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.