Despite my best intentions, this is not a fully specified problem, but rather a research direction.

Competitive online learning is one of the more compelling pieces of learning theory because typical statements of the form “this algorithm will perform almost as well as a large set of other algorithms” rely only on fully-observable quantities, and are therefore applicable in many situations. Examples include Winnow, Weighted Majority, and Binomial Weighting. Algorithms with this property haven’t taken over the world yet. Here might be some reasons:

**Lack of caring**. Many people working on learning theory don’t care about particular applications much. This means constants in the algorithm are not optimized, usable code is often not produced, and empirical studies aren’t done.**Inefficiency**. Viewed from the perspective of other learning algorithms, online learning is terribly inefficient. It requires that every hypothesis (called an expert in the online learning setting) be enumerated and tested on every example. (This is similar to the inefficiency of using Bayes law as an algorithm directly, and there are strong similarities in the algorithms.)

For an example of (1), there is a mysterious factor of 2 in the Binomial Weighting algorithm which has not been fully resolved. Some succesful applications also exist such as those based on SNoW.

The way to combat problem (2) is to introduce more structure into the hypothesis/experts. Some efforts have already been made in this direction. For example, it’s generally feasible to introduce an arbitrary bias or “prior” over the experts in the form of some probability distribution, and perform well with respect to that bias. Another nice piece of work by Adam Kalai and Santosh Vempala discusses how to efficiently handle several forms of structured experts. At an intuitive level, further development discovering how to efficiently work with new forms of structure might payoff well.

The basic research direction here is to address the gap between theory and practice, possibly by solving the above problems and possibly by discovering and addressing other problems.