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Note: The survey took me 20 mins (but also note selection effects on leaving this comment)

Here's a fun thing I noticed:

There are 16 boolean functions of two variables.  Now consider an embedding that maps each of the four pairs {(A=true, B=true), (A=true, B=false), ...} to a point in 2d space.  For any such embedding, at most 14 of the 16 functions will be representable with a linear decision boundary.

For the "default" embedding (x=A, y=B), xor and its complement are the two excluded functions.  If we rearrange the points such that xor is linearly represented, we always lose some other function (and its complement).  In fact, there are 7 meaningfully distinct colinearity-free embeddings, each of which excludes a different pair of functions.[1]

I wonder how this situation scales for higher dimensions and variable counts.  It would also make sense to consider sparse features (which allow superposition to get good average performance).

  1. ^

    The one unexcludable pair is ("always true", "always false").

    These are the seven embeddings:

Oops, I misunderstood what you meant by unimodality earlier. Your comment seems broadly correct now (except for the variance thing). I would still guess that unimodality isn't precisely the right well-behavedness desideratum, but I retract the "directionally wrong".

The variance of the multivariate uniform distribution  is largest along the direction , which is exactly the direction which we would want to represent a AND b.

The variance is actually the same in all directions.  One can sanity-check by integration that the variance is 1/12 both along the axis and along the diagonal.

In fact, there's nothing special about the uniform distribution here: The variance should be independent of direction for any N-dimensional joint distribution where the N constituent distributions are independent and have equal variance.[1]

The diagram in the post showing that "and" is linearly represented works if the features are represented discretely (so that there are exactly 4 points for 2 binary features, instead of a distribution for each combination).  As soon as you start defining features with thresholds like DanielVarga did, the argument stops going through in general, and the claim can become false.

The stuff about unimodality doesn't seem relevant to me, and in fact seems directionally wrong.

  1. ^

    I have a not-fully-verbalized proof which I don't have time to write out

Maybe models track which features are basic and enforce that these features be more salient

Couldn't it just write derivative features more weakly, and therefore not need any tracking mechanism other than the magnitude itself?

When you describe the "emailing protein sequences -> nanotech" route, are you imagining an AGI with computers on which it can run code (like simulations)?  Or do you claim that the AGI could design the protein sequences without writing simulations, by simply thinking about it "in its head"?

It would still be interesting to know whether you were surprised by GPT-4's capabilities (if you have played with it enough to have a good take)

POV: I'm in an ancestral environment, and I (somehow) only care about the rewarding feeling of eating bread. I only care about the nice feeling which comes from having sex, or watching the birth of my son, or being gaining power in the tribe. I don't care about the real-world status of my actual son, although I might have strictly instrumental heuristics about e.g. how to keep him safe and well-fed in certain situations, as cognitive shortcuts for getting reward (but not as terminal values). 

Would such a person sacrifice themselves for their children (in situations where doing so would be a fitness advantage)?

Any idea why "cheese Euclidean distance to top-right corner" is so important?  It's surprising to me because the convolutional layers should apply the same filter everywhere.

Agreed.  To give a concrete toy example:  Suppose that Luigi always outputs "A", and Waluigi is {50% A, 50% B}.  If the prior is {50% luigi, 50% waluigi}, each "A" outputted is a 2:1 update towards Luigi.  The probability of "B" keeps dropping, and the probability of ever seeing a "B" asymptotes to 50% (as it must).

This is the case for perfect predictors, but there could be some argument about particular kinds of imperfect predictors which supports the claim in the post.

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