.

因此，在这里你可以看到基于能量的方法和概率方法之间的区别。

So here you can see the difference between energy based and probabilistic.

对于概率方法，这可能是高斯混合模型。

For a probabilistic method, this might be a Gaussian mixture model.

但即使高斯混合模型使用的是高斯分布，其全局归一化通常也困难得多。

But the Gaussian mixture models are usually much harder to globally normalize even though it's Gaussian.

所以这仍然相当简单。

So it's still pretty easy.

不错。

Cool.

处理多输出的一种方法是使用潜在变量。

So one way to handle multiple outputs is to is through the use of a latent variable.

如果我们用确定性函数构建机器，那么让机器对单一输入产生多个输出的方法，就是通过潜在变量来参数化输出集合。

So if we're going to build our machine out of deterministic functions, the the way to allow machine to produce multiple outputs for a single input is to parameterize the set of outputs through a latent variable.

因此，典型的架构会类似于这样。

So the typical architecture would look something like this.

你有一个输入变量x，它经过一个预测器，提取出该x变量的表示。

You have an x variable that goes through a predictor that extracts a representation of that x variable.

这个表示与一个潜变量一起通过解码器，从而产生预测。

And that representation together with a latent variable go through a decoder, which produces the prediction.

当你让潜变量在一个集合上变化时，它会使预测在一个相似维度的集合上变化。

When you vary the latent variable over a set, it makes the prediction vary over set of similar dimension.

当然，关键在于找到、构建并训练这个机器，使得潜变量能够代表输出变化的独立解释因素。

And the trick of course is to find, build the machine and train it in such a way that the latent variable represent independent explanatory factors of variation of the output.

所以他的意思是，我们如何能采用一个确定性模型？

So he's saying how can we take a deterministic model?

这有点像变分自编码器，因为想象一下你有一个数据流形，你想设计一种架构，使得潜在变量能够描述该流形的定义域。

This is kind of like variational autoencoders because imagine you have some data manifold and you want to design an architecture such that the latent variable can describe the domain of the manifold.

是的。

Yeah.

这几乎就是变分自编码器的示意图了。

This almost exactly a drawing of a variational autoencoder now.

而且，没错，基本上就是通过这个潜在变量来控制输出的生成方式。

And, yeah, it's basically you have this latent variable that controls how the output is made.

所以你正在生成的是一个完整的流形，这在某种意义上，又是一件非常相似的事情，你只是在学习从潜在变量到流形的这种映射。

So what you're producing is an entire manifold, which in some sense, again, is a very similar thing where you're learning just this mapping from the latent variable to the manifold.

所以那将是一种不依赖于x的条件。

So that would be sort of not conditional on the x.

所以那可能是一个之前的模型，它只是一个关于y的函数f。

So that might be a model before where it's just an f of y.

有意思。

Interesting.

你是说，需要最小化潜变量的信息容量。

You're saying that the information capacity of the latent variable needs to be minimized.

否则，所有的信息都会进入其中。

Otherwise, all of the information would go into that.

那么，编码器接收数据并将其放入这个向量中，然后随机变量是如何在进入解码器之前被添加进去的呢？

So the encoder takes the data and puts it into this vector, and then so how is the random variable added to that before it hits the decoder?

所以，随机变量是从潜变量所描述的分布中采样得到的。

So the random variable is sampled from the distribution that is described by the latent variable.

现在，你无法通过这个进行反向传播。

Now, you can't backpropagate through this.

对吧？

Right?

你无法通过参数化分布并从中采样的操作进行反向传播，但针对某些分布，采样操作等同于——这正是关键所在——你实际上是从一个均值为零、标准差为一的高斯分布中采样。

You can't backpropagate through the operation of parameterizing a distribution and sampling from it, but with certain distributions, sampling from it is the same as, and this is where exactly this comes in, so what you technically do is you sample from a uni, like a one zero mean standard deviation of one Gaussian.

然后你乘以编码器给出的标准差，并加上编码器给出的均值。

And then you multiply by the standard deviation that comes in from the encoder and you add the mean that comes in from the encoder.

而这个操作是可以进行反向传播的。

And that operation you can backpropagate through.

这就是你如何将潜在变量（即来自标准高斯分布的样本）与编码器提供的信息结合起来的方法。

So that is how you combine the latent variable, which is the sample from the Gaussian distribution, the standard Gaussian, with what the encoder gives you.

这被称为重参数化技巧。

It's called the reparameterization trick.

我刚开始读博士时，这个技巧非常重要。

It very big when I started my PhD.

现在，许多基于能量的模型实际上都是使用潜变量构建的，你可以通过边缘化或对潜变量进行最小化，将一个带有潜变量的基于能量的模型简化为不带潜变量的模型。

Now, many energy based models are actually built using latent variables and you can reduce a latent variable energy based model to one that doesn't have one by either marginalizing or minimizing with respect to the latent variable.

因此，推理过程当然是通过同时对目标变量 y 和潜变量 z 最小化基本能量函数来实现的。

So inference of course takes place by minimizing the elementary energy function with respect to both y and z, the variable to be predicted on the latent variable.

你可以通过最小化基本能量函数 e 关于 z，或者通过边缘化（这等价于计算此处所指的某种自由能）来重新定义能量函数 f，即对指数负能量的积分取对数，其中积分在 z 的定义域上进行。

You can simply redefine the energy function f by minimizing the elementary energy function e with respect to z or by marginalizing, which is equivalent to computing some sort of free energy as indicated here, the logarithm of the integral of exponential minus the energy where the integral takes place over the domain of z.

当你确实拥有一个带潜变量的 EBM 时，如果你希望

When you do have a latent variable EBM, if you want

仅用 x 和 y 来表述它，而不使用潜变量，你可以选择对潜变量进行最小化，或者进行边缘化。

to formulate it just in terms of the x and the y without the latent variable, you can either minimize with respect to the latent variable or you can marginalize.

而边缘化就是对 z 的整个定义域进行求和，我们之前讨论过吉布斯分布，就是对 z 的所有可能取值进行求和。

And marginalize is where you kind of sum over the you know, we were talking about the Gibbs distribution earlier, you know, over the domain of of zed.

因此，你实际上是在对所有可能的 z 组合进行求和，然后再进行归一化。

So you're kind of summing over all of the possible combinations with the z and then normalizing.

是的。

Yeah.

所以这里有一个非常简单的例子，好吧，虽然它不完全符合x和y的情况，但如果我们只有一个f(y)，这又回到了k均值和高斯混合模型之间的区别。

So so here, if a very simple example of this that, okay, it doesn't fit the x and y, but if we just had an f of y, would be the, again, the distinction between k means and a Gaussian mixture model.

两者都是聚类模型，但一个是硬分配。

Both are clustering models, but one is just the hard assignment.

所以上面这个例子适用于k均值算法，其中你的潜在变量是数据点所属的聚类。

So the top one would be an example for k means, where your latent variable is the cluster that the data point comes from.

因此当你有一个新数据点，并询问你的能量函数（即k均值函数）对这个数据点的满意度时，它会找到最近的聚类中心，然后到该聚类中心的距离就是能量值。

So when you have a new data point and you ask your energy function, your k means function, how happy are you with this data point, what it will do is it will find the closest cluster center, and then the distance to that one cluster center is the energy.

