Gradient Descent - Artificial Intelligence for Robotics Udacity Japanese English

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この勾配降下法についてはフォーラムでかなりの混乱が見受けられました

So we saw a lot of confusion in the forums about what was going on

出された方程式はブラックボックスでしたが

with this gradient descent. The equations that were given to you were presented as a black box,

中身がどんなものか開けて確認してみたいと思います

and I'd like to open up that black box and see what's going on inside.

勾配降下法について理解しようと Wikipediaをご覧になった方は最初にこの式を目にしたでしょう

For those of you who looked at Wikipedia to try to understand what's going on with gradient descent, the first formula you would have encountered was this one.

この方程式は勾配降下法を表しています

This equation describes gradient descent.

最初は少し怖気づきますよね

Now this is a little intimidating at first.

これらのパラメータは何でしょうか？

What do all of these parameters mean?

b＝a－γに価値関数の値aでの勾配を掛けます

b = a - gamma x the gradient of the function of valued at a.

この式で何が行われるのでしょう

Let's figure out what's going on here.

勾配降下法の話に入る前にまずはこの質問に答えてみてください

Before we get into gradient descent, let's first answer the question,

勾配とは何でしょうか？

what is a gradient?

質問に答えるために目隠しした登山者の助けを借ります

To help me answer this question, I'm going to get the help of my blindfolded hill climber.

目隠しした登山者がこの丸みを帯びた小山を登っています

Here's our blindfolded hill climber, and he's climbing this rounded hill.

彼はできるだけ少ない歩数で頂上までたどり着きたいと考えています

He wants to get to the top, and he wants to do it in as few steps as possible.

どんな方法があるでしょうか？

What is his method going to be?

山が急勾配ならば歩幅を大きくとることができます

Well, as long as the hill is steep, he knows he can afford to take a really big step

まだ頂上に近づいていないからです

because he's still not really close to the top.

勾配が緩やかになって平坦に近くなれば歩幅をだんだん小さくします頂上を通り過ぎて山の反対側に出てしまっては困るからです

As the hill smoothes out and becomes more level, he's going to want to take smaller and smaller steps because he's going to be afraid of overshooting the top and winding up on the other side of the hill.

数学的に表すとどうなるでしょう？

How can we describe this mathematically?

私たちは勾配を利用しますがその場合登山者は山の頂上を目指しています

The way we're going to do that is by using the gradient, and when we do that, our climber is going to get to the top of the hill.

この山を横からではなく上から見てみましょう

Let's look at this hill from a top-down view, instead of from the side.

等傾線を描きましょうこれらの線はすべて同じ高度を表しています

I'm going to draw these isocline, so these lines all represent lines of the same altitude.

等高線が集まっているのがわかります

We can see they come together.

最初はとても近接していますが内側にいくにつれてどんどん離れていきます

They're really close together in the beginning and become further and further apart

山の傾斜がだんだん緩やかになっていくからです

as we go inside, and that's because the hill is getting less and less steep.

どの地点においても勾配はベクトルです

So at any point, the gradient is a vector.

この地点の勾配を選択すれば

If I took the gradient at, let's say, this point.

最急降下の方向を指すベクトルが得られます

I would get a vector pointing in the direction of steepest descent,

ベクトルの長さは山の傾斜の程度によって決まるので

and the length of that vector would depend on exactly how steep the hill is.

おそらくこの地点は山のどこかの場所と一致します

So at this point, which maybe corresponds to somewhere down here on the hill,

この山は傾斜が急なのでベクトルはかなり長く直接上り方向を指します

the hill is quite steep, so our vector is quite long, and it points directly uphill.

この場所の勾配を選んでもまだ上り方向を指すでしょうが

If I was over here and taking the gradient, I would still be pointing uphill,

ベクトルはそれほど長くありません

but the vector wouldn't be quite as long now

もう上まで登っているのでそれほど傾斜はないからです

because now we're up here, and the slope isn't so great,

傾斜が減ると勾配も小さくなり目隠しした登山者の歩幅も小さくなります

and with reduced slope comes reduced gradient and a reduced step size for our blindfolded hill climber.

勾配降下法は目隠しをした登山とよく似ています

Gradient descent is very similar to blindfolded hill climbing.

次は山ではなく谷に入って谷底を探しましょう

Now instead of climbing up a hill, we're climbing into a valley and trying to find the bottom of it

関数を最小化するのです

because we want to minimize a function.

