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Subtitle preview

First 50 subtitle cues from .().eng.srt

50 cues

#100:00:00,000 --> 00:00:04,035
In the previous video, we gave a mathematical definition of gradient
#200:00:04,035 --> 00:00:09,046
descent. Let's delve deeper, and in this video, get better intuition about what the
#300:00:09,046 --> 00:00:14,070
algorithm is doing, and why the steps of the gradient descent algorithm might make
#400:00:14,070 --> 00:00:20,063
sense. Here's the gradient descent algorithm that we saw last time. And, just
#500:00:20,063 --> 00:00:26,042
to remind you, this parameter, or this term, alpha, is called the learning rate.
#600:00:26,042 --> 00:00:32,044
And it controls how big a step we take when updating my parameter theta J. And
#700:00:32,044 --> 00:00:41,035
this second term here is the derivative term. And what I want to do in this video
#800:00:41,035 --> 00:00:47,035
is give you better intuition about what each of these two terms is doing and why, when put
#900:00:47,035 --> 00:00:53,007
together, this entire update makes sense. In order to convey these intuitions, what
#1000:00:53,007 --> 00:00:58,046
I want to do is use a slightly simpler example where we want to minimize the
#1100:00:58,046 --> 00:01:03,002
function of just one parameter. So, so we have a, say we have a cost function J of
#1200:01:03,002 --> 00:01:07,029
just one parameter, theta one, like we did, you know, a few videos back. Where
#1300:01:07,029 --> 00:01:11,091
theta one is a real number, okay? Just so we can have 1D plots, which
#1400:01:11,091 --> 00:01:16,041
are a little bit simpler to look at. And let's try to understand why gradient
#1500:01:16,041 --> 00:01:23,093
descent would do on this function. [sound]. So, let's say here's my function.
#1600:01:24,065 --> 00:01:31,069
J of theta one, and so that's my, and where theta one is a real number. Right,
#1700:01:31,069 --> 00:01:39,020
now let's say I've initialized gradient descent with theta one at this location.
#1800:01:39,020 --> 00:01:46,098
So image that we start off at that point on my function. What gradient descent will
#1900:01:46,098 --> 00:01:56,093
do, is it will update. Theta one gets updated as theta one minus Alpha times DD
#2000:01:56,093 --> 00:02:04,069
theta one J of theta one, right? and oh an just as an aside you know this, this
#2100:02:04,069 --> 00:02:11,063
derivative term, right? If you're wondering why I changed the notation from
#2200:02:11,063 --> 00:02:16,013
these partial derivative symbols. If you don't know what the difference is between
#2300:02:16,013 --> 00:02:20,052
these partial derivative symbols and the dd theta don't worry about it. Technically
#2400:02:20,052 --> 00:02:24,049
in mathematics we call this a partial derivative, we call this a derivative,
#2500:02:24,049 --> 00:02:28,029
depending on the number of, of parameters in the function J, but that's a
#2600:02:28,029 --> 00:02:32,042
mathematical technicality, so, you know For the purpose of this lecture, think of
#2700:02:32,042 --> 00:02:36,076
these partial symbols, and DD theta one as exactly the same thing. And, don't worry
#2800:02:36,076 --> 00:02:41,005
about whether there are any differences. I'm gonna try to use the mathematically
#2900:02:41,005 --> 00:02:45,018
precise notation. But for our purposes, these notations are really the same thing.
#3000:02:45,036 --> 00:02:49,062
So, let's see what this, this equation will do. And so we're going to compute
#3100:02:49,062 --> 00:02:54,029
this derivative of, I'm not sure if you've seen derivatives in calculus before. But
#3200:02:54,029 --> 00:02:58,066
what a derivative, at this point, does, is basically saying, you know, let's. Take
#3300:02:58,066 --> 00:03:02,087
the tangent to that point, like that straight line, the red line, just,
#3400:03:02,087 --> 00:03:06,097
just touching this function and let's look at the slope of this red line. That's
#3500:03:06,097 --> 00:03:11,035
where the derivative is. It says what's the slope of the line that is just
#3600:03:11,035 --> 00:03:15,056
tangent to the function, okay, and the slope of the line is of course is just
#3700:03:15,056 --> 00:03:20,078
right, you know just the height divided by this horizontal thing. Now. This line has
#3800:03:20,078 --> 00:03:28,037
a positive slope, so it has a positive derivative. And so, my update to theta is
#3900:03:28,037 --> 00:03:36,025
going to be, theta one gives the update that theta one minus alpha times some positive
#4000:03:36,025 --> 00:03:43,010
number. Okay? Alpha, the learning rate is always a positive number. And so
#4100:03:43,010 --> 00:03:47,093
I'm gonna to take theta one, this update as theta one minus something. So I'm gonna
#4200:03:47,093 --> 00:03:52,064
end up moving theta one to the left. I'm gonna decrease theta one and we can see
#4300:03:52,064 --> 00:03:57,047
this is the right thing to do because I actually went ahead in this direction you
#4400:03:57,047 --> 00:04:02,058
know to get me closer to the minimum over there. So, gradient descent so far seems
#4500:04:02,058 --> 00:04:08,011
to be doing the right thing. Let's look at another example. So let's take my same
#4600:04:08,011 --> 00:04:13,078
function j. Just trying to draw the same function j of theta one. And now let's say
#4700:04:13,078 --> 00:04:19,018
I had instead initialized my parameter over there on the left. So theta one is
#4800:04:19,018 --> 00:04:24,016
here. I'm gonna add that point on the surface. Now, my derivative term, d, d
#4900:04:24,016 --> 00:04:29,056
theta one j of theta one, when evaluated at this point, gonna look at right. The
#5000:04:29,056 --> 00:04:35,003
slope of that line. So this derivative term is a slope of this line. But this