这就是模型对这个特定数据点的满意程度。

That's how happy the model is with this particular data point.

所以它只能告诉你这些。

So it can only tell you that.

但如果你有一个高斯混合模型，并且有一个新的数据点，你需要遍历该混合模型的每一个分量，并询问它们：你给这个数据点分配了多大的概率密度？

But if you have a Gaussian mixture model and you have a new data point, you need to go through every single component of that mixture and ask them what probability density do you assign to that data point?

然后你需要对所有分量进行积分，才能得到整个模型对你的数据点的看法。

And then you need to integrate across all of them in order to get an answer of what the whole model thinks of your data point.

没错。

Exactly.

所以接着刚才说的。

So just just to carry on from that.

所以我们这里看到的这个 f，现在展示了整个流形。

So this f that we see here, this is now showing us the the entire manifold.

它在一定程度上展示了这个流形在潜在变量所有点上的表现方式。

So it's kind of showing us how this is represented over all of the points of the the latent variable.

那么 f 无穷大和 f 贝塔分别代表什么？

So what's the f sub infinity and f sub beta represent?

在我的例子中，上面是 k 均值的代价函数，下面是高斯混合模型的代价函数，或者在这个情况下是能量函数。

The cost function of in my example, the cost function of k means on top and the cost function of a of a, like, a Gaussian mixture model on the bottom or the the energy function in this case.

它是在对所有这些进行积分。

It's integrating across them.

对吧？

Right?

它是在遍历每一个Z变量，它会获取那个特定Z的能量，并试图用该能量进行加权。

It's it's it's going through each variable of zed, and it's it takes the energy of that particular zed, and it tries to weigh it by that energy.

所以，这更像是对所有不同Z值进行积分。

So, is more like an integration across all of the different Zs.

这就像，也许你可以把它解释为在一场扑克游戏中，你有一个同花听牌，然后你问自己，我应该跟注吗？我应该为了继续游戏而跟注吗？

It's like, maybe you can interpret it as in a game, in a poker game, you want to, you maybe have a flush draw and you ask yourself, should I call this, should I call to in order to continue?

你想要做的是考虑所有可能的未来，也就是所有可能的潜在变量。

What you want to do is you want to think of all the possible futures, which are all the possible latent variables.

所以潜变量就是下一张牌是什么。

So the latent variable is which card comes next.

所以你需要对所有可能性进行积分。

So you want to integrate across all of that.

而对于每一种情况，你都需要问自己：我对这个特定结果有多满意？

And for each of these, you want to ask yourself how happy am I with that particular outcome?

这将是该情况下一个合适的能量函数。

And that would be an appropriate energy function for that.

而如果你玩的是国际象棋，那么你不需要想出任何特定的走法。

Whereas if you played the game of chess, then you don't need to come up with any particular move.

你只想知道对手的最佳走法是什么，以及我的走法与之相比如何。

You just want to know what's the best move my opponent plays and how is my move compared to that.

是的，对于每个Y，你都要找到最小的Z，也就是对手能做出的最佳回应走法，然后你想据此找到自己的最佳走法。

Right, for each Y, you're gonna find the minimum Z, which is the best move your opponent can play in response, you want to find your best move according to that.

这个潜变量的信息容量必须被修正或正则化，这是我稍后要讨论的一个主要问题。

The information capacity of this latent variable must be revised or regularized, and this is a main issue that I will discuss later.

但这可能最终会变得不切实际、难以处理，或者只能通过变分方法进行近似。

But this may turn out to be impractical or intractable or only approximated through variational method.

举个例子，假设潜在变量的数据流形是一个椭圆。

So an example of latent variable, let's say, data manifold is an ellipse.

当我们找到一个数据点时，需要通过找到流形上离它最近的点来计算其能量，从而测量到该流形的距离。

What we when we find a data point, we need to compute its energy by finding the point on the manifold that is closest to it so that we measure the distance to the manifold.

而潜在变量就是导致该点的角度，即流形上最近点的角度。

And the latent variable would be the angle that leads to the point, the closest point on that manifold.

在这个简单案例中，当然你可以明确地写出来，但在更复杂的情况下，我们显然需要找到这个流形，而其参数化并非易事。

Now in this simple case, of course, you can write it explicitly, but in more complex cases, of course, we need to find this manifold and the parametrization is not trivial.

我认为这实际上非常有启发性。

I thought this is really instructive actually.

现在我喜欢数据流形这个类比，因为很多人可能根本不会想到他们的狗狗图片其实是拟合在某种高维流形上的，但实际上确实如此。

Now I like the analogy that there is a data manifold because a lot of people probably don't even think of their pictures of dogs and someone is fitting on some kind of high dimensional manifold, but of of course they do.

尽管这是一个人为构造的例子，但它表明在这个椭圆案例中，你的潜在变量z实际上是一个角度。

And even though it's a contrived example, it's showing that your latent variable z is actually an angle in in this ellipse case.

所以椭圆就是这个流形，而你的潜变量只是一个可以用来将数据点推送到这个流形上任意位置的东西。

So the ellipse is the manifold, and your latent variable is just something that you can use to push your data point onto any position on this manifold.

如果出现一个新的样本，它的能量就简单地取决于它距离流形有多近。

And if a new example comes along, its energy is simply a function of how close it is to the manifold.

所以如果它位于流形上，能量就是零。

So if it's if it sits on the manifold, the energy is zero.

而如果这个新样本位于流形之外，那么能量就会更高。

And if the if the new example sits away from the manifold, then the energy will be higher.

是的。

Yeah.

所以引入这个潜变量，仅仅是为了能够拥有不止一个能量为零的点。

So the introduction of this latent variable is is simply to be able to have more than one point that where the energy is zero.

然后，这些点中的每一个都会被分配一个不同的潜变量。

And and and then each of these points will have a different latent variable assigned.

我确实最喜欢那句话：能量就是到椭圆的距离。

And I I do like the line the most that says energy is the distance to the ellipse.

对吧？

Right?

所以在这种情况下，你的能量函数就是你的点距离椭圆有多远。

So your energy function is how far away is your point to the ellipse in this case.

所以这个隐变量实际上并不像一个椭圆。

So the latent so really though, it's not like an ellipse.

它更像是这样一条弯弯曲曲的线，你知道，不是那种可以通过旋转角度就能持续命中点的东西。

Is it it's like this, like, squiggly line, you know, not something that's like the angle you could just rotate it and keep hitting points.

对吧？

Right?

它会不会像一个波浪形的圆呢？

Wouldn't it be like a like, you know, like a squiggly shaped circle?

嗯，现在要看情况了。

Well, that it now now it depends.

对吧？

Right?

在这种情况下，你的真正能量函数是椭圆。

Your your true energy function is the ellipse in this case.

现在，如果你只有若干数据点，并希望学习一个能量函数，你可能会通过某种插值方法来近似它。

Now if you just have a sample of data points and you want to learn an energy function, what you're going to do is probably approximate that through some interpolation.

于是你得到了一个学习到的能量函数，那么是的，它就会像你所说的那种波浪形的圆。

And then you have a learned energy function and then yes, that would be like you do whatever, your squiggly circle.

