Paper review · November 7, 2026
Scaling Laws for SAE Training Data
With total training volume held fixed and the number of unique activations varied, SAE quality saturated at a certain buffer size. Reusing a small buffer repeatedly cut storage by 8–64x.
Problem
Training SAEs (sparse autoencoders) for interpretability research at large scale requires storing terabytes of activations. If every update uses new activations, storage grows in proportion to training volume. It was not known how many unique activations are needed before repeated reuse (replay) can replace new data.
Method
The authors trained BatchTopK SAEs with total training tokens held fixed (128M, 640M, 1.28B) while varying the share of unique activations from 1/128 to 1. The target was layer 15 of Llama-3.1 8B, with dictionary sizes from 16k to 128k. They fit an effective activation count adapted from data-constrained scaling laws (Muennighoff et al.), and a law in which the amount of repetition that can be tolerated depends on dictionary size. They validated the results on other Llama layers, other Qwen3 sizes, TopK and JumpReLU SAEs, and math and code data. They also measured interpretability metrics with SAEBench.
Results
- Explained variance (EV) rose quickly while the buffer was small, then saturated. For example, with a 128M budget and a 16k dictionary, increasing unique activations from 1M to 128M raised EV from 0.5801 to 0.6393.
- The scaling law fitted to the 96 runs had an RMSE of 4.8×10⁻³ EV. AIC favored the version in which the allowed degree of repetition depends on dictionary size.
- At a 1.28B budget, the saturation range was nearly proportional to dictionary size (16k: 20M–40M, 128k: 160M–320M). The 16k dictionary cut storage by up to 64×.
- With a 640M budget and a 32k dictionary, BatchTopK, TopK and JumpReLU all saturated at 40M unique activations (a 16× reduction). FineMath saturated at 80M (8×) and Python code at 40M (16×).
- On SAEBench, reconstruction MSE and AutoInterp did not drop much for buffers chosen by EV. However, feature absorption did not improve monotonically as diversity increased.
Limitations
The full scaling grid was run only on layer 15 of Llama-3.1 8B with fixed sparsity (k=32). For other layers, models and functions, only the saturation pattern was checked at a single width and budget. Saturation points can only be located to an interval between log-grid points. The recommended buffer size depends on the tolerance (ε), the reference EV and the experimental grid. JumpReLU has a different sparsity, so its absolute performance cannot be compared.
Why it matters
It offers a practical rule (pick the smallest buffer that reaches saturation and reuse it repeatedly) that sharply reduces activation storage, a large part of SAE training cost, with almost no loss in quality. This lowers the barrier to entry for interpretability research.
Key terms
- SAE(Sparse Autoencoder)
- An autoencoder that breaks a model's internal activations down into combinations of many features, only a few of which are active at a time, so that people can interpret them.
- Activation buffer replay
- A method that stores only part of the unique activations and reuses them for training across multiple epochs.
- Explained variance (EV)
- A reconstruction quality metric that shows how much of the variance in the original activations the SAE reconstruction explains.
- Effective number of activations
- A value that discounts the worth of reused data and converts it into the equivalent number of new data points.
Original
Authors: Nikita Koriagin, Nikita Balagansky, Daniil Gavrilov · Posted 2026-11-05 · License CC BY 4.0 · Based on the full text
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