You only learn what a fake coin feels like after there are too many fakes to remember.
Most of us were taught that memorization is the enemy of understanding. Rote learning produces parrots; good teaching starts with the why and lets students skip the tedious drilling. The belief feels humane and modern. A small result from machine learning suggests it gets the relationship between the two almost exactly backwards.
Start with a person rather than a network. You are learning to spot counterfeit coins, and your boss shows you five fakes. You memorize every scratch, every nick, the exact placement of the mint mark on each. Asked “is this fake?”, you compare the coin against your stored images. On those five coins you are perfect. On a sixth you are useless. This is overfitting: a lookup table masquerading as a skill.
Then your boss keeps throwing coins at you. Hundreds, then thousands. Memorizing every scratch becomes impossible, and somewhere in that overload your brain stops tracking scratches and notices that all the fakes are slightly lighter. You stop comparing images and just feel the weight. Thousands of memories have collapsed into one rule. The “aha” was not inspiration. It was your mind running out of room and being forced, by pressure alone, to find the structure that let it stop hoarding. Understanding is what happens when memorization becomes too expensive to continue.
What the networks showed
In 2022, researchers at OpenAI reported the same shape in neural networks and called it grokking. A network trained on a small algorithmic task memorizes its training set early, then sits on a plateau where its performance on unseen examples stays near zero for thousands of additional steps. It looks finished, and stuck. Then, sometimes long after a human would have stopped training, performance on unseen examples suddenly jumps. The network no longer just recognizes its training data. It has compressed the dataset into the underlying rule and can now generalize to inputs it has never seen.
Three things have to be present for that jump, and each has a counterpart in the coin room. The network needs massive exposure, enough examples that no finite lookup table fits them all; five coins never force a rule, because memorizing five is cheap. It needs a resource constraint, usually weight decay, a steady penalty for using too many parameters, which makes the bloated memorized solution costly to keep; the coin inspector’s version is the limit on what anyone can hold in their head. And it needs training past apparent stagnation, because the transition only comes if you keep going after the model looks done. Every signal an observer has, from loss curves to the felt sense of progress, says to stop right before the thing you are waiting for arrives.
Put in human terms: memorization is not the opposite of understanding; it is the prerequisite scaffolding. You cannot grok a pattern you have not first been buried under, because the compression has nothing to compress without the raw data. 死读书, dead reading, is what the scaffolding looks like before the collapse. It is also what it looks like if the collapse never comes.
Two ways to miss it
That gives two ways to fail, which are near enemies of understanding because each keeps the appearance of learning while preventing the transition.
The first is pure memorization. The inspector is only ever shown the same five coins and rewarded for recognizing them exactly, so there is never pressure to compress. A student trained this way passes every exam drawn from the familiar set and shatters at the first question phrased differently.
The second is the conceptual-first fix. Skip the drudgery, says the reformer: just tell the new inspector that fakes are lighter. Now they can recite the rule, but their hand has never felt a two-gram difference across a thousand coins, so the rule floats free of anything it could be a rule of. They have the vocabulary of compression without the material that was compressed.
One refuses to leave the overfit, and the other refuses to enter it. The genuine article requires both terrors at once — the tedium of massive rote and the confusion of watching the rote fail to cohere until it suddenly does. The unfashionable conclusion follows: Worse-is-better reality: just force them to do ten thousand equations. It is ugly and intellectually unromantic, but it supplies the volume and pressure the transition needs, which clever curricula that try to skip the drilling do not.
That conclusion has limits. Grokking needs a rule to find. Mathematics, grammar, chess tactics, harmony and movement have latent structure; a list of historical dates does not, and no amount of drilling will collapse it into anything. The raw material also has to be vivid enough to keep a person going: Kieran Egan argues that story, image, rivalry and anomaly keep a learner engaged, and a human training run abandoned out of boredom never reaches the far side of its plateau. And a person is not a network. The analogy explains why understanding can arrive late and suddenly; it does not tell you how many equations a particular student needs.
The long flat stretch
The hardest part of grokking is the plateau, because nothing visible changes until everything does. The Expert Beginner is what happens when a learner mistakes the plateau for the destination: a bowler stuck at 160 who refuses the grip change that would make him worse before it made him better. Most learners stop when they have an answer that sounds right, and most careers stop when they have a skill level that pays.
So the reframe grokking offers is that understanding is not a higher activity than memorization — it is memorization’s collapse under pressure. You cannot bypass the lower level to reach the higher one, and you cannot stay at the lower level and pretend you have arrived. For a learner, that is both harsh and freeing. The tedium, the confusion and the long stretch where practice seems not to pay are not the obstacle in front of understanding. They are where it is being built. Which leaves an uncomfortable question for anyone who quit something on a plateau: how many more coins would it have taken?
References:
- Power, Burda, et al., Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets (2022)
- Nanda, Chan, et al., Progress measures for grokking via mechanistic interpretability (2023)