[taliesinb]

There is an obvious implication: since the initial models were trained without loops, it is exceedingly unlikely that a single stack of consecutive N layers represents only a single, repeatable circuit that can be safely looped. It is much more likely that the loopable circuits are superposed across multiple layers and have different effective depths.

That you can profitably loop some say 3-layer stack is likely a happy accident, where the performance loss from looping 3/4 of mystery circuit X that partially overlaps that stack is more than outweighed by the performance gain from looping 3/3 of mystery circuit Y that exactly aligns with that stack.

So, if you are willing to train from scratch, just build the looping in during training and let each circuit find its place, in disentangled stacks of various depths. Middle of transformer is:

(X₁)ᴹ ⊕ (Y₁∘Y₂)ᴺ ⊕ (Z₁∘Z₂∘Z₃)ᴾ ⊕ …

Notation: Xᵢ is a layer (of very small width) in a circuit of depth 1..i..D, ⊕ is parallel composition (which sums the width up to rest of transformer), ∘ is serial composition (stacking), and ᴹ is looping. The values of ᴹ shouldnt matter as long as they are > 1, the point is to crank them up after training.

Ablating these individual circuits will tell you whether you needed them at all, but also roughly what they were for in the first place, which would be very interesting.

[taliesinb]

And i bet these would be useful in initial and final parts of transformer too. Because syntactic parsing and unparsing of brackets, programming language ASTs, etc is highly recursive; no doubt current models are painfully learning "unrolled" versions of the relevant recursive circuits, unrolled to some fixed depth that must compete for layers with other circuits, since your total budget is 60 or whatever. Incredibly duplicative and by definition unable to generalize to arbitrary depth!

[awwaiid]

Maybe another idea, no idea if this is a thing, you could pick your block-of-layers size (say... 6) and then during training swap those around every now and then at random. Maybe that would force the common api between blocks, specializaton of the blocks, and then post training analyze what each block is doing (maybe by deleting it while running benchmarks).

[taliesinb]

Amusingly, you need only have circuits of prime depth, though you should probably adjust their widths using something principled, perhaps Euler's totient function.