How Small Can We Go? Calibrating Activation-Cache Compression for SAE Training

Abraham Yeung

cs.LGNov 5, 2026

How Small Can We Go? Calibrating Activation-Cache Compression for SAE Training

Authors: Abraham Yeung

Organizations: Stanford University

Abstract

To train a sparse autoencoder (SAE), you need a massive cache of model activations. We look into the effects that compression has on an SAE in two ways: we keep the original fp32-trained SAE and pass the compressed cache into the same SAE to see how many feature activations stay the same (the frozen-SAE test), and we retrain an SAE on the compressed cache and see how many of the original features still exist (the retraining test). We found that in the retraining test, there is no clear scale for retained features. On a GPT-2 cache, rounding everything to fp16 keeps 60.6% of features, but changing one value in a million by the smallest step fp32 allows (a change eight orders of magnitude smaller) keeps only 70.5%. Meanwhile, changing the random seed keeps 23.0%. For the frozen-SAE test, the frozen SAE's correlation with the original increases in order of how precise the codecs are, but plain $L_2$ distance gives nearly the same ordering. We also see the same frozen reading correspond to different percentages retained at different sites. This leaves the question of the right measurement. We build one by calibrating: retraining on identical bytes sets the ceiling, and a reseed sets the floor. We then measure the effect of each compressed codec against its own site's floor. With TopK SAEs on GPT-2 small and Pythia-1.4B at research-scale budgets, naïve per-token int4 loses more features than a reseed everywhere we tested, as do PCA and random projection on GPT-2. Standardizing each channel before quantizing brings int4 back above the floor in terms of retained features, at the same raw file size. At our smallest budget, a bf16 forward pass, which some pipelines use to save memory, loses slightly more features than rotated 8-bit storage. Calibrating a site costs four training runs and two evaluation passes.

Figures & tables

Figure 1: GPT-2 layer 6, seed 0: one ulp on a millionth of values already costs three quarters of fp16’s loss, so the count needs a floor. Bars: alive features a retrained SAE recovers at best match $r>0.9$ (grey: control; blue: tiny perturbations; orange: real codecs); dashed line: seed floor.

Appendix figures & tables34 assets

Supplementary material from the paper’s appendix.

Appendix

codec	B/tok	survived %	FVU	mean $r$	stable %	Hung. cos	frozen %
fp16	1536	$59.3\pm 1.1$	0.1355	0.866	95.7	0.817	100.0
bf16 storage	1536	55.6	0.1355	0.853	93.3	0.799	100.0
rot-int8	772	$52.7\pm 1.1$	0.1356	0.845	94.2	0.786	100.0
int8	772	46.6	0.1356	0.821	91.5	0.751	100.0
int8-o8	780	50.8	0.1355	0.837	93.7	0.773	100.0
rot-int6	580	$45.9\pm 0.7$	0.1355	0.823	92.1	0.752	100.0

Appendix

Table 1: Full GPT-2 codec panel.

\pm

rows are three-seed means (

\pm

half the range), bare cells seed 0, so fp16 reads

59.3\pm 1.1

here and

60.6

(seed 0) in the text. survived: best match above

r=0.9

against fp32 at the same seed; FVU: fraction of variance unexplained; mean

r

: the retrained SAE’s best-match correlation, averaged over alive features (not the frozen reading); stable: survival among the

774

features reproduced in both other fp32 seeds; Hung. cos: mean matched encoder and decoder cosine over all latents, dead ones included; frozen: the thresholded reading through the frozen fp32 SAE. FVU is flat across every viable codec (

0.1355

0.1361

) while seed-0 agreement falls from

60.6

37.6\%

. The floor row’s FVU is the fp32 reference’s own, and its Hungarian cosine averages the two seed-0 pairs.

codec	survived %	$\times$ floor	stable %
fp16 (bit-identical ceiling)	100.0	6.35 [6.00, 6.71]	100.0
rot-int8	43.0	2.73 [2.58, 2.87]	88.0
int8	32.9	2.09 [1.98, 2.20]	84.9
rot-int4	25.8	1.64 [1.55, 1.73]	81.6
int4	2.8	0.18 [0.15, 0.20]	13.7
seed floor	$15.7\pm 0.1$	1.00 $\times$	(def.)

