A small network memorized modular arithmetic, then generalized a million steps later

Memorize first, generalize a million steps later

In 2022, Alethea Power, Yuri Burda, Harri Edwards, Igor Babuschkin, and Vedant Misra, all at OpenAI, trained a small transformer to fill in arithmetic tables and reported a training curve that has been cited ever since.¹ The task was deliberately plain. Take 97 symbols, treat them as the numbers 0 through 96, and write out a binary operation as a table with one entry for every pair of inputs. Their headline case was division modulo 97: divide one number by another in arithmetic that wraps around at 97. They showed the network half of the 9,409 entries and kept the other half hidden, to see whether it had learned the rule or only the answers.¹

Within roughly a thousand optimization steps, the network answered the training half almost perfectly. On the hidden half it sat near chance, and it stayed there far longer than anyone would keep training a model that looked finished. Only near a million steps did its accuracy on the unseen problems climb to near-perfect.¹ Fitting what it was shown took about a thousandth of the training that learning the rule did. The authors named that late rise grokking.

A 400,000-parameter network on ninety-seven symbols

How big was the thing that grokked? The transformer had two layers, a width of 128, four attention heads, and about 400,000 parameters outside its embeddings.¹ A current frontier model has hundreds of billions. The entire problem was 97 symbols and the table of one operation over them. It was trained with AdamW, an ordinary optimizer, with weight decay turned up. Shown half the table, the network needed on the order of a thousand times more steps to generalize than to memorize.¹ This is the scale to hold on to, because the word grokking now travels attached to systems many millions of times larger.

Two further results from the same paper matter for what came later. The effect depended sharply on how much of the table the network saw. Around a quarter to a third of the table, cutting the training data by a single percentage point raised the median time to generalize by 40 to 50 percent.¹ Less to learn from did not merely slow the network a little. It pushed the moment of generalization sharply further out. The authors also compared several ways of training and found one that helped far more than the rest. Weight decay, a standard penalty that pushes a network's weights toward smaller values, was the most effective. It more than halved the number of examples the network needed to generalize.¹

What forms while the test curve stays flat

The curve looks like a switch flipping. The most important thing later work established is that, underneath, nothing flips. In 2023 Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt took the simplest grokking case, addition modulo 113, and worked out exactly what the trained network was computing.²

Because they knew the algorithm, Nanda and colleagues could measure how much of it existed at each moment of training, without waiting for the test score to move. They built two readings from the network's internal frequencies: one for how complete the generalizing circuit is, another for how much the network still leans on rote memorized answers.² Tracked that way, training falls into three stretches.

The circuit starts forming and strengthening thousands of epochs before the test loss drops. The final fall, the part a person watching the curve would call the moment of grokking, is the network discarding the memorized answers it no longer needs. Nanda and colleagues describe grokking as "the gradual amplification of structured mechanisms encoded in the weights, followed by the later removal of memorizing components."² The jump is sudden only in the one number that was being watched.

This is also where the account of weight decay gets sharper. Nanda and colleagues find that weight decay drives that final cleanup, and in their setup they call it necessary for grokking to happen at all. The original paper had put it more mildly, ranking weight decay the most effective of several training choices it compared, the one that more than halved the data the network needed.²¹ Read together, they agree on direction even where they differ on strength: weight decay is the main force moving the network off the memorized solution. Other work reached a similar picture from a different angle. Ziming Liu and colleagues modeled grokking as a slow reorganization of how the network represents the numbers internally. For a transformer, they found, the grokking setting sits close to plain memorization, which is part of why the payoff takes so long to arrive.³

Grokking leaves the toy setting

None of this has stopped the word from traveling. Grokking now gets invoked whenever a large model seems to improve in a jump, as evidence that networks hold abilities that more training will unlock. Two lines of work say how much of that the research supports.

The first pulls the mystery out of the effect. Ziming Liu, Eric Michaud, and Max Tegmark traced grokking to the size of the network's weights, and showed they could turn it on and off at will.⁴ Tuning that one quantity, they induced grokking on images, language, and molecules, well outside arithmetic, and switched it off on the algorithmic tasks where it was first seen.⁴ Weight decay is one of the controls on that quantity, which ties this back to what both earlier papers found. On this account grokking is a property of the training landscape a researcher can dial, not a threshold a network crosses by getting big.

The second tests the scale question directly. In 2025 Ziyue Li, Chenrui Fan, and Tianyi Zhou tracked a 7-billion-parameter model through pretraining and reported delayed generalization at that scale.⁵ It is real, and it widens grokking well past toy networks. It is also not the single switch of the shorthand. What they found was local and staggered: different kinds of data, math or code or commonsense, generalized at different times, the harder and later-memorized ones lagging furthest, with no one moment when the whole model grokked.⁵ They had to detect it through internal measures rather than a test score, because there was no single curve to watch jump. The finding extends the phenomenon and cuts against the popular reading at the same time: at scale, grokking looks less like a hidden switch and more like many small, unsynchronized transitions.

The researchers who study grokking are careful about the leap. Writing for Quanta on the state of the field, the science journalist reports the effect has been studied in "only extremely small networks," and quotes the mathematician Mikhail Belkin calling the modular-arithmetic tasks where grokking lives "a drop in the ocean" next to what current networks do.⁶ The effect is genuine and reproducible, and it is still a long way from explaining why frontier models improve.