Thanks! Yeah that’s right. My colleague Nischal Mainali uses the term cavity method for this bulk-system distinction (what I call “background-foreground” in the body). I think the term originally meant something a little more specific and spin-glassy but has become the term of art for all mean field settings, at least in certain stat-phys contexts?
And great question. If you have a large D-dimensional space of fields associated to neurons, you might a priori think that you would need something like exp(D) neurons to “fully sample” the distribution (i.e. get something that looks dense like my point cloud in the relevant space of fields). But in practice, mean field methods require much much fewer particles to be valid. This happens in physics of course (where one has an infinite-dimensional or huge space of macroscopic observables, but predictions from the infinite-dimensional limit are true already for pretty small systems).
In the NN context I’m working on a paper that explains why in mean field you actually need only polynomially many neurons (in the sample size or some complexity parameter) for the mean field prediction to be true to high order. A useful intuition here is that while the “cloud” of neurons is high-dimensional in general, the thing we ultimately care about for e.g. generalizability is accuracy on a random test input. Reductively, this means that the cloud is one-dimensional and the law of large numbers kicks in very soon. So say we abstractly know that a reasonable mean-field distribution of neurons exists, and the output is additive in the single-neuron field from this distribution to leading order (the standard cavity method assumption). Then it’s a distribution in some high-dimensional function space and might have low-probability regions, may require exponentially many neurons, etc. But if we have some sampler of this space and have sampled N neurons, we immediately have accuracy to within log(N)/\sqrt{N} on almost all inputs (just by usual CLT arguments—here it’s convenient to use bounded activations, which is why we’re using these in experiments). The nontrivial thing is actually proving that a “good cavity method distribution” exists and isn’t too crazy.
In the superposition case things are particularly nice. In the superposition setting I’ve looked at here https://www.lesswrong.com/posts/siu22scEfuKxpSgfK/a-tale-of-three-theories-sparsity-frustration-and, the different superposition components are just independent theories that interact via a mass term that encodes interferences. When the width is small we’re actually not at all in the usual mean field setting (the effective mean field is heavily modified to account for the small width), but the heuristic story is there
I (kinda) recognized it because my partner (who actually did the recognizing) uses it to study finite-sized microbial ecology models. In their case they add an additional “cavity” species to an existing community and solve for self consistency. I’m very excited to see the superposition work.
Thanks! Yeah that’s right. My colleague Nischal Mainali uses the term cavity method for this bulk-system distinction (what I call “background-foreground” in the body). I think the term originally meant something a little more specific and spin-glassy but has become the term of art for all mean field settings, at least in certain stat-phys contexts?
And great question. If you have a large D-dimensional space of fields associated to neurons, you might a priori think that you would need something like exp(D) neurons to “fully sample” the distribution (i.e. get something that looks dense like my point cloud in the relevant space of fields). But in practice, mean field methods require much much fewer particles to be valid. This happens in physics of course (where one has an infinite-dimensional or huge space of macroscopic observables, but predictions from the infinite-dimensional limit are true already for pretty small systems).
In the NN context I’m working on a paper that explains why in mean field you actually need only polynomially many neurons (in the sample size or some complexity parameter) for the mean field prediction to be true to high order. A useful intuition here is that while the “cloud” of neurons is high-dimensional in general, the thing we ultimately care about for e.g. generalizability is accuracy on a random test input. Reductively, this means that the cloud is one-dimensional and the law of large numbers kicks in very soon. So say we abstractly know that a reasonable mean-field distribution of neurons exists, and the output is additive in the single-neuron field from this distribution to leading order (the standard cavity method assumption). Then it’s a distribution in some high-dimensional function space and might have low-probability regions, may require exponentially many neurons, etc. But if we have some sampler of this space and have sampled N neurons, we immediately have accuracy to within log(N)/\sqrt{N} on almost all inputs (just by usual CLT arguments—here it’s convenient to use bounded activations, which is why we’re using these in experiments). The nontrivial thing is actually proving that a “good cavity method distribution” exists and isn’t too crazy.
In the superposition case things are particularly nice. In the superposition setting I’ve looked at here https://www.lesswrong.com/posts/siu22scEfuKxpSgfK/a-tale-of-three-theories-sparsity-frustration-and, the different superposition components are just independent theories that interact via a mass term that encodes interferences. When the width is small we’re actually not at all in the usual mean field setting (the effective mean field is heavily modified to account for the small width), but the heuristic story is there
I (kinda) recognized it because my partner (who actually did the recognizing) uses it to study finite-sized microbial ecology models. In their case they add an additional “cavity” species to an existing community and solve for self consistency. I’m very excited to see the superposition work.