lecture 8: understanding bias

[ilya239]

The VC dimension is single number that is a property of the hypothesis set.
But, what is "bias of a hypothesis set"? Bias seems to depend also on dataset size and the learning algorithm, since it depends on ; depends on the learning algorithm, and the set of datasets over which the expectation is taken depends on dataset size.

Slide 4 says that bias measures "how well can approximate ". Does this mean "with a sufficiently large dataset and a perfect learning algorithm"?
Is the bias of a (hypothesis set, learning algorithm) combination a single value -- the asymptote of the learning curve? Or is there some notion of bias that is a property of a hypothesis set by itself? If the hypothesis set contains the target function, that does not mean the bias is zero, does it? The beginning of the lecture seems to imply otherwise, but if there is no restriction on the learning algorithm, what guarantees that the average function will in fact be close to the target function for large enough dataset size?
Or is it assumed that the learning algorithm always picks a hypothesis which minimizes ?

[yaser]

Originally Posted by ilya239

The VC dimension is single number that is a property of the hypothesis set.
But, what is "bias of a hypothesis set"? Bias seems to depend also on dataset size and the learning algorithm, since it depends on ; depends on the learning algorithm, and the set of datasets over which the expectation is taken depends on dataset size.

Your observation is correct that the bias-variance analysis is not as general as the VC analysis. The bias does depend on the learning algorithm. It also depends on the number of examples, usually slightly.

Slide 4 says that bias measures "how well can approximate ". Does this mean "with a sufficiently large dataset and a perfect learning algorithm"?
Is the bias of a (hypothesis set, learning algorithm) combination a single value -- the asymptote of the learning curve? Or is there some notion of bias that is a property of a hypothesis set by itself? If the hypothesis set contains the target function, that does not mean the bias is zero, does it? The beginning of the lecture seems to imply otherwise, but if there is no restriction on the learning algorithm, what guarantees that the average function will in fact be close to the target function for large enough dataset size?
Or is it assumed that the learning algorithm always picks a hypothesis which minimizes ?

Good questions . What you are saying would hold if we were using the best approximation of in as the vehicle for measuring the bias. We are not. We are using a "limited resource" version of it that is based on averaging hypotheses that we get from training on a finite set of data points. This version is often close to the best approximation so that's why we can take that liberty.

[ilya239]

Originally Posted by yaser

The bias does depend on the learning algorithm. It also depends on the number of examples, usually slightly.
...
This version is often close to the best approximation so that's why we can take that liberty.

Thanks for the explanation.
In HW4 #4 the average hypothesis is measurably shifted from the hypothesis set member giving the lowest mean squared error. Probably because two-point dataset is too small, i.e. this is not representative of realistic cases?

[yaser]

Originally Posted by ilya239

Thanks for the explanation.
In HW4 #4 the average hypothesis is measurably shifted from the hypothesis set member giving the lowest mean squared error. Probably because two-point dataset is too small, i.e. this is not representative of realistic cases?

Indeed, the fewer the number of points, the more likely that the average hypothesis will differ from the best approximation. The difference tends to be small, though.

[gah44]

Originally Posted by ilya239

Thanks for the explanation.
In HW4 #4 the average hypothesis is measurably shifted from the hypothesis set member giving the lowest mean squared error. Probably because two-point dataset is too small, i.e. this is not representative of realistic cases?

Well, it is also that the two point data set is small relative to the two parameter hypotheses. If you have 100 points, and 99th degree polynomials, it would also have large variance. I will guess that minimizing bias plus variance happens with the number of fit parameters near the square root of the number of points per data set.

[ilya239]

Originally Posted by gah44

Well, it is also that the two point data set is small relative to the two parameter hypotheses. If you have 100 points, and 99th degree polynomials, it would also have large variance. I will guess that minimizing bias plus variance happens with the number of fit parameters near the square root of the number of points per data set.

Large variance, sure. I was trying to understand why large bias. If you take a huge number of 100-point datasets, learn a hypothesis from each, and take the average value of these, why might it be far from the target function's value?
On the other hand, I'm not sure how to prove that it won't be far

[yaser]

Originally Posted by ilya239

I was trying to understand why large bias. If you take a huge number of 100-point datasets, learn a hypothesis from each, and take the average value of these, why might it be far from the target function's value?
On the other hand, I'm not sure how to prove that it won't be far

It is unlikely (as a practical observation) to be far, but it is likely to be different.

[gah44]

Well, when I wrote that one I was remembering the first time I tried using a polynomial fit program. (It was in Fortran 66, as a hint to how long ago that was.)

I fit an N degree polynomial to N points.

Even so, I believe if you fit a 99th degree polynomials to sets of 100 points you will have a large variance, as did the 1st degree to two points. It won't be easy at all to visualize, though.

[ilya239]

Originally Posted by yaser

It is unlikely (as a practical observation) to be far, but it is likely to be different.

Sorry to be harping on this question, but I just wanted to ask: is there any intuitive way to see that the average hypothesis will be close to the best hypothesis from the hypothesis set, beyond "practical observation"? E.g. for hypothesis sets satisfying certain well-behavedness criteria, such as being parameterized by a finite number of parameters, containing only continuous functions, etc. The lectures rely in crucial ways on this assumption and it would help to get some more intuition for why it is true for the typically used hypothesis sets, if possible.

[magdon]

In general one cannot say anything analytical about bias and variance. For example the average hypothesis can be very far from the best hypothesis in the model for arbitrarily constructed hypothesis sets and learning algorithms. For example, the average function need not even be in the hypothesis set. However, what we say about the average function being a good approximation to the best you can do is not that far off for general models used in practice.

Problem 4.11 takes you through one of the few situations where one can say something reasonably technical. We can extrapolate (without proof) the conclusions to the more general setting as follows:

(1) When the model is well specified: this means that the hypothesis set contains the target function or a good approximation to it;

(2) When the noise has zero mean and is well behaved, for example having finite variance;

(3) When the learning algorithm is reasonably "stable", which means that small perturbations in the data set lead to small "proportionate" changes in the learned hypothesis (the learning algorithm version of a bounded first derivative);

Then, the average learned function will be approximately the one you would learn from a data set having zero noise; this zero noise hypothesis will (for reasonable N) be close to the optimal function you could learn and will become more so very quickly with increasing N (think trying to learn a polynomial with noiseless data). The conditions above are reasonably general. It is the 3rd condition that is most important, and one can mostly relax the well specified requirement in practice.

Originally Posted by ilya239

Sorry to be harping on this question, but I just wanted to ask: is there any intuitive way to see that the average hypothesis will be close to the best hypothesis from the hypothesis set, beyond "practical observation"? E.g. for hypothesis sets satisfying certain well-behavedness criteria, such as being parameterized by a finite number of parameters, containing only continuous functions, etc. The lectures rely in crucial ways on this assumption and it would help to get some more intuition for why it is true for the typically used hypothesis sets, if possible.