没错，但我想他想表达的是，这里的真正能量函数应该是椭圆本身，作为一个流形概念。

It's true, but I guess what he wants to say, your true energy function here would be the ellipse itself as concept, as a manifold.

现在，我想他稍后会讨论我们如何学习能量函数，以及对比方法。

And now, I think later he's gonna talk about how do we learn energy functions and there's the contrastive methods.

然后还有一些方法是对事物进行正则化的。

And then there are the methods where you regularize things.

在这种情况下，如果我已经知道它是一个椭圆，我可以正则化我的模型，规定它只能生成椭圆。

And in this case, if I already know it's an ellipse, I could regularize my model to simply say you're only allowed to produce ellipses.

那么，如果给我这组数据点，它就不会变成一个歪歪扭扭的圆圈。

And then if I'm given this set of data points, it would not turn out to be a squiggly circle.

实际上它会非常接近这个椭圆。

It would actually come to be pretty close to this ellipse.

对我来说，有点像，我在整个深度学习的学习过程中一直看到数据流形这个术语。

For me, kind of like, I've been seeing this term data manifold, like, throughout my entire study of deep learning.

我觉得这张图终于帮我理解了数据流形到底是什么。

I feel like this picture is finally helping me understand what the heck a data manifold is.

所以它就像是连接数据之间的这条高维空间中的路径。

So it's like it's like this path in this high dimensional space that is connecting the data to each other.

是的

Yeah.

流形只是子空间的一种 fancy 说法，但通常子空间会让人联想到线性或其他类似的东西。

Manifold is just a fancy way of saying subspace, but usually subspace is associated with it being linear or something like this.

但流形可以是任何你想要的形式。

But manifold can be whatever you want.

它就是，嗯，这里有数据，那里也有数据，数据在这里到处都是，或者可以出现在任何地方。

Just It be, well, here's data and here's data and data is everywhere here or can be anywhere here.

这是一个美妙的想法，因为流形在各个地方都会出现。

It it's a beautiful idea because manifolds come up all over the place.

例如，如果你对向量进行 L2 归一化，那么它们都会存在于一个叫做单位超球面的流形上。

If if you, for example, do an l two normalization on your vectors, then they all exist on a manifold called the unit hypersphere.

因为如果你仔细想想，它就是所有长度为一的可能向量的集合。

Because if you think about it, it's all the possible vectors that have a a length of one.

在机器学习中，有很多例子都是固定流形的情况。

And there are many examples in machine learning where it's a fixed manifold.

如果你看地理数据，例如，流形就是一个球面，而卷积神经网络（CNN）则是在平面流形上工作。

If you if you look at geo data, for example, the manifold is a sphere, and CNN's work on the planar manifold.

实际上，如果你仔细想想，在某些更高维的空间中，存在一个几乎所有数据都位于其上的流形，而且许多类型的分析和机器学习，甚至像TISNI和UMAP这样的方法，都是从数据所在的流形角度来思考的。

And, actually, if you think about it, in some higher dimensional space, there exists a manifold that almost all data sits on and and many of the types of analysis and machine learning even like TISNI and UMAP think about data in terms of the manifold that it sits on.

是的。

Yeah.

不过我觉得，在高维意义上，它看起来太复杂了，这样思考的意义何在呢？

I I guess though it just seems to me like in the high dimensional sense, it's so complicated that it what's the point in thinking of it like that?

比如，如果它是一个单位超球面，比如说像L2归一化的参数向量，它们具有如此巨大的维度，我只是不明白这样思考的意义何在。

Like, if if if it's a unit hypersphere of say like l two normalized parameter vectors that have like this massive dimensionality, I just don't get what the point is of thinking about it like this.

嗯，这实际上与能量函数有非常紧密的联系，通常你假设你的流形在某种程度上是连续且平滑的，而能量函数基本上就是你到那个流形的距离，是一一对应的。

Well, it's a very close connection to energy functions actually in that usually you assume your manifold is somewhat continuous and smooth, and the energy function is is one to one, basically your distance to that manifold.

所以它之所以极其通用，原因与能量函数极其通用是一样的。

So it is incredibly general for the same reason that the energy function is incredibly general.

它只是简单地说，当数据良好时能量函数表现良好，当数据不佳时则不然。

It simply says the energy function is happy when the data is good and not when the data isn't.

我真的很想找出一些例子，说明这种思路在我们感兴趣的模型上是如何运作的，因为这个流形是人为构造的。

I'm I'm really interested to come up with examples of how this works on the kind of models that we get excited about because this is a contrived manifold.

当我们谈论自然语言处理和BERT这样的东西时，甚至语言本身也位于某个高维流形上，我认为思考哪些例子位于这个流形上、哪些不位于其上，以及能量如何围绕这些例子被提升以进行学习，是非常有启发性的。

When we talk about something like natural language processing and and BERT, even language fits on some higher dimension manifold, and I think it's quite instructive to think of examples that do and don't sit on that manifold, and energy is being pushed up around those examples as a way of learning.

是的。

Yeah.

我的意思是，归根结底，我们可能永远无法直接描述这个流形本身，但我们可以构建这些能量函数，告诉你离流形有多远。

I mean, it comes down to the fact that we probably will never be able to describe the manifold as manifold, but what we can do is we can build these energy functions that tell you basically how far you're away from the manifold.

所以，如果你能构建出这样的能量函数，至少你就可以通过让能量函数变低，以合理的概率接近这个流形。

So if you can build an energy function like this, then at least you can hit the manifold with a reasonable probability by simply making the energy function happy.

不错。

Cool.

我会采用一种方法，找到指向流形上最近点的方向。

I will do the angle that leads to the point, the closest point on that manifold.

当然，在这个简单的情况下，你可以显式地写出它，但在更复杂的情况下，我们需要找到这个流形，而参数化绝非易事。

Now in this simple case, of course, you can write it explicitly, but in more complex cases, of course, we need to find this manifold and the parametrization is not trivial.

好的。

Okay.

那么，我们如何训练基于能量的模型呢？

So how do we train energy based models?

我们需要确保数据样本的能量低于数据流形之外的能量。

What we need to do is make sure the energy for data samples is lower than the energy outside of the data manifold.

为此有两种方法：对比方法会明确降低数据点的能量，同时提高数据流形之外点的能量，或者对这些点的提升较弱。

And there's two types of methods for this contrastive methods that explicitly push down on the data points and push up on other points outside the data manifold or maybe on this, but less strongly.

另一种是正则化架构方法，本质上是限制低能量所允许的空白空间体积，从而自动地像收缩包装一样贴近数据流形，而无需主动提升能量。

And then there is regularizing architectural methods that essentially limit the volume of space in white space that can take low energy and therefore kind of shrink-wrap the data manifold automatically without having to push up.

这里有一些非常有趣的地方。

There's some really interesting things here.

当我们训练这个能量函数 f 时，我们希望对于给定的 y，其能量低于训练集中所有其他 y 的能量。

So when we train this energy function f, we want it to be a lower energy for the given y than all of the other y's in the training set.

这看起来是合理的。

So that seems to make sense.

我们希望函数足够平滑，以便能够使用基于梯度的方法。

We want the function to be smooth so that we can use gradient based methods.

