Inside the Loop: A Mechanistic Study of Weight-Tied Transformers on Depth-Bound Algorithmic Tasks

When does a weight-tied looped transformer -- one block applied T times -- implement an actual algorithm? We answer with four findings from controlled populations on group word problems. (1) The budget law: free training installs a linear computation frontier, a mechanism that solves v positions per loop, whose speed is priced by the training contract: v ~ n_train/T_train (exponent 0.98 +/- 0.04, R^2=0.99), exactly unity under T=n training. SGD selects a frontier matching the minimum the contract demands; granting more test-time loops than ever trained rescues late positions at fixed input length, yielding a principled halting rule T* = ceil(n / v-hat). (2) Architecture prior, not expressivity, picks the algorithm: standard-depth transformers learn parallel scans on this family; weight tying flips the selection to the serial frontier, even when positional addressing for a log-depth scan is supplied. At matched depth and parameters, untied models extrapolate worst and fail to learn A5 at all. (3) The walls are not where circuit complexity says: NC1-completeness costs nothing (A5 generalizes fully), while group order does (S5's 120x120 operator deadlocks joint learning) -- and an operator-first curriculum dissolves the wall in every seed. (4) Mechanisms are portable, not mandatable: warm-starting across budget contracts transfers the algorithm in every seed, re-pricing its speed, while imposing seriality through the input schedule fails where free training succeeds. These results are invisible to standard instruments, which provably saturate at the fixed points trained loops converge to. We introduce a head instrument, the convergence-time scaling tau(n,i), validate it causally via damage cones whose slope reproduces v, and show in-distribution head measurements predict out-of-distribution fate where tail metrics do not. Results replicate on the public easy-to-hard benchmark.

Looped Transformers scale sequential computation by applying a compact stack of physical blocks for multiple rounds, increasing unrolled depth without increasing stored parameters. This reuse changes the residual-scaling problem: in an untied Transformer, each residual branch receives and applies its own parameter update, whereas in a looped Transformer one shared update aggregates gradients from repeated visits and is read back by those same visits in the next linearized forward pass. We formalize this tied-depth effect through a first-order perturbation bound controlled by a visit-alignment coefficient $κ_R$. The bound recovers the DeepNorm exponent when visits decorrelate, but in the conservative aligned regime it requires the exponent to increase from $1/4$ to $1/2$ as loop count grows at fixed physical depth. The resulting method, DeepLoop, keeps the Post-LN DeepNorm architecture and sets $α=(2N)^{1/2}$ and $β=(8N)^{-1/2}$ for unrolled depth $N$. On GPT-style looped language models at GPT-2 small and GPT-2 medium scale, DeepLoop is neutral when no physical block is revisited and improves validation loss and downstream accuracy once recurrent depth is activated. These results show that stable recurrent depth requires residual scaling rules that account for parameter visits, not only nominal layer count.

Looped Transformers have drawn growing attention because looping improves multi-step reasoning and increases the executed depth of a model without adding parameters. They achieve this by reusing a stack of $L$ layers for $N$ loops. However, increasing the loop count $N$ not only destabilizes training but also shifts the best hyperparameters, so they must be searched again for every loop count. Inspired by Depth-$μ$P (Yang et al., 2024), which multiplies each residual branch by $L^{-1/2}$ for depth $L$ so that a learning rate tuned on a shallow untied network transfers to a deep one, we study how the residual branches and the learning rate should scale with the loop count $N$. We find that loops cannot simply be treated as extra depth, because the loops share weights. At initialization, distinct layers add nearly uncorrelated increments to the residual stream, so their sum grows as $\sqrt{L}$, whereas a shared layer adds positively correlated increments at every loop, so their sum grows linearly in $N$. For a single layer looped $N$ times, we show in the infinite-width limit that multiplying the residual branch by $1/N$ and keeping the learning rate independent of $N$ is the only parameterization with both a stable forward pass and maximal residual-stream updates. We then generalize the analysis to a stack of $L$ distinct layers looped $N$ times to obtain a parameterization we call Loop-$μ$P, in which the loop factor $1/N$ combines with the depth factor $L^{-1/2}$. Empirically, we find in language modeling that the base learning rate tuned on the smallest model stays best under Loop-$μ$P for loop counts up to $8$ and executed depths up to $384$ layers. These results make Loop-$μ$P a practical recipe for scaling looped Transformers.