AF - A Robust Natural Latent Over A Mixed Distribution Is Natural Over The Distributions Which Were Mixed by johnswentworth

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Welcome 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: A Robust Natural Latent Over A Mixed Distribution Is Natural Over The Distributions Which Were Mixed, published by johnswentworth on August 22, 2024 on The AI Alignment Forum.
This post walks through the math for a theorem. It's intended to be a reference post, which we'll link back to as-needed from future posts.
The question which first motivated this theorem for us was: "Redness of a marker seems like maybe a natural latent over a bunch of parts of the marker, and redness of a car seems like maybe a natural latent over a bunch of parts of the car, but what makes redness of the marker 'the same as' redness of the car? How are they both instances of one natural thing, i.e. redness? (or 'color'?)".
But we're not going to explain in this post how the math might connect to that use-case; this post is just the math.
Suppose we have multiple distributions P1,…,Pk over the same random variables X1,…,Xn. (Speaking somewhat more precisely: the distributions are over the same set, and an element of that set is represented by values (x1,…,xn).) We take a mixture of the distributions: P[X]:=jαjPj[X], where jαj=1 and α is nonnegative.
Then our theorem says: if an approximate natural latent exists over P[X], and that latent is robustly natural under changing the mixture weights α, then the same latent is approximately natural over Pj[X] for all j.
Mathematically: the natural latent over P[X] is defined by (x,λP[Λ=λ|X=x]), and naturality means that the distribution (x,λP[Λ=λ|X=x]P[X=x]) satisfies the
naturality conditions (mediation and redundancy).The theorem says that, if the joint distribution (x,λP[Λ=λ|X=x]jαjPj[X=x]) satisfies the naturality conditions robustly with respect to changes in α, then (x,λP[Λ=λ|X=x]Pj[X=x]) satisfies the naturality conditions for all j.
"Robustness" here can be interpreted in multiple ways - we'll cover two here, one for which the theorem is trivial and another more substantive, but we expect there are probably more notions of "robustness" which also make the theorem work.
Trivial Version
First notion of robustness: the joint distribution (x,λP[Λ=λ|X=x]jαjPj[X=x]) satisfies the naturality conditions to within ϵ for all values of α (subject to jαj=1 and α nonnegative).
Then: the joint distribution (x,λP[Λ=λ|X=x]jαjPj[X=x]) satisfies the naturality conditions to within ϵ specifically for αj=δjk, i.e. α which is 0 in all entries except a 1 in entry k. In that case, the joint distribution is (x,λP[Λ=λ|X=x]Pk[X=x]), therefore Λ is natural over Pk. Invoke for each k, and the theorem is proven.
... but that's just abusing an overly-strong notion of robustness. Let's do a more interesting one.
Nontrivial Version
Second notion of robustness: the joint distribution (x,λP[Λ=λ|X=x]jαjPj[X=x]) satisfies the naturality conditions to within ϵ, and the gradient of the approximation error with respect to (allowed) changes in α is (locally) zero.
We need to prove that the joint distributions (x,λP[Λ=λ|X=x]Pj[X=x]) satisfy both the mediation and redundancy conditions for each j. We'll start with redundancy, because it's simpler.
Redundancy
We can express the approximation error of the redundancy condition with respect to Xi under the mixed distribution as
DKL(P[Λ,X]||P[X]P[Λ|Xi])=EX[DKL(P[Λ|X]||P[Λ|Xi])]
where, recall, P[Λ,X]:=P[Λ|X]jαjPj[X].
We can rewrite that approximation error as:
EX[DKL(P[Λ|X]||P[Λ|Xi])]
=jαjPj[X]DKL(P[Λ|X]||P[Λ|Xi])
=jαjEjX[DKL(P[Λ|X]||P[Λ|Xi])]
Note that Pj[Λ|X]=P[Λ|X] is the same under all the distributions (by definition), so:
=jαjDKL(Pj[Λ,X]||P[Λ|Xi])
and by
factorization transfer:
jαjDKL(Pj[Λ,X]||Pj[Λ|Xi])
In other words: if ϵji is the redundancy error with respect to Xi under distribution j, and ϵi is the redundancy error with respect to Xi under the mixed distribution P, then
ϵijαjϵji
The redundancy error of the mixed distribution is a...