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-   -   Hw 6 q1 (http://book.caltech.edu/bookforum/showthread.php?t=477)

lucifirm 05-11-2012 03:42 AM

Hw 6 q1
 
The problem question is:
"In general, if we use H′ instead of H, how does deterministic noise behave?"

My doubt is:
Is this for a fixed N? A small N or large enough to get rid of overfitting?

mikesakiandcp 05-11-2012 11:04 AM

Re: Hw 6 q1
 
Quote:

Originally Posted by lucifirm (Post 2041)
The problem question is:
"In general, if we use H′ instead of H, how does deterministic noise behave?"

My doubt is:
Is this for a fixed N? A small N or large enough to get rid of overfitting?

The size of the training set (N) is more related to overfitting of stochastic noise. Deterministic noise is the ability of the hypothesis set to fit the target function.

elkka 05-11-2012 12:59 PM

Re: Hw 6 q1
 
I am also confused about this term "in general". Does it mean - in absolutely any situation. Or does it mean in most situations? Or - in all reasonable situations, excluding cases when we try to fit 10 degree polynomials to 10 points, as in this lecture's example?

mikesakiandcp, I think N has to do with deterministic noise, at least as described in the lecture. Yes, it is the ability of the hypothesis set to fit the target function, measured as expected difference between the "best" hypothesis and the target. But the way we defined the expected hypothesis, as an expectation over infinite number of data sets of specific size N - that depends on N very much. Slide 14, Lecture 11, illustrates the connection.

AqibEjaz 05-11-2012 02:11 PM

Re: Hw 6 q1
 
@elkka: Well the "deterministic noise" is actually independent of N, refer to lecture 08 slide 20, You can see that the "bias" remains the same no matter how large N becomes. With increasing N it is the variance that becomes smaller and hence overall Eout becomes smaller. As I understand it, if you have infinite training sets, then it does not matter whether you have 10 points in each set or 10,000 points, the average hypothesis will remain the same. In case of 10 points, the different hypotheses we get from each training set will be spread all over the place but they will be "centered" around the same hypothesis (i.e. the average hypothesis). In case of 10,000 points, the individual hypotheses will be less spread out but again they will be centered around the same hypothesis as that in the 10 points. "Bias" only depends upon the mismatch between the target function and the modelling hypothesis.

mikesakiandcp 05-11-2012 02:39 PM

Re: Hw 6 q1
 
Quote:

Originally Posted by elkka (Post 2051)
I am also confused about this term "in general". Does it mean - in absolutely any situation. Or does it mean in most situations? Or - in all reasonable situations, excluding cases when we try to fit 10 degree polynomials to 10 points, as in this lecture's example?

mikesakiandcp, I think N has to do with deterministic noise, at least as described in the lecture. Yes, it is the ability of the hypothesis set to fit the target function, measured as expected difference between the "best" hypothesis and the target. But the way we defined the expected hypothesis, as an expectation over infinite number of data sets of specific size N - that depends on N very much. Slide 14, Lecture 11, illustrates the connection.

You are right, N is related to the deterministic noise. What I meant to say is that we have no control over N (since it is the number of inputs in our training set, which we have no control over). Given a fixed training set (and thus a fixed N), we are interested in how well the hypothesis set can approximate the target function.

gjtucker 05-11-2012 03:00 PM

Re: Hw 6 q1
 
It seems like it depends on the definition of deterministic noise. If we define it as E_x[(g^{bar}(x) - f(x))^2] (as was done in the lecture slides) and we assume that g^{bar} is the best hypothesis in H, then it is independent of N.

Where the finite N comes in is through the variance term. With few N, the more complicated model will have a harder time finding the best hypothesis and have high variance, which what we see in the plots in lecture. But, as N increases, my guess is that E_x[(g^{bar}(x) - f(x))^2] says approximately the same, while the variance term goes down. I suppose this wouldn't be too hard to check numerically.

dudefromdayton 05-11-2012 04:55 PM

Re: Hw 6 q1
 
Heads up from the textbook: exercise 4.3 on page 125!

elkka 05-11-2012 06:35 PM

Re: Hw 6 q1
 
Thanks, but I don't have the book

vasilism 05-12-2012 03:37 AM

Re: Hw 6 q1
 
Some questions,
What does it means that a function (H') is a subset of another function (H)?
H' is picked from the same data model we use for H?

yaser 05-12-2012 03:57 AM

Re: Hw 6 q1
 
Quote:

Originally Posted by vasilism (Post 2063)
Some questions,
What does it means that a function (H') is a subset of another function (H)?
H' is picked from the same data model we use for H?

{\cal H} and {\cal H}' are not functions, but rather sets of functions (the hypotheses h\in{\cal H} are functions).


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