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Equation 7 · Part 1 · What Interpretability Actually Costs to Do at Scale

Symbol k

kk
kk

What this part means

the because published topk configurations keep.

Its job in the formula

k is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Where the article explains it

Because published TopK configurations keep k in the tens to low hundreds while n runs into the millions, n ≫\gg k and the encoding term dominates almost entirely: CSAEC_{\mathrm{SAE}} ≈\approx 2dnT .

The passage around this formula

and the specific variant OpenAI’s interpretability team used to push this to frontier scale, the TopK autoencoder, replaces the soft ℓ1\ell_1 penalty with an explicit constraint: exactly k latents fire, chosen by magnitude, and the rest are hard-zeroed. Gao and colleagues introduced this variant, established scaling laws relating autoencoder size and sparsity to reconstruction error, and — this is the operative fact for a cost accounting — trained a sixteen-million-latent autoencoder on GPT-4’s activations over forty billion tokens [ 1 ] .

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Sources cited in the surrounding passage

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