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English•SRT•14.9 KB•Dec 10, 2014•IMDb tt0000000

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Subtitle preview

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

50 cues

#100:00:00,000 --> 00:00:04,093
We've previously defined the cost function J. In this video I want to tell you about
#200:00:04,093 --> 00:00:09,063
an algorithm called gradient descent for minimizing the cost function J. It turns
#300:00:09,063 --> 00:00:14,027
out gradient descent is a more general algorithm and is used not only in linear
#400:00:14,027 --> 00:00:18,091
regression. It's actually used all over the place in machine learning. And later
#500:00:18,091 --> 00:00:23,079
in the class we'll use gradient descent to minimize other functions as well, not just
#600:00:23,079 --> 00:00:27,084
the cost function J, for linear regression. So in this video, I'm going to
#700:00:27,084 --> 00:00:32,055
talk about gradient descent for minimizing some arbitrary function J. And then in
#800:00:32,055 --> 00:00:37,040
later videos, we'll take those algorithm and apply it specifically to the cost
#900:00:37,040 --> 00:00:43,033
function J that we had to find for linear regression. So here's the problem
#1000:00:43,033 --> 00:00:48,011
setup. We're going to see that we have some function J of (theta0, theta1).
#1100:00:48,011 --> 00:00:52,077
Maybe it's a cost function from linear regression. Maybe it's some other function
#1200:00:52,077 --> 00:00:56,080
we want to minimize. And we want to come up with an algorithm for
#1300:00:56,080 --> 00:01:01,017
minimizing that as a function of J of (theta0, theta1). Just as an aside,
#1400:01:01,017 --> 00:01:05,079
it turns out that gradient descent actually applies to more general
#1500:01:05,079 --> 00:01:10,099
functions. So imagine if you have a function that's a function of
#1600:01:10,099 --> 00:01:16,019
J of (theta0, theta1, theta2, up to some theta n). And you want to
#1700:01:16,040 --> 00:01:21,079
minimize over (theta0 up to theta n) of this J of (theta0 up to theta n).
#1800:01:21,079 --> 00:01:26,057
It turns out gradient descent is an algorithm for solving
#1900:01:26,057 --> 00:01:31,036
this more general problem, but for the sake of brevity, for the sake of
#2000:01:31,036 --> 00:01:35,093
your succinctness of notation, I'm just going to present only two parameters
#2100:01:36,011 --> 00:01:41,009
throughout the rest of this video. Here's the idea for gradient descent. What we're
#2200:01:41,009 --> 00:01:45,088
going to do is we're going to start off with some initial guesses for theta0 and
#2300:01:45,088 --> 00:01:50,078
theta1. Doesn't really matter what they are, but a common choice would be if we
#2400:01:50,078 --> 00:01:55,045
set theta0 to 0, and set theta1 to 0. Just initialize
#2500:01:55,045 --> 00:02:00,032
them to 0. What we're going to do in gradient descent is we'll keep changing
#2600:02:00,032 --> 00:02:05,025
theta0 and theta1 a little bit to try to reduce J of (theta0, theta1)
#2700:02:05,025 --> 00:02:10,057
until hopefully we wind up at a minimum or maybe a local minimum. So, let's see
#2800:02:10,079 --> 00:02:16,010
see pictures of what gradient descent does. Let's say I try to minimize this
#2900:02:16,010 --> 00:02:20,084
function. So notice the axes. This is, (theta0, theta1) on the horizontal
#3000:02:20,084 --> 00:02:25,077
axes, and J is a vertical axis. And so the height of the surface shows J, and we
#3100:02:25,077 --> 00:02:30,058
want to minimize this function. So, we're going to start off with (theta0, theta1)
#3200:02:30,058 --> 00:02:35,037
at some point. So imagine picking some value for (theta0, theta1), and that
#3300:02:35,037 --> 00:02:39,093
corresponds to starting at some point on the surface of this function. Okay? So
#3400:02:39,093 --> 00:02:44,020
whatever value of (theta0, theta1) gives you some point here. I did
#3500:02:44,020 --> 00:02:48,081
initialize them to 0, but sometimes you initialize it to other
#3600:02:48,081 --> 00:02:53,094
values as well. Now. I want us to imagine that this figure shows a hill. Imagine
#3700:02:53,094 --> 00:02:59,017
this is like a landscape of some grassy park with two hills like so.
#3800:02:59,017 --> 00:03:04,061
And I want you to imagine that you are physically standing at that point on the
#3900:03:04,061 --> 00:03:09,099
hill on this little red hill in your park. In gradient descent, what we're
#4000:03:09,099 --> 00:03:15,077
going to do is spin 360 degrees around and just look all around us and ask, "If I were
#4100:03:15,077 --> 00:03:20,042
to take a little baby step in some direction, and I want to go downhill as
#4200:03:20,042 --> 00:03:25,031
quickly as possible, what direction do I take that little baby step in if I want to
#4300:03:25,031 --> 00:03:29,068
go down, if I sort of want to physically walk down this hill as rapidly as
#4400:03:29,068 --> 00:03:34,046
possible?" Turns out that if we're standing at that point on the hill, you look all
#4500:03:34,046 --> 00:03:39,018
around, you find that the best direction to take a little step downhill
#4600:03:39,018 --> 00:03:44,003
is roughly that direction. Okay. And now you're at this new point on your hill.
#4700:03:44,003 --> 00:03:49,043
You're going to, again, look all around, and then say, "What direction should I step in order
#4800:03:49,043 --> 00:03:54,069
to take a little baby step downhill?" And if you do that and take another step, you
#4900:03:54,069 --> 00:03:59,069
take a step in that direction, and then you keep going. You know, from this new
#5000:03:59,069 --> 00:04:04,083
point, you look around, decide what direction will take you downhill most