然后他谈到了两类学习方法，这部分变得非常有趣。

And then he talks about two classes of learning methods, and this is where it gets really interesting.

他讨论了对比方法，以及所谓的正则化和架构方法，后者指的是像PCA、k均值聚类等等。

He talks about contrastive methods and so called regularized and architectural methods by which he means things like PCA and k means and and so on.

我认为这里有一个真正的使命。

And there's a real mission here, I think.

他并没有讨论传统的那种监督分类神经网络。

He doesn't talk about traditional kind of supervised classification neural networks.

是的。

Yeah.

所以，首先，它实际上比那还要更普遍。

So so first, it is actually even more more general than that.

你不仅希望让你的数据点比数据集中的其他点能量更低，而是比任何其他点都低，对吧？

You don't just want to make your point have a lower energy than every other point in your dataset, but than any other point, right?

这只是不同方法之间的区别，而且我们之前也打算讨论的，就是他们是如何提出这种对比度量的。

It is just the difference between the different methods, and that was gonna talk about as well, is how they come up with this contrastive measure.

所以如果你想想GAN，被压低的是真实数据集中的点，而被抬高的是生成器能想出来试图欺骗判别器的所有东西，对吧？

So if you think of a GAN, the points that are pushed down are the points in the true dataset, and the points that are pushed up is everything the generator can come up with to try to fool the discriminator, right?

所以它甚至不完全是数据集本身的点。

So it's not even points in the dataset per se.

另一件事是，你实际上可以以这种方式来思考传统的监督学习。

And the other thing is you can actually think of the traditional supervised learning in this way.

所以，如果x是你的输入，y是你的标签，你在做什么呢？

So if x is your input and y is your label, what are you doing?

你是在提升正确的标签，同时压低所有其他的，比如说所有逻辑回归的输出，因为你这是通过一个softmax分类器来运行的。

You're pushing up the label that is correct, and you're pushing down all the other, all the logits, let's say, because you run this through a softmax classifier.

所以，通过提升一个标签，你立刻就压低了其他标签。

So immediately by pushing one label up, you push the others down.

这样你就得到了你的能量。

And there you have your energy.

简单来说，这个的负值就是监督学习的能量函数。

Simply the negative of that is your energy function for supervised learning.

是的。

Yeah.

所以，我们用一种智力框架来讨论许多机器学习算法中的现象，这真的很有趣。

So so it's really interesting that we're using a kind of intellectual scaffold to talk about what's happening in a lot of machine learning algorithms.

正如你所说，你刚才提到的方法中，能量被推高或推低，但杨立昆进一步指出，在这种架构方法中，这是一种略有不同的范式。

And as you say, the approach you just spoke about, energy is being pushed up and being pushed down, but Yann goes on to say that in this architectural method, it's a slightly different paradigm.

也就是说，不是通过推高或推低，而是像收缩包装一样，将功能空间围绕流形包裹起来，这样就不需要推低了。

That's when rather than pushing things up and down, you kind of shrink-wrap the functional space around the manifold so you you don't need to push down.

数据样本的能量低于数据流形之外的能量。

For data samples is lower than the energy outside of the data manifold.

对于这种对比方法，有两种类型：一种显式地推低数据点，推高其他点。

And there's two types of methods for this contrastive methods that explicitly push down on the data points and push up on other point.

我不会逐条阅读这些内容，但有一大类经典方法可以在这个框架下被解释为对比方法或架构方法。

I'm not gonna read through all of this, but the big list of classical methods that can be interpreted in the in this context either as contrastive methods or architectural method.

最大似然法在处理难以归一化的分布时，实际上属于对比方法的一部分。

Maximum likelihood insist in distributions that are not easily normalized is actually a part of contrastive methods.

这正是我首先想要讨论的内容。

And this is what I'm I'm gonna talk about first.

因此，概率方法存在一个问题，你可以对此提出质疑。

So there is an issue with probabilistic methods which you can offer.

这非常有趣，因为我们最近一直在大量讨论对比方法。

This is really interesting because we've been talking a lot about contrasted methods recently.

几周前，我们讨论过用于强化学习的对比无监督表征，本质上是在谈像MoCo和SimCLR这样的方法。

We had the contrasted unsupervised representations for reinforcement learning chap on the other week, and essentially, that was talking about things like MoCo and SimCLR.

这些方法会选取对比样本，对正样本对进行压制，同时对正负样本对进行提升。

And what those approaches do is they take these contrastive examples, and they kind of push down on the positive positive pair, and they push up on the positive negative pair.

通过在函数空间中这样做，你实际上是在学习所有图像所处的流形位置。

And by doing so in the functional landscape, you're kind of learning where the manifold of all of the images are.

那么，为什么我们要如此严格地限制潜在向量z的信息容量呢？

So why do we wanna limit the information capacity of that latent z vector so much?

我认为这并不一定与此相关。

Well, I think that doesn't necessarily come into this.

我认为z的作用是在你希望在整个流形上生成新样本时使用的。

I think the z is when you want to be able to generate new examples across the manifold.

但在Siamese网络或SIMCLR类型的算法中，其实不需要潜在变量。

But I think in in the case of these Siamese networks or the the SIM CLR type algorithms, then there's no need for a latent variable.

我认为这里的关键是，你的能量函数不应显式依赖于潜在变量。

Well, I think the thing here is when you your energy function should not, let's say, depend explicitly on the latent variable.

这意味着，如果你在玩扑克，你的出牌应该独立于下一张牌真正会是什么。

That means if you're poker in game, your move should be independent of what the next card is really truly going to be.

如果你在下棋，你的能量函数仅由你所下的棋步决定，因为z只是对手的最小值。

And if you're in a chess game, your the energy function is just determined by what move you're doing because the z is just going to be whatever is the minimum for the other player.

我认为这仅仅是一种表达方式，意思是这些信息不应该泄露到你的代价函数中。

I think it's just an expression to say that this the the information shouldn't basically leak into your into this cost.

有意思。

Interesting.

它还谈到了BERT，我认为这是目前许多人非常感兴趣的，它正在将流形外的点映射到流形上的点。

It talks about BERTS as well, which is I I think it's of huge interest to so many people at the moment, and that is mapping points off the manifold to points which are on the manifold.

而那些流形外的点，不过是我们已知在流形上的点的噪声版本。

And the points that are off the manifold are just noised versions of ones that we know are on the manifold.

所以轻微地将它们推离流形，这确实引发了关于应该将它们推离流形多远的问题。

So slightly pushing them off the manifold, it does raise questions about how far should they be pushed off the manifold.

而如果你把它们推得太远，是否在某个时刻会变得有问题呢？

And if you push them too far, does that at some point become problematic?

如果噪声强度太高，就意味着离流形太远了，大概是这个意思。

If the noise strength is too high, it's too high off the manifold, that kind of idea.

是不是因为他举了混合模型的例子，这些模型实际上很糟糕，因为它们想要一个像峡谷一样的流形，一旦偏离流形，能量几乎就变得无限高。

Would it because he gave the example of these mixture models actually being really bad because they want to have a manifold which is it's like a canyon and infinitely high in energy almost as soon as you get off the manifold.

