THEORY-071: A memorising and a generalising circuit compete on logits per unit norm, and which one wins flips when the data makes memorising more expensive

Source

Varma et al. (2023), LIT-539 — ARXIV-2309.02390. The account is this paper’s: the three ingredients, the dataset-size argument for D_crit, and the ungrokking and semi-grokking predictions it derived and then observed.

The account

Three ingredients, claimed sufficient:

1. C_gen generalises and C_mem does not;
2. C_gen is more efficient — it produces equivalent cross-entropy at lower parameter norm;
3. C_gen is learned more slowly.

Then the story is short. Early on, C_mem is built fast and training loss goes to near zero. After that the cross-entropy term has almost nothing left to say and the regularizer does the talking: gradient descent keeps reducing total loss by moving parameter norm from C_mem to C_gen, and when enough has moved, test accuracy transitions. Nothing is discovered late. The generalising circuit was being built all along; what changes is which circuit is cheaper to keep.

The dataset-size argument. A classifier trained on D ∪ {(x, y*)} cannot be more efficient than one trained on D, so efficiency is non-increasing in dataset size on average. If the model already generalises to the new point, nothing changes — so C_gen’s efficiency is flat in D. If it does not, the memorising circuit must spend more norm — so C_mem’s efficiency falls. They cross at D_crit.

The index is not dataset size

That argument is about examples, and it is the part LIT-667 corrects by name. Wang et al. train on a mixture of atomic facts and facts deduced from them, and separate two knobs the algorithmic setting fuses:

• Hold the inferred/atomic ratio φ and scale the data. Nothing happens — not the gap between the train and test curves, not the level reached. There is no D_crit in this setting to measure.
• Hold the size and raise φ. Grokking accelerates monotonically, and at φ = 18.0 it is gone.

The efficiency argument survives this, and in fact the paper runs it: C_mem must store the inferred facts as well as the atomic ones, while C_gen stores the atomic facts twice at most, so N_mem grows with φ and N_gen is bounded. Scaling the data at fixed φ scales both, and the ratio of the two is what the regularizer sees. The crossover is real; dataset size was a proxy for it in a setting where every example is an inferred fact and there is nothing else to trade against.

So this document’s claim is now stated by what the crossover is indexed on — whatever makes memorising dearer while leaving generalising alone — which is the training fraction in modular addition and the derived-fact ratio in a knowledge graph. Both are the same quantity seen through different data.

This is the part of SOTA-402 the account explains: why added data should go to derived facts rather than to more atomic ones. Raising φ makes C_mem store more while C_gen’s burden stays bounded, so the crossover comes sooner; adding atomic and derived facts together at fixed φ scales both circuits’ costs and leaves the ratio the regularizer sees where it was. It does not explain that practice’s other limit — that no φ buys out-of-distribution composition — and it needs the weight decay that practice’s setup includes.

One caution, because it cuts the other way too: the reverse inference is not available. Modular addition has no atomic/inferred split, so φ is not defined there, and nothing here shows that the size dependence LIT-538 and LIT-085 measured was secretly a distribution effect. What it shows is that those experiments could not have told the difference, because they moved size and composition with one knob.

Why the evidence is the strongest in this cluster

The crossover implies two behaviours nobody had reported, and the paper went and found them.

Ungrokking. Take a network that grokked at D > D_crit and keep training it on D′ < D_crit. C_mem is now the cheaper circuit, so norm should move back and test accuracy should fall. It does — with a sharp transition in D′ around D_crit, on removal of examples alone rather than the introduction of new ones, and with a final test accuracy independent of weight decay, because D_crit is. That last sub-prediction is the one hardest to hit by accident.

Semi-grokking. At D ≈ D_crit the circuits are similarly efficient, so a mixture is possible and delayed generalisation should arrive at middling accuracy. The paper states that its theory permits either this or a clean winner and does not predict which, then reports observing semi-grokking.

All four predictions confirmed on 1-layer transformers with AdamW on modular addition at P = 113, with nine further algorithmic tasks in the appendix.

Why it is Proposed anyway

It needs weight decay, and LIT-537 exhibits grokking without any — on modular arithmetic, with a two-layer MLP, and with the parameter norm rising through the transition. The three ingredients are claimed sufficient rather than necessary, so this is not an internal contradiction; it does mean the account cannot be the general explanation, which is what its title reads as.

And the circuits are constructions. C_mem-only networks come from training on random labels, C_gen-only from large datasets with a check that > 95% of logit norm lies in the trigonometric subspace. The efficiency curves are measured on these, not on the mixed networks that actually grok.

LIT-667 does not close that. It traces a generalising circuit in a non-algorithmic setting, which the previous version of the promote_when asked for, and then argues the efficiency ordering by counting facts — how many each circuit must store — rather than by measuring norm per unit logit. A counting argument over a traced circuit is a better thing to have than a counting argument over a hypothesised one, and it is still not the measurement. That is what the rewritten promote_when asks for, and it now asks for it in the setting the account was built in rather than somewhere else.

The status is about scope. On evidential shape — risky predictions, made first, confirmed after — this is the best-supported account here, and THEORY-070, which contradicts it, leaves both of its novel phenomena unexplained. LIT-667 adds a second domain and a third confirmed prediction of the same shape (raising weight decay accelerates grokking, which the efficiency story implies and the paper then measured), which is why the correction above is an amendment and not a demotion.

A note on what this cost. The previous promote_when asked for “a measured D_crit … outside algorithmic data”. The paper that went outside algorithmic data reported that there is no D_crit there to measure. A promotion condition can be met in spirit and refuted in letter at the same time, and the honest response is to fix the condition rather than to read the paper as satisfying it — the same call as THEORY-032 v3, for a different reason: there the result was the right shape and the wrong measurement, here the condition named a quantity that does not exist in the setting it demanded.