Appendix

Table 2: Pythia-1.4B layer 8, same construction as Table 1 . Codec runs are seed 0 only, so gaps under the

2

pp tie band are unresolved; the bracketed intervals resample features against the seed-0/seed-1 floor pair, not seeds. fp16 is bit-lossless under the bf16 master, so its row is a same-seed ceiling.

codec	survived %	$\times$ floor	stable %
fp16 (bit-identical ceiling)	100.0	5.44 [5.17, 5.70]	100.0
rot-int8	38.4	2.09 [1.99, 2.18]	89.3
int8	27.7	1.51 [1.44, 1.57]	83.4
rot-int4	26.1	1.42 [1.35, 1.48]	83.0
int4	0.8	0.05 [0.03, 0.06]	5.1
seed floor	$18.4\pm 0.1$	1.00 $\times$	(def.)

Appendix

Table 3: Pythia-1.4B layer 16. Identical construction, interval convention and bit-identical fp16 ceiling reading to Table 2 . Source: runs_m4_L16/eval .

codec	GPT-2 $r$	retr.%	L8 $r$	retr.%	L16 $r$	retr.%
fp16	0.99997	60.6	1.00000	100.0	1.00000	100.0
rot-int8	0.99919	53.8	0.99922	43.0	0.99924	38.4
int8	0.99632	46.6	0.99488	32.9	0.98916	27.7
rot-int4	0.98536	37.7	0.98678	25.8	0.98696	26.1
int4	0.84005	15.6	0.56333	2.8	0.15633	0.8
seed floor	—	23.0	—	15.7	—	18.4

Appendix

Table 4: Frozen-SAE mean

r

beside retrained agreement at three sites. Within a site the frozen SAE ranks the codecs as retraining does; across sites the same reading carries no level (rot-int8 reads within

0.00005

everywhere and retrains to

2.09

–

2.73\times

the local floor). Codec rows are seed 0; floors are three-pair means.

codec	survived %	mean $r$
fp16	26.4	0.716
rot-int8	25.3	0.708
rot-int4	26.1	0.708
int4	13.7	0.623
seed floor	23.7–24.2	0.699

Appendix

Table 5: JumpReLU panel (GPT-2, calibrated

\lambda=8.68

). The viable codecs land within a few points of the floor (fp16

26.4

, rot-int4

26.1

, rot-int8

25.3

against

23.7

24.2

), too close together for a graded verdict. mean

r

: the retrained best-match correlation averaged over alive features. The fp32 and viable-codec runs end with about a third of latents dead (the naïve int4 run about a quarter), and the naïve int4 run converged to

L_{0}=38.2

against fp32’s 18.6, so its row is shown but excluded from any verdict as confounded.

budget	stab. thresh	fp16 %	rot-int8 %	rot-int4 %	gap pp
$1\times$ (33M visits)	0.900	95.7	94.2	88.8	7.0
$3\times$ (98M)	0.936	95.7	94.7	93.5	2.2
$10\times$ (328M)	0.954	95.7	96.3	94.2	1.6

Appendix

Table 6: Budget scaling on a fixed stable set (

N=774

features at every tier, binomial SE about

1.1

pp). fp16 is flat, the rotated codecs improve. fp16’s count is

741

774

at all three tiers by coincidence: the three pairs are distinct runs whose all-alive agreement is

60.6

49.2

and

50.6\%

. Budget was scaled by re-reading one 20M-token cache (up to about 16 passes over the same tokens), so this isolates optimization from data diversity but not from convergence on a fixed corpus; at

3\times

, fresh text gives the same floor (Table 25 ).

