My Wild & Crazy Idea: Causal Computational Learning Theory

Scott Aaronson

But first: what ever happened to my WACI from a few years ago: a Web 2.0 mathematics discussion site and conjecture/theorem repository?

There now exists such a site, Mathoverflow.net, which is everything I hoped for and more.In just ~2 years, it’s noticeably changed the practice of mathematics.I had nothing to do with its creation.

PAC (Probabilistically Approximately Correct) Learning

“Computational complexity theory meets statistics”

PAC-learning is a hugely successful model—but like most statistics, it doesn’t care about the distinction between correlation and cause

Given a collection of labeled examples (x1,f(x1)),…,(xm,f(xm)) drawn independently from some unknown

distribution D, problem is to output a hypothesis h such that h(x)=f(x) for

most x~D with high probability

The result: PAC explains how banks predict who will repay their loans, but not how Einstein predicted the bending of starlight by the sun

vs

To predict what will happen in novel situations, you need to know something about causal mechanisms—which often requires controlled experiments (together with prior knowledge about temporal direction, autonomy of subsystems, etc.)

Best theory of causality we currently have:Judea Pearl’s “do-calculus”

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My WACI Challenge for TheoryTraditional statistics : PAC-learning

:: Pearl’s do-calculus : what?Potential applications: Debugging, reconstructing gene regulatory networks…

Existing work in the direction I’m talking about: - PAC-learning with membership and equivalence queries- Angluin, Aspnes, Chen, Wu: “Learning a circuit by injecting values,” STOC’2006- Pearl’s IC algorithm- Leakage-resilience in cryptography