我认为这里的总体讨论是，我们需要拥有能更好地泛化到未见数据的模型，并且具有平滑的能量函数，以某种方式描述流形，这种方式能代表我们数据分布中可能合理预期的任何类型数据。

I think that the general discussion here is that we need to have models that generalize better to previously unseen data and have smooth energy functions that describe the manifold in a way that represents any type of data that we could, you know, reasonably expect to see in our data distribution.

但正如我们所知，深度学习的风险在于模型只会内插，而不会外推。

But just as we all know, because of the the perils of deep learning are that the models interpolate, they don't extrapolate.

如果你在流形上制造一个凹陷，你希望的是在流形上形成大量凹陷，从而在你希望流形所在的位置周围形成一个峡谷。

If you make a depression in the manifold, what you want is lots and lots of depressions in the manifold to form a canyon around your around where you want your manifold to be.

但如果凹陷很小，没有在你的流形周围形成一个良好的峡谷，如果问题空间过于稀疏，那么你实际上并没有学到关于流形的任何东西。

But if the depressions are small and they don't form a nice canyon around your manifold, if the problem space is too sparse, then you're not really learning anything about your manifold.

是的，我认为这正是所有这类问题的症结所在。

Well, exact I think that is that is the exact problem with any of these things.

因此，在监督学习方面，一方面，我们确切知道哪些东西需要被推高或压低。

So in supervised learning on the one hand, have the exact knowledge which things are there to push up and down.

这就像随便什么，你的10个类别或者ImageNet中的一千个类别。

It's just whatever, your 10 classes or your thousand classes in ImageNet.

但是，比如说，在完全无监督学习中，你必须考虑每一个可能存在的数据点。

But in, let's say, fully unsupervised learning, you have to consider every single possible data point there is.

如果你只是想学习一个语言模型，就像我们一开始说的，你必须考虑每一个可能的标记序列。

If you just want to learn a language model, as we said at the beginning, you have to consider every possible sequence of tokens there is.

而我们的模型就是没有能力去学习那个。

And our models are just don't have the capacity to learn that.

所以我们试图做的是，我们只是想告诉它们：看，这里有些地方不太对。

So what we're trying to do is we're just trying to tell them, look, here is something that's slightly wrong.

请把它纠正过来。

Please make it correct.

这样它就能做的，是在真正的语言内容和这些被破坏的输入之间形成一道沟壑。

So it can what it can do is it can form that valley between the what's truly in the language and these corrupted inputs.

我们只是希望这就足够了。

And we're just kind of hoping that that's enough.

对吧？

Right?

我们不考虑那之外的任何东西。

We we don't consider anything outside of that.

我们不考虑任何胡言乱语或任何其他东西。

We don't consider any gibberish or anything.

是的。

Yeah.

正如你所说，多远算太远，多近算太近。

As you said, how far is too far and how far is too close.

所以如果我们现在开始只是替换单个单词，那甚至可能是一个原本正确的句子。

So if we now start just replacing single words, that could even be a sentence that is actually correct.

而且，你知道，所以

And, you know, so

让我们把BERT的类比再稍微推进一步。

Let's let's push the BERT analogy just a little bit further.

所以我们已经学习了这个语言流形，并且使用一个BERT模型，它接收比如说500个标记，实际上少于500个，因为通常有一个分隔符标记。

So we've learned this language manifold, and with a BERT model that takes in, let's say, 500 tokens, it's less than 500 because, generally, there's a separator token.

但该流形上的每一个点都是一段有效的语言。

But every single point on that manifold is a piece of valid language.

所以我认为这又回到了课程学习中垫脚石的概念，我们有这些不同的流形，或者说这个高维空间中的子集，我们就像是把它交给模型，然后说：'来，试着理解这个庞大流形或者说整个空间的这个子集。'

So I think it comes back to this idea of stepping stones in curriculum learning and that we have these different manifolds or like subsets of this high dimensional space that we're like giving it and we're like here, try to understand this subset of this massive manifold or like entire space.

所以你试图找到一条路径，将你看到的各个流形彼此连接起来。

So you try to find a path that connects the manifold that you're seeing to each other.

所以，也许这就是为什么你会先进行一个预训练任务，然后接着下一个，再下一个，因为在这个庞大的空间中连接这些流形会更容易。

So maybe that's why like you do one pre training task and then the next one and the next one because it's easier to connect these manifolds in this like massive space.

是的。

Yeah.

我同意。

I I agree.

因为你基本上所做的就是，通过每个预训练任务，你都在寻找一种不同的方式来定义流形上或流形外的点。

Because what you're what you're basically doing is with each pretraining task, you're finding a different way of defining points off and on the manifold.

对吧？

Right?

所以，如果你的预训练任务是遮蔽部分标记，那么你在流形之外找到的点就是那些仅仅缺少一些词语的句子。

So if if you have the pretraining task of masking some of the tokens, the points that you find off the manifold are ones that are just missing some words.

但另一个任务则是交换一些标记。

But then the other task is swapping some tokens.

所以，这为你提供了另一种寻找流形之外点的方法，因为那些无效的负样本——想象一下，如果你的BERT输入仅仅是随机抽取500个标记，然后直接输入进去。

So that just gives you a different way of finding points off the manifold because what's not gonna work is as negative samples have Just imagine if your BERT input was just You just sample 500 random tokens, and then you give that.

然后你会说，你能多好地从这个重建这个训练样本呢？

And then you say, How well can you reconstruct this training example right here from this?

这离流形太远了。

That is too far off the manifold.

你打算怎么做？

How are you gonna do that?

而且，这是一个有趣的点，因为现在你有了你的训练样本。

And also, this is an interesting point because now you have your training samples.

所以从技术上讲，你应该输入并问：你能多好地重建我数据集中的任何一个样本？

So technically what you should do is you should give the input and say, How well can you reconstruct any of the samples in my dataset?

任何一个。

Any.

但因为我们取了一个样本并掩码了其中的词，我们可以相当确定流形上最近的点就是这个特定的样本。

But since we took one and we masked out words from this one, we can be pretty sure that the closest point on the manifold is actually that particular sample.

但如果你离流形更远，情况就不再是这样了。

But that's no longer the case if you go further off the manifold.

你不再知道该用哪个训练样本了，因此在训练这个模型时会遇到问题，因为你不清楚应该让它朝哪个方向学习。

You don't know anymore which of your training so you're gonna have a problem with training this thing because you don't know what to train it towards.

你必须让模型针对数据集中的每一个样本进行训练。

You would have to train it towards every single thing in your dataset.

我对BERT也有些困惑，因为它有两个目标，第一个我认出是掩码或去噪自编码器，因为它说的是：这里有一个输入，我会随机掩码掉一些词，然后让你重建它。

I'm also a bit confused with BERT because it has two objectives, and the first one I recognize as being the masked or the denoising autoencoder because it's saying here is an input, I'm going to mask out some random words, and then I want you to reconstruct it.

这是一种情况。

That's one thing.

而下一个任务是下一句预测，这完全是另一回事。

And then the next thing is the next sentence prediction, which is completely different.

这根本不是自编码器。

That's not an autoencoder.

所以你是在动态地在这两个任务之间切换，难道这两个任务不会在内部构建出完全不同的流形吗？

So you're switching between those two tasks dynamically, and wouldn't those two tasks build a completely different manifold internally?

