Machine Learning (Theory)

10/26/2005

Fallback Analysis is a Secret to Useful Algorithms

The ideal of theoretical algorithm analysis is to construct an algorithm with accompanying optimality theorems proving that it is a useful algorithm. This ideal often fails, particularly for learning algorithms and theory. The general form of a theorem is:

If preconditions Then postconditions
When we design learning algorithms it is very common to come up with precondition assumptions such as “the data is IID”, “the learning problem is drawn from a known distribution over learning problems”, or “there is a perfect classifier”. All of these example preconditions can be false for real-world problems in ways that are not easily detectable. This means that algorithms derived and justified by these very common forms of analysis may be prone to catastrophic failure in routine (mis)application.

We can hope for better. Several different kinds of learning algorithm analysis have been developed some of which have fewer preconditions. Simply demanding that these forms of analysis be used may be too strong—there is an unresolved criticism that these algorithm may be “too worst case”. Nevertheless, it is possible to have a learning algorithm that simultaneously provides strong postconditions given strong preconditions, reasonable postconditions given reasonable preconditions, and weak postconditions given weak preconditions. Some examples of this I’ve encountered include:

  1. Sham, Matthias and Dean showing that some Bayesian regression is robust in a minimax online learning analysis.
  2. The cover tree which creates an O(n) datastructure for nearest neighbor queries while simultaneously guaranteeing O(log(n)) query time when the metric obeys a dimensionality constraint.

The basic claim is that algorithms with a good fallback analysis are significantly more likely to achieve the theoretical algorithm analysis ideal. Both of the above algorithms have been tested in practice and found capable.

Several significant difficulties occur for anyone working on fallback analysis.

  1. It’s harder. This is probably the most valid reason—people have limited time to do things. Nevertheless, it is reasonable to hope that the core techniques used by many people have had this effort put into them.
  2. It is psychologically difficult to both assume and not assume a precondition, for a researcher. A critical valuable resource here is observing multiple forms of analysis.
  3. It is psychologically difficult for a reviewer to appreciate the value of both assuming and not assuming some precondition. This is a matter of education.
  4. It is neither “sexy” nore straightforward. In particular, theoretically inclined people 1) get great joy from showing that something new is possible and 1) routinely work on papers of the form “here is a better algorithm to do X given the same assumptions”. A fallback analysis requires a change in assumption invalidating (2) and the new thing that it shows for (1) is subtle: that two existing guarantees can hold for the same algorithm. My hope here is that this subtlety becomes better appreciated in time—making useful algorithms has a fundamental sexiness of it’s own.

10/20/2005

Machine Learning in the News

Tags: General jl@ 10:55 am

The New York Times had a short interview about machine learning in datamining being used pervasively by the IRS and large corporations to predict who to audit and who to target for various marketing campaigns. This is a big application area of machine learning. It can be harmful (learning + databases = another way to invade privacy) or beneficial (as google demonstrates, better targeting of marketing campaigns is far less annoying). This is yet more evidence that we can not rely upon “I’m just another fish in the school” logic for our expectations about treatment by government and large corporations.

10/19/2005

Workshop: Atomic Learning

Tags: General jl@ 12:23 pm

We are planning to have a workshop on atomic learning Jan 7 & 8 at TTI-Chicago.

Details are here.

The earlier request for interest is here.

The primary deadline is abstracts due Nov. 20 to jl@tti-c.org.

10/16/2005

Complexity: It’s all in your head

One of the central concerns of learning is to understand and to
prevent overfitting. Various notion of “function complexity” often
arise: VC dimension, Rademacher complexity, comparison classes of
experts, and program length are just a few.

The term “complexity” to me seems somehow misleading; the terms never
capture something that meets my intuitive notion of complexity. The
Bayesian notion clearly captures what’s going on. Functions aren’t
“complex”– they’re just “surprising”: we assign to them low
probability. Most (all?) complexity notions I know boil down
to some (generally loose) bound on the prior probability of the function.

In a sense, “complexity” fundementally arises because probability
distributions must sum to one. You can’t believe in all possibilities
at the same time, or at least not equally. Rather you have to
carefully spread the probability mass over the options you’d like to
consider. Large complexity classes means that beliefs are spread
thinly. In it’s simplest form, this phenomenom give the log (1\n) for
n hypotheses in classic PAC bounds.

In fact, one way to think about good learning algorithms is that they
are those which take full advantage of their probability mass.
In the language of Minimum Description Length, they correspond to
“non-defective distributions”.

So this raises a question: are there notions of complexity (preferably finite,
computable ones) that differ fundementally from the notions of “prior”
or “surprisingness”? Game-theoretic setups would seem to be promising,
although much of the work I’m familiar with ties it closely to the notion
of prior as well.

10/13/2005

Site tweak

Tags: General jl@ 10:43 am

Several people have had difficulty with comments which seem to have an allowed language significantly poorer than posts. The set of allowed html tags has been increased and the markdown filter has been put in place to try to make commenting easier. I’ll put some examples into the comments of this post.

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