EA - How to do theoretical research, a personal perspective by Mark Xu

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: How to do theoretical research, a personal perspective, published by Mark Xu on August 19, 2022 on The Effective Altruism Forum. "Where do new [algorithms] come from? I keep reading about someone who invented [an algorithm] to do something-or-other but there's no mention of how." A shrug of robed shoulders. "Where do new books come from, Mr. Potter? Those who read many books sometimes become able to write them in turn. How? No one knows." "There are books on how to write -" "Reading them will not make you a famous playwright. After all such advice is accounted for, what remains is mystery. The invention of new [algorithms] is a similar mystery of purer form." - Harry Potter and the Methods of Rationality Most of the content in this document came out of extensive conversations with Paul Christiano. (This document describes one way of thinking about how to do one particular type of research. There are other ways to productively do this kind of research, and other productive kinds of research. Consider this document peppered with phrases like “from my perspective”, “I think”, “my sense is”, etc.) A lot of people have a vague mental picture of what empirical research looks like, which consists of exploring data, articulating hypotheses about the data, and running experiments that potentially falsify the hypothesis. I think people lack a similarly mechanistic picture of what theoretical research looks like, which results in them not knowing how to do theory, being skeptical of the possibility of theoretical progress, etc. I do think the difficulty of getting high-quality real-world feedback makes theoretical research more difficult than empirical research, but I think it’s possible to get enough real-world feedback when doing theory that you can still expect to make steady progress. Currently, I think many people think of theory as “someone sits in a room and has a brilliant insight to solve the problem.” Instead, I think a more accurate picture is very similar to the picture one has for empirical research: the theorist explores some data, gradually builds an intuitive sense of what’s going on, articulates hypotheses that capture their intuition, and falsifies their hypotheses by testing them against data, all the while iteratively building up their understanding. The key difference between the empirical researcher and the theoretical researcher is while the empirical researcher can build intuition and falsify hypothesis by considering real-world data, the theorist, although ultimately be grounded in the real world, must build intuition and falsify hypothesis by considering thought experiments, simple toy examples, computations, etc. Even mathematics, the purest of intellectual pursuits, roughly follows this process of iteration. Terrance Tao: [A]ctual solutions to a major problem tend to be arrived at by a process more like the following (often involving several mathematicians over a period of years or decades, with many of the intermediate steps described here being significant publishable papers in their own right): Isolate a toy model case x of major problem X. Solve model case x using method A. Try using method A to solve the full problem X. This does not succeed, but method A can be extended to handle a few more model cases of X, such as x’ and x”. Eventually, it is realised that method A relies crucially on a property P being true; this property is known for x, x’, and x”, thus explaining the current progress so far. Conjecture that P is true for all instances of problem X. Discover a family of counterexamples y, y’, y”, . to this conjecture. This shows that either method A has to be adapted to avoid reliance on P, or that a new method is needed. Take the simplest counterexample y in this family, and try to prove X for this special case. Meanwhile, try to see wheth...