task	codec	$\Delta$ acc	direction cos
sst2	fp16	+0.0011	1.000
sst2	rot-int8	+0.0000	1.000
sst2	rot-int4	+0.0011	0.949
sst2	int4-o16	-0.0080	0.778
sst2	int4	-0.0608	0.489
sst2	pca256	-0.0046	0.500

Appendix

Table 7: Linear probes: accuracy change against the fp32-trained probe, and learned-direction cosine, each scored against a seven-split resample floor (

0.008

0.002

accuracy and

0.80

0.77

cosine on sst2/ag_news, mean over splits, with cosine minima

0.788

0.759

; probe_floor.json ). Accuracy resolution is one validation example, so read the deltas as counts. PCA-256 keeps accuracy on sst2 but not on ag_news (its drop there is outside the floor) while its directions collapse to about

0.5

on both; int4-o16 fails the sst2 direction floor (

0.778

against a floor minimum of

0.788

) on a test the retraining test passes. The two other codecs with probe runs, int8 and rot-int6, clear both floors on both tasks (

0.991

0.995

and

0.998

0.998

cosine) and are omitted for width.

	$>0.001$	$>0.002$	$>0.005$	$>0.01$	$>0.02$
gpt2_l6
n alive	4860	2927	1421	654	201
floor %	23.1	27.0	33.7	40.1	47.5
rot-int4 $\times$ floor	1.63	1.50	1.38	1.28	1.13
pythia_l8
n alive	6573	4053	1523	506	100

Appendix

Table 8: The alive cut, swept. Tightening it raises agreement and the floor together and lowers rot-int4’s ratio monotonically (

1.63

1.13\times

on GPT-2); our

0.001

cut is the loosest of the five. rot-int4 only, so this bounds a ratio, not an ordering; the floor here averages the two seed-0 pairs at each site (

23.1

against the three-pair

23.0

on GPT-2).

sparsity	seed floor %	rot-int8 %	$\times$ floor	stable-only %	$n_{\text{alive}}$	stable frac %
$k=16$	$28.6\pm 0.8$	49.1	1.72 $\times$	92.8	2,912	21.0
$k=32$	$23.0\pm 0.1$	53.8	2.34 $\times$	94.2	4,860	15.9
$k=64$	$14.7\pm 0.2$	47.4	3.23 $\times$	97.1	8,269	10.1

Appendix

Table 9: Sparsity moves the floor without touching a byte: over a

1.95\times

swing in the floor, rot-int8’s ratio spreads

1.72

–

3.23\times

while its survival among seed-stable features spreads only

92.8

–

97.1\%

treatment	B/tok	rel-L2	frozen $r$	survived %	$\times$ floor	stable %	Hung. cos	FVU
fp16 storage	1536	0.00020	0.99997	60.6	2.63	95.7	0.817	0.1355
rot-int8 storage	772	0.00551	0.99919	53.8	2.34	94.2	0.786	0.1356
int8-o8 storage	780	0.00928	0.99868	50.8	2.21	93.7	0.773	0.1355
bf16 forward	3072	0.00980	0.99819	51.2	2.22	94.2	0.775	0.1356
int8 storage	772	0.02660	0.99632	46.6	2.02	91.5	0.751	0.1356
seed floor	—	—	—	23.0	1.00	(def.)	—	—

Appendix

Table 10: A treatment that is not a storage codec: computing the cache with a bf16 forward (stored in fp32) beside the storage codecs. Over three seeds the bf16 forward retains

49.7

–

51.2\%

against rot-int8’s

51.6

–

53.8

, disjoint but with

n=3

(a sign test cannot go below

p=0.25

; within-seed deltas

-2.7

-1.8

-3.0

pp). At

3\times

budget the comparison reads

44.4

against

45.3\%

(one seed each), inside the tie band. It runs

2.21\times

faster.

codec	raw B/tok	zstd B/tok	ratio
fp16	1536	1419.0	1.08 $\times$
int8	772	510.9	1.51 $\times$
rot-int8	772	709.1	1.09 $\times$
int6	580	431.9	1.34 $\times$
rot-int6	580	557.0	1.04 $\times$
int4	388	146.4	2.65 $\times$

