Ben Edelman

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(Told this to Jenny in person, but posting for the benefit of others)

AI safety is a young, pre-paradigmatic area of research without a universally accepted mathematical formalism, so if you're after cool math, my suggestion is to learn the basics of one or two well-established fields that are mathematically mature and have a decent chance of being relevant to AI safety.

In particular, I think Learning Theory and Causality are areas with plenty of Aesthetic Math™.

Learning theory

Statistical learning theory is the mathematical study of inductive reasoning—how can we make generalizations from past observations to future observations? It's an entire mathematically rich field devoted to formalizing Occam's razor.

Computational learning theory imposes the further restriction that learning algorithms be computationally efficient. It has rich connections to other parts of theoretical computer science (for example, there is a duality between computational learning theory and cryptography—positive results for one translate to negative results for the other!) And there are many fun problems of a combinatorial puzzle flavor.

Most of learning theory assumes that observations are drawn i.i.d. from a distribution. Online Learning asks what happens if we eliminate this assumption. Incredibly, it can be shown that inductive reasoning can be successful even when observations are handcrafted by an adversary. The key is to measure success in relative rather than absolute terms: how did you perform in comparison to the best member of a pre-specified class of predictors? There are beautiful connections to convex analysis.



I don't know this area as well, but the material I have learned has been mathematically beautiful. In particular, I suggest learning about Judea Pearl's theory of causality, which has been very influential in computer science, statistics, and some of the natural and social sciences. (There are a few competing formalisms for causality, but Pearl's is the most mathematically beautiful as far as I can tell.) Pearl's theory generalizes the classical theory of probability to allow for reasoning about cause and effect, using a framework that involves manipulations of directed acyclic graphs.

Reading: Causality, by Pearl.