Meta-memorization and memorization scaling laws in transformers

Larger transformers can memorize more data, with a capacity (in bits) that scales roughly linearly with parameter count. Counterintuitively, larger models also memorize *faster*, requiring fewer repetitions of a sequence during training before it is perfectly memorized. We reproduce both memorization scaling laws in transformers trained on unstructured sequences, demonstrating that efficient memorization is not due to a larger model's superior ability to learn structure in data. Rapid memorization is due to a *meta-memorization* effect, where models trained to memorize a large set of random sequences learn to memorize novel sequences more quickly. The rate $r(t)$ at which models memorize grows with training time $t$ as a power-law $r(t)\sim t^{\beta }$ with a fixed exponent $\beta$ but an increasing upper cutoff across model scales. In one-layer models, we trace the origin of meta-memorization to structured attention heads that convey length-invariant sequence information to MLPs where memorized sequences are stored. Meta-memorization is explained using a simplified model of a dense associative memory where retrieval is constrained by an attentional bottleneck. Put together, our results suggest transformers dynamically acquire reusable mechanisms that enable highly efficient storage and retrieval of memories.