Appendix

Table 11: Lossless zstd-3 over the quantized payloads, measured on one shard per codec. Rotated codecs compress only

1.04

–

1.26\times

further; naïve scaling leaves headroom that zstd recovers. Standardization without the rotation lands with the rotated codec (

303.8

against

307.5

B/token).

codec	B/tok read	disk MB/s	decode MB/s	loader tok/s
fp32	3072	3094	10416	383k
fp16	1536	3085	4748	445k
int8	772	2743	754	319k
rot-int8	772	2917	242	205k
int6	580	2796	208	228k
rot-int6	580	2998	113	147k

Appendix

Table 12: Serial loader throughput per store (prefetch

=0

, warm page cache, one Apple M5). The

1\times

training run consumes

10{,}017

tokens/s, so even the slowest store (int4-o16, 55k tokens/s) outruns training; the standardize-and-rotate decode path costs real time (rot-int8 reads at

64\%

of int8’s rate). One-machine numbers. In a paired idle-machine measurement (two interleaved runs each), std-int4 decodes at 88 MB/s against rot-int4’s 89; our std-int4 keeps the rotation matmul with an identity matrix, so this does not isolate the rotation’s own cost.

	$r>0.7$	$r>0.8$	$r>0.85$	$r>0.9$	$r>0.95$
GPT-2 L6
floor (survival)	59.4	42.4	32.9	23.0	10.9
fp16	1.45 $\times$	1.83 $\times$	2.15 $\times$	2.63 $\times$	3.78 $\times$
rot-int8	1.39 $\times$	1.71 $\times$	1.97 $\times$	2.34 $\times$	3.19 $\times$
rot-int4	1.22 $\times$	1.39 $\times$	1.50 $\times$	1.63 $\times$	1.90 $\times$
int4	0.83 $\times$	0.79 $\times$	0.75 $\times$	0.68 $\times$	0.61 $\times$

Appendix

Table 13: Ratio to floor as the counting threshold moves. Levels change with the threshold; the ordering and the below-floor verdict for naïve int4 do not.

perturbation	rel-L2	frozen $r$	survived %	pp/decade
zero (bit-exact)	0	1.000000	100.0	—
noise $10^{-2}$	1.00e-02	0.998551	51.2	+7.82
noise $10^{-3}$	1.00e-03	0.999847	59.0	+2.38
fp16 (codec)	2.03e-04	0.999969	60.6	+0.07
noise $10^{-4}$	1.00e-04	0.999985	60.6	+1.19
noise $10^{-5}$	1.00e-05	0.999999	61.8	+1.69

Appendix

Table 14: The noise levels and sparse changes (seed 0; pp/decade is the change in agreement per decade of relative

L_{2}

to the next row): isotropic noise at an exact relative

L_{2}

and one-ulp changes to a fixed count of values, all stored in fp32 on both sides. Everything from

n=77

15{,}335

is flat; the one-value change also differs in when it reaches training (Appendix J ). The fp16-scale noise and the millionth-of-values change were re-run twice with fresh randomness (

60.6

60.4

60.8

and

70.5

69.7

71.6\%

); the table prints the original runs.

arm	rel-L2	all alive %	seed-stable %
noise $10^{-7}$	1.01e-07	66.0	96.4
noise $10^{-6}$	1.00e-06	63.5	97.0
noise $10^{-5}$	1.00e-05	61.8	96.1
noise $10^{-4}$	1.00e-04	60.6	95.3
noise $10^{-3}$	1.00e-03	59.0	95.9
noise $10^{-2}$	1.00e-02	51.2	93.4

Appendix

Table 15: The noise levels and codecs on the seed-stable stratum (

n=774

) beside all alive features. The response is flatter still on the stable stratum (

0.35

pp per decade from noise

10^{-7}

10^{-4}

, against

2.6

from fp16 to rot-int4), and the ratio of the two rates depends on the window, so we quote no single ratio.