最小化はこのマイナス記号でエンコードされ勾配を足すのではなく引き算します

That minimization is encoded in this - sign so instead of adding the gradient,

これらの変数の意味を説明します

we're going to subtract it. Let's talk about what each of these variables mean.

aは現在の位置です

A is our current position.

私たちが登山者ならここから出発します

If we were the hill climber, it would be right here where he starts off.

bは次の位置なので登山者がどのように進むかが決まります

B is going to be his next position so this is how the hill climber decides where to go next.

この勾配の項は最急上昇の方向を示します

This gradient term tells us the direction of steepest ascent.

このマイナス記号は反転しますでは実際に最急降下の方向を探索してみましょう

This - sign flips that around. It says okay, let's actually find the direction of steepest descent.

γは重み係数です　式の説明はこれで終わりです

This gamma is just a waiting factor.

山を下りてこの式を新しい位置として使ってみましょう

That's all this formula is doing. It's saying, let's take a step down the hill and use that as our new equation-- our new position. Sorry.

詳しい説明には少し微積分を使わなければいけません

Now to elaborate on this, we're going to need to use a little bit of calculus,

微積分を知らなくても怖がらず

so if you don't know calculus, don't be intimidated.

理解できるようにベストを尽くしてください

Try your best to follow along, and if you can't, that's okay too.

これは勾配降下法で行われることの本質を捉えていると思います

I think this captures the essence of what's going on with gradient descent.

まず最小化しようとしている関数の正体を理解します

First, we have to figure out what exactly is the function we're trying to minimize.

講義の目的として最小化しようとした関数はyiでした重み係数αに（xi－yi）²を掛けたものと

For the purposes of the lecture, the function we're trying to minimize was a function of yi, and it was equal to some waiting alpha x xi - yi squared,

重み係数βに（yi－yi＋1）²を掛けたものを足します

+ some waiting beta x yi - yi +1 squared,

y座標についても前のy座標と同様に計算するので

so the next y coordinate, and we also do the same for the previous y coordinate,

yi－1になります

これを勾配降下法を使って最小化したいと思います

We want to minimize this, and we're going to use gradient descent.

勾配降下法では十分に緩やかだと言える下部の傾斜が得られるまで

Well, gradient descent says that, we should just iterate over this process until we get

このプロセスを繰り返す必要があります

to a sufficiently shallow slope that we're confident are in the bottom.

最小値が見つかれば1歩ごとに計算していきます

We've really found a minimum. So at each step, we're going to follow this.

新しい位置bはyi’になります

B, our new location becomes yi prime.

これは前の位置yiと等しくなりこの勾配を引くと

That's going to equal our old location, yi, and then - this gradient,

ここの勾配はyiに対する微分になります

and the gradient here is just going to be the derivative with respect to yi.

これはとても簡単な微積分なので心配しないでください

If you don't know calculus again, don't worry about it, and if you do, this is a fairly simple derivative.

簡略化して書きます　途中のステップをすべて説明するわけではありません

I'm going to writie it simplified. I'm not going to show you all the intermediate steps.

これをきれいにするとyi’＝yi＋2α（xi－yi）に 2β（yi＋1＋yi－1－2yi）を足したものになります

But once we do this out, we find that yi prime = yi + 2 alpha x (xi - yi) + 2B(yi + 1 + yi - 1 - 2yi).

皆さんが講義で与えれた方程式とほぼ同じです

This almost, almost looks like the equation that you were given in class.

少し問題があるのはこれら2つなので

There's a little problem, and that's these 2s.

この2つを消してしまいます

I'm going to do something where I just erase them.

こんなことしてはいけないと思うかもしれませんが

You may be saying, hey, you're not allowed to do that,

これらのパラメータは元々α／2と β／2だったことにすれば何も問題ありません

but let's just pretend I'd originally called these parameters alpha over 2 and beta over 2, then everything will work out just fine.

これが勾配降下法です

That's gradient descent.

このような場合は最新のステップから生じて同時に更新されます

That's where this occasion comes from in our update step. This would be simultaneous update.

勾配降下法のもう1つのよい利用方法です

Another good way of using gradient descent.

この解説では実装方法に触れていませんが問題ありません

It's not how we implement it in lectures, but that's okay.

お役に立てれば幸いです

I hope this was helpful. Good luck!

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