案例学习：预测房价

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来自 University of Washington 的课程

机器学习：回归

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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.

- Emily FoxAmazon Professor of Machine Learning

Statistics - Carlos GuestrinAmazon Professor of Machine Learning

Computer Science and Engineering

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But before we get to how to minimize residual sum of squares in particular,

let's talk about optimization in a more abstract sense.

In terms of just a generic function and thinking about how do we minimize it,

what are the properties of the minimum of a function and algorithms for

achieving that minimum?

One of the first important concepts that we can talk about is the notion of convex

and concave functions.

So here, I have a picture of a concave function over

just one variable, we'll call it w.

And this function here, this is g(w).

And this is a concave function, and the way I remember concave versus convex,

is concave it looks like you're in a cave, so it looks like the arch of a cave.

And convex is just the other one.

So, the way we can think about defining what a concave

function is, we can look at any two values of w.

Let's call it a and b.

And look at the value of the function at a and b.

So g(a) and g(b).

And let's draw a line between g(a) and g(b).

So that's this green line here.

And what we see is that this line

lies below g(w) everywhere.

And a concave function is function where for any value of a and any value

of b that you choose, when you go and draw this little line between the points of

the function, it's always gonna lie below the actual curve of the function itself.

A convex function,

on the other hand, let's just draw a little picture of a convex function.

It has exactly the same, but opposite, so not the same.

It has exactly the opposite property.

Where if you choose any a and any b, so

just to be clear this is w, this is g(w).

If I go and connect g(a) with

g(b), with just a line, and

this line is above g(w), everywhere.

Okay, so that's the very intuitive geometric definition of what a concave or

convex function is.

But then of course there are functions that are neither concave nor convex.

So, let me just draw two examples here.

Here's an example of a function that is not concave nor convex.

And how do we see that?

Well, let's look at a point a and

a point b, and draw a line between these points.

And we see that this line lies both below the curve here and above the curve here.

So it doesn't satisfy either the concave or convex criteria.

And likewise this function Has the same property, a and b.

Draw our little line, which okay I didn't draw that well at all.

Let me try and draw a straight line.

But I think you get the point, that neither of these functions are concave,

nor convex.

Okay, so that's the notion of defining what a concave or convex function is.

But we're looking at an optimization objective,

where either our goal is to find the minimum or maximum of a function.

So typically if we're looking at a concave function, our interest is

in finding the maximum, and for convex it's in finding the minimum.

So let's look at, what is the maximum of this concave function?

Well, that's just this point right here.

So that is the maximum.

And [COUGH] here, what I'm saying is

our goal is max over all w, g(w).

In terms of the notation that we introduced a little bit earlier.

And so what's the property of this concave function at this point?

Well what we know is that this is the point where the derivative is 0.

So there's no rate of change of this function at this point.

And what about a convex function?

If we're seeking to find min over all w g(w),

well the minimum is clearly this point right here.

And again, what's the property?

Same thing, derivative = 0.

And note that for the concave and convex functions,

there's only one place where the derivative = 0.

In contrast, when we look at the two examples for functions that were neither

concave nor convex that we showed on the previous slide.

What we see is in this case, for example,

they're multiple locations where you have the derivative equaling 0.

So, let's write that explicitly and

say that there are multiple

solutions to derivative = 0.

In contrast for this function there's actually no solution to derivative = 0.

The derivative is nowhere near 0.

The function continuously has a rate of change.

So, no solution

To derivative = 0.

Okay, so the key take home message here,

is the reason we're talking about concave and convex functions is the fact that when

we have an objective to find the minimum or the maximum.

Minimum of a convex, maximum of a concave.

The solution to that is gonna be unique, and we know that it's gonna exist.

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