Gradient descent: multidimensional hill descent

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来自 University of Washington 的课程

机器学习：回归

4023 个评分

University of Washington

机器学习：回归

4023 个评分

课程 2（共 4 门，Specialization Machine Learning）

案例学习：预测房价

从本节课中

Simple Linear Regression

Our course starts from the most basic regression model: Just fitting a line to data. This simple model for forming predictions from a single, univariate feature of the data is appropriately called "simple linear regression".<p> In this module, we describe the high-level regression task and then specialize these concepts to the simple linear regression case. You will learn how to formulate a simple regression model and fit the model to data using both a closed-form solution as well as an iterative optimization algorithm called gradient descent. Based on this fitted function, you will interpret the estimated model parameters and form predictions. You will also analyze the sensitivity of your fit to outlying observations.<p> You will examine all of these concepts in the context of a case study of predicting house prices from the square feet of the house.

Gradients: derivatives in multiple dimensions5:37

Gradient descent: multidimensional hill descent6:00

与讲师见面

Emily Fox
Amazon Professor of Machine Learning
Statistics
Carlos Guestrin
Amazon Professor of Machine Learning
Computer Science and Engineering

[MUSIC]

So that defines a gradient.

But one thing that's gonna be useful

is just a different way to visualize the surfaces that we're optimizing over.

So instead of looking at these 3D mesh plots that we've been looking at, we can

look at a contour plot, where we can kind of think of this as a bird's eye view.

[SOUND] It's

a bird's eye view of this function that we're examining,

where here we've now taken this 3D mesh and just transformed it to a 2D plane.

So we have, just in case you can't see it in the slides, this is w0 here.

And this is w1, that's a very big zero.

Okay, and what each one of these curves is.

Again, I'm gonna switch colors to red so it's more visible.

So this curve, for example,

Represents just taking this mesh on the left hand side and slicing it.

So it's just a slice through the 3D function.

So.

Sorry, the pen is really misbehaving.

A slice of the 3D surface where

all values here have the same

value of this function.

Sorry, our functions are called g(w0, w1).

So let me just step up and say what I'm trying to say here,

which is every w0, w1 pair along

this ellipse here Has the same value of the function g

because it was just a flat slice through that 3D contour that we're looking at.

Okay, so each of these rings have, in this case, they are increasing

values of the function as we go from blue, blue means a low value of the function,

all the way out to red, that means a high value of the function.

We're looking at different slices.

We're slicing the function at different values, and

that creates these different contours.

Okay, so this is what's called a contour plot, and it's useful because

it's easier to work with 2D when we're on a 2D surface here.

So drawing things will be easier with this representation.

Okay, so that was just a little detour into contour plots.

So that I can talk about the gradient descent algorithm, which is the analogous

algorithm to what I call the hill decent algorithm in 1D.

But, in place of the derivative of the function,

we've now specified the gradient of the function.

And other than that, everything looks exactly the same.

So what we're doing, is we're taking

we now have a vector of parameters, and we're updating them all at once.

We're taking our previous vector and

we're updating

with our sum, adda times our gradient which was also a vector.

So, it's just the vector analog of the hill descent algorithm.

But, if I wanna show this a little bit in pictures here.

Again switching back to red because it'll be easier to see on this plat.

Well if i'm out here at a point

the gradient is actually it's pointing in the direction of steepest assent.

So that's up hill.

It's pointing this way.

But we're moving in the negative gradient direction.

So let me specify that this thing here is our gradient,

gradient direction, but

then our steps are gonna be in the opposite direction.

So let me actually draw- sorry to take up a little time here but

I think it's worthwhile for clarity.

Let me just happen to draw the gradient so that it's a purple vector so

it's different from the vectors I'm going to be drawing right now.

Okay.

Cuz the other vectors that I'm gonna be drawing right now

are the steps of my gradient descent algorithm.

So the actual steps I'm taking

are gonna be moving here,

Towards this optimal value.

So, it's exactly like what we saw in the 1D case,

but now we're moving it in a 2D space.

Or really any dimensional space but what I'm drawing is just a 2D space.

And in terms of assessing convergence in this case well in place of looking at

the absolute value of the derivative we're going to look at

the magnitude of the gradient.

And when the magnitude of the gradient is less that sum epsilon that we're fixing,

we're gonna say that the algorithm has converged.

>> [MUSIC]

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