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Regression vs. Classification as a Primitive

For learning reductions we have been concentrating on reducing various complex learning problems to binary classification. This choice needs to be actively questioned, because it was not carefully considered.

Binary clasification is learning a classifier c:X -> {0,1} so as to minimize the probability of being wrong, Prx,y~D(c(x) <> y).

The primary alternative candidate seems to be squared error regression. In squared error regression, you learn a regressor s:X -> [0,1] so as to minimize squared error, Ex,y~D (s(x)-y)2.

It is difficult to judge one primitive against another. The judgement must at least partially be made on nontheoretical grounds because (essentially) we are evaluating a choice between two axioms/assumptions.

These two primitives are significantly related. Classification can be reduced to regression in the obvious way: you use the regressor to predict D(y=1|x), then threshold at 0.5. For this simple reduction a squared error regret of r implies a classification regret of at most r0.5. Regression can also be used to reduce to classification using the Probing algorithm. (This is much more obvious when you look at an updated proof. ) Under this reduction, a classification regret of r implies a squared error regression regret of at most r.

Both primitives enjoy a significant amount of prior work with (perhaps) classification enjoying more work in the machine learning community and regression having more emphasis in the statistics community.

The (nonhistoric) reasons for preferring classification seem to be:

  1. Aesthetically, what could be a more natural primitive than predicting a single bit?
  2. Classification is (potentially) inherently more representationally concise. When translated into transistors, a classification algorithm might use fewer transistors than a regressor, simply because the natural representation is bits rather than reals (~= floats).

There are several reasons to consider regression over classification:

  1. More uniform convergence. For a single squared error regressor, the rate of convergence of the estimate of squared error to the expected squared error goes as 1/m, assuming IID samples. For a single binary classifier, the rate of convergence of the estimate of binary loss to the expected binary loss goes as a function varying between 1/m and 1/m0.5.
  2. There is a computational inequivalence between the primitives, as far as we know. In particular, the Probing algorithm must call a classifier several times in several ways to make a high precision regression prediction. On the other hand, classification via regression requires one call to the underlying regressor.
  3. Squared error regression learning is often easier than 0/1 classification learning. This is becaue squared error regression is convex, but 0/1 loss is not. Note: this does not imply that squared error regression is convex (It isn’t for general regression algorithms). Instead, it just means that nonconvexity is not enforced by the loss function.

The mathematical evidence points toward squared error regression as a better primitive, although doubts still seem reasonable to entertain.

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Who is having visa problems reaching US conferences?

Many of the large machine learning conferences were in the US this summer. A common problem which students from abroad encounter is visa issues.
Just getting a visa to visit can be pretty rough: you stand around in lines, sometimes for days. Even worse is the timing with respect to ticket buying. Airplane tickets typically need to be bought well in advance on nonrefundable terms to secure a reasonable rate for air travel. When a visa is denied, as happens reasonably often, a very expensive ticket is burnt.

A serious effort is under way to raise this as in issue in need of fixing. Over the long term, effectively driving research conferences to locate outside of the US seems an unwise policy. Robert Schapire is planning to talk to a congressman. Sally Goldman suggested putting together a list of problem cases, and Phil Long setup an email address immigration.and.confs@gmail.com to collect them.

If you (or someone you know) has had insurmountable difficulties reaching a conference in the US, please send an email with:

Name:
Email address:
Conference:
Difficulty:
Details: (be brief please)

We expect most of the problem cases are students, so don’t be shy.

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New Models

How should we, as researchers in machine learning, organize ourselves?

The most immediate measurable objective of computer science research is publishing a paper. The most difficult aspect of publishing a paper is having reviewers accept and recommend it for publication. The simplest mechanism for doing this is to show theoretical progress on some standard, well-known easily understood problem.

In doing this, we often fall into a local minima of the research process. The basic problem in machine learning is that it is very unclear that the mathematical model is the right one for the (or some) real problem. A good mathematical model in machine learning should have one fundamental trait: it should aid the design of effective learning algorithms. To date, our ability to solve interesting learning problems (speech recognition, machine translation, object recognition, etc…) remains limited (although improving), so the “rightness” of our models is in doubt.