criterion	perturbed %	floor %	margin (pp)	below ceiling (pp)
$10^{-7}$ dense
$r>0.50$	95.6	83.8	+11.8	4.4
$r>0.70$	88.3	59.4	+28.9	11.7
$r>0.85$	76.2	32.9	+43.3	23.8
$r>0.90$ (ours)	66.0	23.0	+43.0	34.0
$r>0.95$	46.3	10.9	+35.4	53.7

Appendix

Table 16: How far the two smallest perturbations sit above the seed floor at each counting threshold. The margin is positive everywhere (

11

–

43

pp dense,

12

–

47

sparse) but its size is threshold-dependent.

codec	direction cos	causal cos	effect/floor
ag_news (floor 0.9995 / 0.9998)
fp16	1.0000	1.0000	0.002
rot-int8	1.0000	1.0000	0.007
rot-int4	1.0000	1.0000	0.053
pca256	0.9998	0.9999	0.276
int4	0.7901	0.8294	13.713 FAIL

Appendix

Table 17: The causal test: a difference-in-means steering vector injected into the residual stream, scored against the variation between two disjoint halves of the same data (the half-versus-half floor runs about twice a full-data comparison, so it is conservative only for a fail; the like-for-like referent is half the floor). Naïve int4 fails at least

399

400

resample splits on both tasks; PCA-256 passes on average but fails

51

400

ag_news splits against the like-for-like referent; task-matched prompts and a strength sweep leave every verdict unchanged.

arm	B/tok	rel- $L_{2}$	frozen $r$	survived %	$\times$ floor
naïve int4	388	0.4553	0.84005	$16.0\pm 0.3$	0.69 $\times$
std-int4 (standardize only)	388	0.1033	0.98466	$38.0\pm 0.9$	1.65 $\times$
rot-int4 (standardize $+$ rotate)	388	0.0999	0.98536	$37.7\pm 0.3$	1.64 $\times$

Appendix

Table 18: Which step repairs 4-bit. std-int4 is byte-identical to rot-int4 with the rotation set to the identity, so the two differ by exactly the rotation: standardization alone gives

38.0\pm 0.9

%, adding the rotation

37.7\pm 0.3

%, inside the tie band. All three arms are three-seed means over same-seed pairs. The ablation keeps the rotation matmul, so it isolates the rotation’s effect, not its decode cost.

codec	lost/seed %	P(lost s1 $\|$ lost s0) %	indep. %	lost all 3 %	indep. %
rot-int8	6.1	17.8	6.3	0.13	0.023
rot-int4	11.1	25.3	12.0	0.90	0.134
int4	41.0	75.5	42.9	24.16	6.876

Appendix

Table 19: Is codec damage a shared bias or seed-like noise? On the fp32 seed-stable set (

n=774

), losses are enriched above independence for every codec, so a shared component exists, but under rot-int8 only

0.13\%

of seed-stable features are lost in all three seeds against

24.2\%

under naïve int4. Recomputed from released pair files; no training.

codec	$r>0.8$ %	$r>0.9$ %	$r>0.95$ %	Hungarian %
fp16	90.1 / 46.1	95.7 / 54.0	97.3 / 57.9	80.7 / 52.7
rot-int8	86.0 / 38.0	94.2 / 46.2	98.8 / 50.5	76.4 / 44.9
rot-int4	73.8 / 19.9	88.8 / 28.0	95.5 / 33.3	62.9 / 27.7
int4	41.3 / 2.9	58.9 / 7.4	73.6 / 11.3	26.9 / 11.1

Appendix

Table 20: Survival among seed-stable / seed-unstable features under four definitions of stability: activation-space matching at three thresholds, and weight-space Hungarian matching on encoder

+

decoder cosine. Under every definition the viable codecs keep far more of the stable features than of the unstable ones (stable 62.9 to 98.8%, unstable 19.9 to 57.9%), and naïve int4 is lowest on both; the levels move with the definition, so a stratified percentage is only interpretable with its definition attached. On Pythia the viable codecs keep 18.6 to 38.3% of the unstable features ( runs_m4_L8 , runs_m4_L16 stability files). GPT-2 L6, seed 0. Sources: stability_fp32_s0_t0p*.json , stability_hungarian.json .

