Scaling Laws for SAE Training Data

Abstract

Activation storage is often a bottleneck in sparse autoencoder (SAE) training. We study how many unique activations an SAE needs when its total training budget is fixed. Quality improves rapidly as the buffer grows, then reaches a plateau where replaying the buffer is nearly as effective as using fresh activations throughout training. Beyond this point, additional unique activations bring little benefit to reconstruction or interpretability, so the buffer need not grow with the training budget. We describe this tradeoff between diversity and repetition with a scaling law for limited data. We validate it across dictionary sizes and token budgets, early, middle, and late layers of Llama-3.1, Qwen3 model sizes, and BatchTopK, TopK, and JumpReLU SAEs. Sweeps using mathematical text and Python code test the effect of changing the activation source. Evaluation covers explained variance (EV), reconstruction MSE, feature absorption, automated interpretability, and sparse probing. These results give a practical training rule: choose the smallest buffer that reaches the quality plateau and replay it for the remaining budget. In our experiments, this reduces activation storage by $8\text{--}64\times$ with negligible quality loss.