也许下一句预测任务会把这些中间标记都抓取过来，以某种方式把它们全部聚拢在一起，我不太确定。

Maybe it the next sentence prediction task grabs all these intermediate tokens, pushes them all up together in a way I'm not sure.

但也许关键在于形成，因为我的意思是，我只是在三维空间里思考这个问题。

But maybe it's about forming because you're like I mean, I'm just thinking about it in, like, a three-dimensional space.

就像，有一个立方体，立方体里有数据的子集。

Like, there's this cube and there's subsets of data in the cube.

如果我能抓住一大堆点并把它们一起移动，也许这比单独移动更有帮助，如果这说得通的话。

If I get to grab a whole bunch of points and move them together, maybe that helps than just moving individually, if that makes any sense.

是的。

Yeah.

我觉得...我觉得这个类比相当好，因为...那么，再次说明，作为输入，好吧。

I think I think I think the the analogy is pretty good because so, again, what As an input Okay.

我们的输入空间现在是两个连续的句子，对吧？

Our input space is now two consecutive sentences, right?

我们已经讨论过，掩码语言模型只会告诉你这两个句子中哪个...这样你就可以将这两个句子重构为一个。

We've already discussed that the masked language model will simply tell you which of the two so you can reconstruct these double sentences as one.

但现在你有了另一种从流形外取点的方式，那就是构建一个非连续的双句。

But now you have a different way of making points off the manifold and that is to construct a non consecutive double sentence.

所以这只是另一种构建偏离流形点的方式。

So it's just another way of constructing points that are off the manifold.

也就是说，你在学习一个能量函数，它表明如果两个句子相互衔接，你应该非常满意；如果两个句子不连贯，你就不应该满意。

So you're telling, you're learning an energy function that says you should be very happy if two sentences follow each other, and you should be not happy if two sentences don't follow each other.

因此，你正在为那些偏离流形的特定点学习一个能量函数，将这些点推高，同时将真正的双句对推低。

So you're learning an energy function for those particular points off of the manifold, push those up, and push the true double sentences down.

并且你希望你的模型能够学到关于语言的一些有意义的东西。

And you hope you hope that your model will learn something meaningful about language.

但是，你不觉得这两个任务之间存在分歧吗？还是你认为它们

But don't don't you think that there is a divergence in the in the two tasks, Or do you think that they

是的。

yeah.

对。

Yeah.

这正是关键所在。

That's that's the point.

对吧？

Right?

关键是，每增加一个任务，目的都是找到更多位于流形之外的点。

The point is with every additional task you introduce, the point is to find more ways to find points off of the manifold.

但在哪种情况下，你会认为一个任务能帮助另一个任务表现得更好？还是你觉得它们能互相促进？

But in which case do you think that one task makes the other one work better, or do you think that they help each other work well?

是的。

Yeah.

它们确实会共享特征。

They they do feature sharing.

最终目标就是让特征得以共享。

That's the ultimate goal is that that that features are shared.

语言特征中有一些东西能够同时帮助这两个任务。

That there is something about language features that will help both tasks.

通过平等对两个任务进行梯度下降，这些特征可能会发展得更好。

And by doing gradient descent on both tasks equally, these features might develop better.

然后，相同的特征会迁移到你随后进行微调的其他任务上。

And then the same features will transfer to other tasks that you then fine tune on.

这正是我首先将要讨论的内容。

Which is what I'm I'm gonna talk about first.

概率方法存在一个问题，当然，你几乎总是可以通过吉布斯分布将一个能量函数转化为概率分布。

So there is an issue with probabilistic methods, which you can of course almost always turn an energy function into a probability distribution using a Gibbs distribution.

你可以使用最大似然估计，但如果你想获得密度估计，基本上就必须使用最大似然。

You can do maximum likelihood, but you basically have to do maximum likelihood if you want estimates of densities.

问题是，估计密度并不一定是个好主意，因为通过最大似然，系统会试图给数据点赋予尽可能低的能量，而给数据流形外的点赋予尽可能高的能量，这会导致系统形成极其深而窄的峡谷。

The problem is that estimating densities is not necessarily a good idea because by doing maximum likelihood, what the system wants to do is give the lowest possible energy to data points and the highest possible energy to point two points just outside of the data manifold, which leads the system to create to creating extremely deep narrow canyons.

而这些峡谷对于推理来说并没有特别的用处。

And those are not particularly useful for inference.

我们需要的是平滑的函数。

We need smooth functions.

因此，这些函数需要通过先验或其他正则化方法进行正则化。

So those functions would need to be regularized, for example, by a prior or another regularizer.

所以我的第一个问题是，尽管我看到了峡谷，但在我看来它很平滑。

So my first question is even though I see the Canyon, but it looks smooth to me.

我想他是想用这些超级陡峭的墙壁来演示，但我明白他的意思，不过看起来这似乎只是那个温度参数的一个特性，这些东西最终会变得多平滑、多陡峭。

I think he wants to demonstrate with these super steep walls, but I get what he's saying, but it seems to be just, you know, a property of that of that temperature parameter, how smooth and how steep these things are really gonna turn out to be.

我认为这些东西的主要问题在于，是的，你必须进行全局归一化，这意味着输入空间中的每一个点都必须被赋予某个非零概率，这可就难办了。

I think the the main problem with these things is that, again, yeah, you have to globally normalize, which means that every single point in your input space must be assigned some nonzero probability, and good luck.

因为它在最后一个要点上提到了，但既然如此，为什么要使用概率模型呢？

Because it it says on the last bullet point, but then why use a probabilistic model?

所以我认为，显然，Yann LeCun 并不属于概率模型阵营。

So I think, clearly, Yann LeCun is not in the probabilistic models camp.

是的。

Yeah.

可能不是。

Probably not.

对。

Yeah.

这只是因为你不想对分母上所有那些项进行求和吗？

Is that just because you don't wanna have to sum over all those all the terms on the bottom though?

如果你有一个概率分布，并且你试图将大部分概率分配给其中一个，而其他部分则相应减少，这就会造成一个额外的深谷，我想。

If you're if you have this probability distribution and you're, you know, trying to assign a bunch of probability to one and the others in that kinda way, it causes this extra, like, deep valley, I guess.

是的。

Yeah.

我想通常发生的情况是，当你的数据点在这里时，它会尽量将大量概率质量分配给该点，这会自动减少其他地方的概率质量，包括该数据点周围的区域。

I guess what usually happens is it just it will just when your data point is here, it will just try to assign as much probability mass to that point, and that will automatically take away mass from other places, including things around that data point.

所以，如果你没有正确正则化，最终只会得到在数据点上直接分布，而点与点之间什么都没有。

So what you're if you don't regularize properly, you're just going to end up with like a direct distribution everywhere on your data points and nothing in between.

但同样，我认为这取决于这个温度参数。

But again, I think it depends on this temperature thing.

但那样我们就失去了实际估计密度的优势，我们不再估计密度了。

But then we lose the advantage of actually estimating densities, we're not estimating densities anymore.

所以为什么不彻底放弃隐私框架，直接用能量函数学习依赖关系呢？

So why not throw away the privacy framework altogether and just learn dependencies with energy function.