site	$n$	frozen $r$ $\rho$	rel- $L_{2}$ $\rho$	cosine $\rho$
GPT-2 L6	15	0.982	0.979	0.982
Pythia L8	5	1.000	1.000	1.000
Pythia L16	5	1.000	1.000	1.000

Appendix

Table 21: Spearman rank agreement of three cheap readings with retrained agreement over every released condition per site. The 15 GPT-2 conditions are the 13 storage codecs of Table 1 plus the bf16 forward and the std-int4 ablation. None of the three needs a retrained SAE, and none ranks better than the others.

codec	CV AUC	mean act	peak act	rate
rot-int6	0.836	0.758	0.713	0.552
rot-int4	0.844	0.755	0.711	0.570
int4-o16	0.831	0.749	0.711	0.564
int6	0.838	0.747	0.709	0.573

Appendix

Table 22: Predicting per-feature survival from fp32 statistics. CV AUC: a logistic model on all three, 5-fold cross-validated; the other columns are single-statistic AUCs. Mean activation is the strongest single predictor, unlike the peak activation reported for weight quantization.

condition	$r>0.9$	same effect	same $\|\,r>0.9$	effect cos	$\Delta$ logp $r$	top-50 ctx
identical bytes	100.0	100.0	100.0	1.000	1.000	1.00
1 ulp on $10^{-6}$ of values	73.0	79.7	94.5	0.971	0.966	0.82
fp16	63.7	68.0	89.5	0.942	0.935	0.78
std-int4	38.7	38.7	73.3	0.842	0.816	0.62
naïve int4	15.7	12.7	46.8	0.651	0.584	0.38
reseed, seed 1	26.0	22.7	53.8	0.748	0.687	0.48

Appendix

Table 23: Do matched features do the same thing? 300 sampled alive fp32 features, each removed from the residual stream on its 32 strongest held-out rows, then its best match in each retrained SAE on the same rows. Same effect: cosine above

0.9

between the two changes in next-token log-probabilities; the last three columns are medians. std-int4 exceeds reseeding by 15.7 pp (95% CI 10.5 to 20.8); among pairs matched at

r>0.9

its effects are as close as a reseed’s (median difference

+0.014

, CI

-0.018

+0.042

). Seed 0, one sample.

condition	$m{=}1$	$m{=}4$	$m{=}8$	lost, $m{=}4$	set EV
identical bytes	100.0	100.0	100.0	—	—
1 ulp on $10^{-6}$ of values	$70.4\pm 1.1$	$74.0\pm 1.2$	$75.2\pm 1.1$	$13.2\pm 1.2$	$0.89\pm 0.00$
fp16	60.1	65.6	67.3	14.0	0.91
rot-int4	37.7	46.7	50.0	14.8	0.93
std-int4	$37.9\pm 0.8$	$46.6\pm 0.8$	$49.7\pm 0.8$	$14.3\pm 0.1$	$0.91\pm 0.01$
naïve int4	15.6	22.4	26.2	8.5	0.73

Appendix

Table 24: Is a lost feature gone or re-cut? Share of alive fp32 features recovered on held-out tokens by a least-squares map from their

m

best-correlated partners (

R^{2}>0.81

, so

m{=}1

is the retraining count re-scored on these tokens). Recovering from four partners keeps std-int4 13.4 pp above reseeding (three seeds each). Lost: of the features lost at

m{=}1

, the share four partners recover, 12–15% in every condition but naïve int4; four random partners recover at most 0.12%. Set EV: the lost features’ top-64 decoder subspace, applied to the retrained SAE’s unmatched features, as a fraction of its within-SAE value [ 6 ] .

