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Long ago, I noticed in computer science ‘approximation’ is often defined w.r.t. some optimal value. In cognitive science this type of value-approximation has limited theoretical important. Instead, structure-approximation is often more relevant. Yet, AI approaches to cogsci ignore this. 19/🧵

Iris van Rooij 💭

5 авг., 00:59

Value- vs structure-approximability are *fully dissociable*. IOW, an output may be close to the optimal value, yet arbitrarily off in terms of structure, and vice versa, an output may be close to the required structure, yet arbitrarily off in terms of value. Illustrations in the figures. 20/🧵

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Hel Eline

5 авг., 01:40

*nods contemplatively at words and figures beyond my comprehension* "yup, looks like witchcraft."

Ответ для Hel Eline

😅🫶

One upshot is that it *really* matters what kind of approximation, with what kinds of guarantees (e.g., value, structure, how close), one is making claims about. One cannot just claim “this intractable f is approximable with this A_approx", without precise definitions and formal proof. 21/🧵

Ответ для Iris van Rooij 💭

Iris van Rooij 💭

5 авг., 01:14

I am also curious about your intuitions about how often an intractable function may be tractably approximable. What do you think? Pick one of the below options: a) always b) often c) sometimes d) seldom e) never 22/🧵

Ответ для Iris van Rooij 💭

Roger Watt

5 авг., 11:10

This is interesting. My own understanding might be out of date. It used to be that many functions normpdf(), sin() etc in almost any programming language were a polynomial approximation. It was only because their behaviour was exactly known that this was acceptable.