因此，放弃概率框架在某种程度上让我们能更自由地选择目标函数。

So throwing away the probabilistic framework sort of allows us to use more freedom in sort of deciding on what objective function to use.

目标函数的特征是：它必须是数据点能量的增函数，同时是数据流形外点能量的减函数，并且可能通过某种依赖于这两类点的间隔来实现。

The characteristic of the objective function is that it must be an increasing function of the energy of data points and the decreasing function of the energy of points outside the data manifold and perhaps through some sort of margin that depends on those two points.

所以他是在说，让我们放弃概率框架。

So he's saying let's throw away the probabilistic framework.

现在他引入了机器学习领域中常见的损失函数类型。

And now he's introducing the kind of loss functions that we see across the machine learning world.

所以像合页损失，以及 presumably 平方损失之类的函数都会包含在内。

So things like hinge loss and presumably things like squared loss would be in there.

他所说的非常直观：如果样本远离数据流形，那么它的能量应该更高。

And he what he's saying is quite intuitive that it should be that if the example falls away from the data manifold, then it should have a higher energy.

是的。

Yep.

没错。

True.

但是，我们能否

But can we

我们能不能先退一步，把话说清楚，这是在和概率方法做比较。

just, like, backtrack and just, like, be very clear about this is in comparison to probabilistic methods.

具体区别到底在哪里？

What would where exactly is the difference?

区别仅仅在于不需要对所有可能的'为什么'进行求和吗？

Is it just this idea of not summing over all of the possible whys?

这就是关键区别吗？

Is that the key distinction?

是的。

Yeah.

事实上，你失去了对数据点可能性进行数值预测的能力。

The fact is here, you lose the ability to make a numerical prediction about how likely a data point is.

你只是将其与其他数据点进行比较。

You simply compare it to others.

现在你只能做到这些了。

That's all you can do now.

这是否触及了频率学派与贝叶斯学派的核心区别？

Does this get to the nub of Frequentist versus Bayesian?

贝叶斯方法是否包含这样一种观念：将事物置于所有可能发生的情况的背景下，从而能够讨论置信度和可能性等？

Is about the Bayesian approach has this notion of seeing things in the context of all of the possible things that could occur, therefore, you can reason about confidence and and its likelihood, etcetera?

我不确定。

I'm not sure.

我认为频率学派仍然会归一化他们的分布，但可能可以类比为：概率模型总是将结果与输入的全局空间相关联。

I think a frequentist would still normalize their distributions, but it could maybe be compared to that in that a probabilistic model will always put it in relation to the global space of of inputs.

而能量函数只是给你一个数值。

Whereas energy functions just give you the number.

例如，对于这些能量函数，我不会详细展开，但多年来它们已被用于各种场景，比如科学网络、对称学习、排序或嵌入。

For instance, for those energies, I'm not gonna go through the details, but they've been using various context over the over the years, either for things like science networks, symmetric learning, or for ranking or or or embedding.

而最近，出现了一种目标函数，它使用的不是一对点，而是一个完整的集合。

And then more recently, there's been a objective function that use not just a pair of points, but a whole set.

所以很明显，自监督学习如今已有非常成功的应用，尤其是在自然语言处理领域。

So obviously, there's very successful applications of self supervised learning today, in particular in the context of natural language processing.

大家都知道BERT。

Everybody knows about the BERT.

在此之前，有一类技术使用了某种形式的去噪自编码器，即你输入一个数据，将其破坏，然后训练系统区分干净版本和破坏后的版本。

This was preceded by the sort of techniques, which used a form of denoising autoencoder where you take an input, you corrupt it, and then you train the system to distinguish between the clean version and the corrupted version.

在去噪自编码器中，你训练系统将破坏后的版本映射到干净版本。

In denoising auto encoder, you train the system to map corrupted version to clean versions.

因此，现在对于被破坏点的重构误差，就是被破坏点与这个干净版本之间的距离。

Therefore, now the reconstruction error for corrupted points is the distance between the corrupted point and this clean version.

所以，你自动获得了一个能量曲面，它会随着到流形（如右下角所示）的距离增大而增长。

And so you have automatically an energy surface that grows with a distance to the manifold as represented here on the bottom right.

这代表了由去噪自编码器产生的能量函数的基本梯度场的矢量场。

This represents the vector field of the basically, the gradient field of the energy function produced by denoising autoencoder.

所以这真的很有趣。

So this is really interesting.

我认为正如我们之前所说，我们还没有真正可视化像BERT这样的东西。

I think as we were saying before, we haven't really visualized something like BERT.

当然，这不是BERT的流形。

Of course, this isn't the manifold for BERT.

这只是一个简单的流形，但它确实准确地展示了去噪自编码器在这里发生的情况。

This this is just a a simple manifold, but it does show exactly what's happening here with this denoising autoencoder.

我们只是在寻找偏离流形的例子，并与位于流形上的例子进行比较。

We're just finding examples that are off the manifold and comparing to ones that are on the manifold.

如果它真的是BERT的流形，那可就太有意思了。

It would be so funny if it actually is the manifold of BERT.

就像，如果未来某个数学家证明人类语言是一种螺旋结构，那杨立昆简直会被奉为预言之神。

Like, if some mathematician in the future proves that human language is a spiral, it just would be like Yann LeCun would be treated as a god for predicting.

我也就是在这儿瞎想想。

I'm just just imagining things here.

是的。

Yeah.

对。

Yeah.

归根结底，就是这么回事。

Ultimately, it's exactly that.

对吧？

Right?

所以你需要找到一种方法，将点抛出流形，然后自动将它们映射回来，这就得到了能量函数。

So you want to find some method of throwing points off the manifold and then mapping them back on automatically gives you this energy function.

而且你找到将点从流形上移除的方法越多，你的能量函数就会越好。

And the more ways you find of knocking points off the manifold, the better your energy function is going to be.

可能值得再讨论一下这张图。

And it it might be just be worth talking about this diagram again.

所以他使用了演示开头提到的视觉概念，也就是自监督。

So he's using the visual concept from the beginning of the presentation, which is the self supervision.

所以在这个空间中，x 有点不连贯，然后它进入一个确定性的预测器解码器。

So the the x is kind of disjointed in this space, and it goes into a deterministic predictor decoder.

这个c是什么？

What's this c?

它是在一个红色方框里。

Is this is in a red square.

那是损失函数。

That's the loss.

在...在这个情况下，它接收的是y hat，也就是波浪线符号。

In in the in this case, it's the so it it takes the y hat, which is or tilde.

我能从这里看到它。

I can see it from here.

它接收的是BERT预测的y。

It is it takes the y that BERT predicts.

对吧？

Right?

BERT会给出它认为你在移除所有标记之前的句子是什么，然后将其与你实际移除所有标记之前的句子进行比较。在BERT的情况下，它会对每个标记应用一个分类损失。

BERT says here is what I think the sentence was before you took out all the tokens, and it compares it to the actual sentence before you took out all the tokens, and then it just applies in Bert's case, it applies a classification loss on each token.

我明白了。

I see.

所以如果解码器在流形上预测了某个内容，那么它会降低能量吗？

So if the decoder has predicted something on the manifold, then it will push the energy down?

如果解码器正好在流形上预测了那个内容，那么损失就会是零。

Well, if the decoder has predicted that exact thing on the manifold, then the loss will be zero.