If our mathematical models are bad, the simple mechanism of research above can not yield the end goal. (This should be agreed on even by people who disagree about what the end goal of machine learning is!) Instead, research which proposes and investigates new mathematical models for machine learning might yield the end goal. Doing this is hard.

  1. Coming up with a new mathematical model is just plain not easy. Some sources of inspiration include:
    1. Watching carefully: what happens succesfully in practice, can often be abstracted into a mathematical model.
    2. Swapping fields: In other fields (for example crypto), other methods of analysis have been developed. Sometimes, these methods can be transferred.
    3. Model repair: Existing mathematical models often have easily comprehendable failure modes. By thinking about how to avoid such failure modes, we can sometimes produce a new mathematical model.
  2. Speaking about a new model is hard. The difficulty starts with you in explaining it. Often, when trying to converge on a new model, we think of it in terms of the difference with respect to an older model, leading to a tangled explanation. The difficulty continues with other people (in particular: reviewers) reading it. For a reviewer with limited time, it is very tempting to assume that any particular paper is operating in some familiar model and fail out. The best approach here seems to be super explicitness. You can’t be too blunt about saying “this isn’t the model you are thinking about”.
  3. Succeeding with new models is also hard. When people don’t have a reference frame to understand the new model, they are unlikely to follow up, as is necessary for success in academia.

The good news here is that a succesful new model can be a big win. I wish it was an easier win: the barriers to success are formidably high, and it seems we should do everything possible to lower the barriers to success for the sake of improving research.

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The Stock Prediction Machine Learning Problem

…is discussed in this nytimes article. I generally expect such approaches to become more common since computers are getting faster, machine learning is getting better, and data is becoming more plentiful. This is another example where machine learning technology may have a huge economic impact. Some side notes:

  1. We-in-research know almost nothing about how these things are done (because it is typically a corporate secret).
  2. … but the limited discussion in the article seem naive from a machine learning viewpoint.
    1. The learning process used apparently often fails to take into account transaction costs.
    2. What little of the approaches is discussed appears modeling based. It seems plausible that more direct prediction methods can yield an edge.
  3. One difficulty with stock picking as a research topic is that it is inherently a zero sum game (for every winner, there is a loser). Much of the rest of research is positive sum (basically, everyone wins).
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MaxEnt contradicts Bayes Rule?

A few weeks ago I read this. David Blei and I spent some time thinking hard about this a few years back (thanks to Kary Myers for pointing us to it):

In short I was thinking that “bayesian belief updating” and “maximum entropy” were two othogonal principles. But it appear that they are not, and that they can even be in conflict !
Example (from Kass 1996); consider a Die (6 sides), consider prior knowledge E[X]=3.5.
Maximum entropy leads to P(X)= (1/6, 1/6, 1/6, 1/6, 1/6, 1/6).
Now consider a new piece of evidence A=”X is an odd number”
Bayesian posterior P(X|A)= P(A|X) P(X) = (1/3, 0, 1/3, 0, 1/3, 0).
But MaxEnt with the constraints E[X]=3.5 and E[Indicator function of A]=1 leads to (.22, 0, .32, 0, .47, 0) !! (note that E[Indicator function of A]=P(A))
Indeed, for MaxEnt, because there is no more ‘6′, big numbers must be more probable to ensure an average of 3.5. For bayesian updating, P(X|A) doesn’t have to have a 3.5 expectation. P(X) and P(X|a) are different distributions.
Conclusion ? MaxEnt and bayesian updating are two different principle leading to different belief distributions. Am I right ?

I don’t believe there is any paradox at all between MaxEnt (perhaps more generally, MinRelEnt) and Bayesian updates. Here, straight MaxEnt make no sense. The implication of the problem is that the ensemble average 3.5 is no longer an active constraint. That is, we no longer believe the contraint E[X]=3.5 once we have the additional data that X is an odd number. The sequential update using minimum relative entropy is identical to Bayes rule and produces the correct answer. These two answers are simply (correct) answers to different questions.