\pm

rows are three runs.

regime	SAE	agreement %	$\times$ floor	stable %	fixed-774 %
replay (20M, $\sim$ 5 passes)	floor (fp32 reseed)	$33.3\pm 0.1$	1.00	—	—
	fp16	$49.0\pm 0.7$	1.47	$92.0\pm 0.2$	$95.8\pm 0.4$
	std-int4	$38.7\pm 0.6$	1.16	$87.8\pm 1.9$	$93.7\pm 1.7$
fresh (98.5M, one pass)	floor (fp32 reseed)	$33.4\pm 0.3$	1.00	—	—
	std-int4	$38.8\pm 0.3$	1.16	$88.0\pm 1.5$	$93.6\pm 1.2$

Appendix

Table 25: Fresh text against replay at the

3\times

budget (98.3M token-visits). Floors are three seed pairs; codec rows are same-seed pairs at three seeds, all

\pm

half the range. Stable: features matched above

r=0.9

in both other fp32 seeds of the same regime; fixed-774: the top 774 by cross-seed agreement, as in Table 6 . The fresh floor sits +0.09 pp from the replay floor and std-int4 keeps the same lead in both regimes.

candidate	tokens (M)	stored GB	agreement %	$\times$ ref
fp32, fresh (reference)	98.5	303	33.4 (33.2–33.7)	1.00
fp16, replayed	20.0	30.7	32.6 (32.0–33.0)	0.974
std-int4, more text	79.0	30.6	32.6 (31.4–33.3)	0.974
fp32, replayed †	20.0	61.4	32.8 (31.8–33.2)	0.980
naïve int4, replayed †	20.0	7.8	15.9 (15.7–16.1)	0.476

Appendix

Table 26: Equal stored bytes at

3\times

: fp16 on the 20M-token cache against std-int4 on the first 79.0M tokens of the fresh store, in 99.76% of fp16’s bytes. Agreement is cross-seed against fresh fp32 (six pairs per candidate, mean and range), so no candidate shares an initialization or data order with its reference; the reference row is fresh fp32 against itself across seeds (three pairs). The two candidates differ by +0.02 pp: a tie. † Post-hoc resolution check, not pre-registered (naïve int4: seed 0 only, two pairs).

format	decode	local	busy	volume	busy
fp32	CPU (paper)	117.0	0.83	75.1	0.99
	GPU	118.5	0.51	119.5	0.53
fp16	CPU (paper)	114.6	0.89	115.1	0.84
	GPU	121.0	0.28	119.7	0.33
std-int4	CPU (paper)	92.9	0.99	50.1	1.00
	CPU, no matmul	92.6	0.99	90.9	0.99

Appendix

Table 27: Training throughput on an A100 (thousand tokens/s, full

1\times

run, mean of two fresh containers) by cache format and where the loader decodes; busy is the share of training time the loader spent reading, decoding and shuffling. Every decode path of a format trains bit-identical weights. Best std-int4 over best fp32: 1.09 from local disk, 1.04 from the network volume, under the pre-registered

1.10

for a material gain. Volume reads varied 60–840 MB/s between runs.

codec	frozen mean $r$	model $\rho$	statistic $\|\rho\|$	crossing %
fp16	1.0000	0.736	0.644	–
bf16	0.9998	0.888	0.734	–
int8	0.9963	0.989	0.843	–
int8-o8	0.9987	0.966	0.818	–
int6	0.9832	0.991	0.861	–
int6-o8	0.9948	0.979	0.838	–

Appendix

Table 28: The frozen-SAE test explained by TopK margins (GPT-2 layer 6, the frozen test’s own SAE and tokens). model: Spearman correlation, over alive features, between predicted and measured frozen disagreement

1-r

; the prediction uses only the second moment of the codec’s error, estimated on held-out tokens, and the SAE’s margins on clean activations. statistic: the largest

|\rho|

of log firing rate, log mean or log peak activation with

1-r

. crossing %: median share of a degraded or damaged feature’s squared error that comes from tokens where it is active on one side only (–: not reported for the noise-like codecs, which have at most 6 degraded or damaged features each). Below the rule, codecs that remove part of the signal rather than add noise. Sources: geom_margins.json , geom_summary.json .