对吧？

Right?

在这种情况下，你正在学习你的预测器和解码器函数，以使能量变低。

And you're learning in this case, you're learning your predictor and decoder function to make the energy low.

是的。

Yeah.

所以你可以看到这两个输入。

So so you can see the two inputs.

所以它说这是提取的文本。

So it says this is a of text extracted.

所以这是被破坏的版本，然后我们希望预测器和解码器能预测出未被破坏的版本。

So this is this is being corrupted, and and then we want the predictor and the decoder to predict the non corrupted version.

如果比较器发现它们是相同的，就会在流形上该点的内部表示处降低能量。

And if the comparator finds them as being the same, it will push down the energy on the internal representation at that point in the manifold.

是的。

Yes.

所以这就像是注入了一个先验，比如：这是你可以将点抛出流形的方式。我在想，真正困扰我的是它如何映射到这个流形里，因为你是在同时训练特征，就像我们讨论的把它推出这个流形。

So it's like we inject this prior of, like, here's how you can throw points off the manifold, and then I guess, like, what's really bothering me with thinking about this is how it is mapped into this manifold because you're training the features as you're simultaneously, like, talking about pushing it off this manifold.

所以它是在学习这个流形，通过被告知什么在上面、什么不在上面，在整个训练过程中发生了如此大的转变。

So it's learning this manifold as it being told what's on or off of it transforms so much throughout the training.

对吗？

Right?

这里涉及多个流形吗？

Are multiple manifolds going on here.

对吗？

Right?

我认为在这种情况下，在鸟类的例子中，我们只是在考虑自然语言的流形，特别是双句的流形，但本质上是自然语言的流形。

I I think in in this case, in the bird case, it we're just thinking about the manifold of natural language of and specifically double sentences, but just the manifold of natural language.

我们正在把它从流形中移开。

And we're throwing it off.

我们只是通过破坏它，就把某些东西从流形中移开了。

We're throwing something off the manifold by simply corrupting it.

然后我们学习这个解码器函数，将它映射回流形。

And then we're learning this decoder function to map it back onto the manifold.

能量函数衡量的是与流形的距离。

And the energy function measures the distance to the manifold.

对吧？

Right?

在这种情况下，我们不需要学习这个能量函数，因为它已经给定了。

And in this case, we don't have to learn this energy function because that's a given.

能量函数就是 y 和 y hat 之间的损失，但现在我们正在学习这个解码器和预测器模型，以最小化这个能量函数。

The energy function is the loss between the y and the y hat, but we're now learning this decoder and predictor model in order to minimize that energy function.

所以就像我一开始说的，并不总是需要学习能量函数本身。

So it's as I said at the beginning, it's not always that you learn the energy function per se.

你学习的是那个距离，但有时你实际上学习的是最小化距离的那个东西。

You learn that distance, but sometimes you actually learn the thing that minimizes the distance.

把这些算法看作是表示学习算法很有意思，但延伸开来，它们也是流形学习算法。

It's interesting to think of these algorithms are representation learning algorithms, but they are also manifold learning algorithms by extension.

大多数时候，我们并没有限制可以学习的流形类型。

And most of the time, we are not constraining the type of manifold that could be learned.

我们在这里约束的是将事物从流形上移除的方式。

What we're constraining here is the way we throw things off the manifold.

也就是说，我们这样做是因为如果不加以约束，我们就无法知道流形上原本是什么。

That is that is and and we're doing that because if we wouldn't constrain that, we would have no idea of what was on the manifold.

因为我们能在这里训练任何东西的唯一原因是，我们知道这是流形上的数据样本，并且最接近那个被破坏的y。

Because the only reason we can train anything here is because we know this is the data sample that is on the manifold and that is closest to the y, to the corrupted one.

但是，它学习的流形在训练过程中发生变化，这一点是否相关呢？

But would it is it relevant that the the manifold that it's learning is changing during training?

是的，我明白。

Well Yeah.

我懂了。

I get it.

你指的是哪个流形？

On which manifold you tell me.

情况真的是这样吗？

Well, is that the case?

当BERT模型收敛时，是否存在一个流形？

Is is it that when when the BERT model converges, there is a manifold?

但这个流形会变化吗？

But does that manifold change?

确实存在一个流形。

There is a manifold.

自然语言的流形永远不会改变。

Manifold of natural language never changes.

对吗？

Right?

那只是自然语言的流形，而我们的训练集提供了这个流形上的数据点。

That is just the manifold of natural language, and we have data points on this manifold given in our training set.

我们所做的就是知道这些点位于流形上，而我们能做的只是稍微扰动它们。

And all we're doing is we know that these points are on the manifold, and all we can do is we're throwing them off a bit.

我们实际上并不确定，但我们认为我们是在扰动它们。

We don't actually know, but we think we're throwing them off.

我们相当确定，因为我们屏蔽了这些词，这应该会给你一些不在流形上的东西，然后我们学习将它们映射回去。

We're pretty sure because we mask out these words, and that should give you something that's not on the manifold, and then we're learning to map them back.

但在这一点上，我同意你的看法。

But on that point, I agree with you.

很可能存在某种概念上的流形，能完美地代表英语语言。

There may well be some notional manifold which perfectly represents the English language.

但神经网络中的每一层都将一个流形转换为另一个，我们可以很容易地将语言转换为一个球形流形，只需进行L2归一化即可。

But we could every single layer in a neural network transforms one manifold into another, and we could quite easily transform language into a spherical manifold just to just doing an l two normalization.

那么BERT的所有输入都将存在于这个球面上。

So then all of the inputs to BERT would exist on on the sphere.

所以我想说的是，语言可能存在于许多不同的流形上，而对于BERT这样的典型神经网络，流形在训练过程中会发生变化和演化吗？

So I guess what I'm saying to you is that there are many possible manifolds that language could exist on, and and given a typical neural network for BERT, does the manifold change and evolve during training?

我认为确实如此。

I think it does.

BERT所代表的那个肯定是这样的。

The one that it that the BERT represents for sure.

是的。

Yeah.

没错。

Yeah.

是

Is

是否也值得考虑它在通过网络时也在一定程度上缩小了流形的维度？

it also worth thinking that it's shrinking the dimensionality of the manifold as it goes through the network too a little bit?

所以，也许把它从流形上抛出去，是在试图告诉我们，在压缩过程中要保持它们某种程度上的重叠？

And so maybe, like, throwing it off the manifold is trying to tell it as we compress it, keep them overlapping in some way?

这是个好问题。

That's a good question.

BERT 在一开始能生成的东西，那个空间是更高维的，还是只是不同？

Is the things that BERT can produce at the beginning, is that space somehow higher dimensional, or is it just different?

对吧？

Right?

这是个很好的问题，真的是个好问题。

It is a good it is a really good question.

所以，如果这是它应该学习的语言流形，那么在开始时，它是在学习输出所有可能的东西，还是只是在学习输出其他东西，而我们只是在这么做？

So if this is the manifold of language that it's supposed to learn, at the beginning, is it learning to output every possible thing, or is it just learning to output something else and and we're just kinda doing that?

作为我，我完全不知道。

As a I have no idea.

谁知道呢？

Who knows?