target	positive %	statistics	geometry	both
$1\times$ , $k=32$ (fit)	15.9	0.846	0.829	0.868
$3\times$	24.9	0.787	0.741	0.794
$10\times$	32.9	0.762	0.696	0.752
$3\times$ fresh text	24.9	0.793	0.738	0.794
$k=16$	21.0	0.817	0.801	0.837
$k=64$	10.1	0.899	0.842	0.902

Appendix

Table 29: Predicting from one SAE which features retraining recovers (GPT-2 layer 6, AUC). Above the rule the target is seed stability (best match

r>0.9

in both other seeds); the logistic model is fit once on the

1\times

k=32

SAE (its row is cross-validated) and applied unchanged to the other regimes’ own SAEs. Below the rule, the same seed-fit model predicts which features survive retraining on each codec. positive %: seed-stable or retained share of alive features. statistics: log firing rate, log mean and log peak activation. geometry: co-firing share with the best partner, decoder coherence, co-firing of the nearest decoder atom, TopK margin relative to the activation, encoder-decoder cosine. Predictors use held-out eval tokens. Source: geom_identifiability.json .

change	step	agreement %	distance	+2,000 steps
$10^{-7}$ , one batch	0	64.7	0.6561	0.2338
	4,000	99.9	0.0830	0.0452
	6,000	100.0	0.0301	0.0301
	7,000	100.0	0.0198	–
	7,500	100.0	0.0107	–
	7,800	100.0	0.0041	–

Appendix

Table 30: In this run, when a perturbation reaches training decides its cost (GPT-2 layer 6, seed 0, one run per row). Each perturbed run is a copy of the reference run, which reproduces Table 1 ’s fp32 seed-0 SAE bit for bit, that receives the change at the listed step(s) and trains on the same batches. agreement: the paper’s retained percentage against the reference. distance: relative Frobenius distance of the decoder from the reference’s, at the end and 2,000 steps after the change. The one-ulp row replays the one-scalar experiment (its altered value enters training twice; the weights are bit-identical to the reference at every logged step from 130 until the second entry; optimizer moments were not compared) and reproduces that SAE bit for bit. The last row is not a copy: it is the separately trained noise7 SAE of Table 14 , where every batch carries the change, scored against the same reference. Sources: geom_summary.json , pair_geomdyn_*.json .

Figure 2: Retrained agreement against perturbation size. Blue: calibrated isotropic noise, fp32 on both sides; orange: real codecs; star and diamond: one ulp on one scalar and on a millionth of them; the shaded band marks fp32’s own precision, below which no dense fp32 noise can go. Retraining barely responds to the size of tiny perturbations but separates the real codecs clearly. GPT-2 L6, seed 0; the fp16-scale noise and the millionth-of-values change were each run three times and are plotted at their original run.

Figure 3: Agreement as a multiple of each site’s own seed floor, against measured bits per dimension, $1\times$ budget (33M token-visits on GPT-2, 24.6M on Pythia). Filled markers: fp16 and the rotated codecs; hollow: naïve per-token at the same rate. The dotted riser is the effect of standardizing and rotating at identical bytes (on GPT-2, standardizing alone gives the same, Table 18 ), and at 4 bits it spans the floor at every site. Read the ordering, not the levels: at $10\times$ budget rot-int4’s ratio is $1.09\times$ . Pythia’s fp16 is its own bf16 master, so it is omitted there.

Figure 4: Agreement on a fixed-size stable set ( $N=774$ , the $1\times$ tier’s stable-feature count, taking the top $N$ by cross-seed stability at each budget) against training budget; error bars are binomial standard errors. fp16 is flat; the rotated codecs improve, so the gap does not widen.

Figure 5: The frozen-SAE reading against the retraining it stands in for, at three sites; the $x$ axis is frozen disagreement $1-r$ , log scale, improving rightward. Every series is monotone (the frozen SAE ranks codecs correctly), but the series are offset from each other (the same reading corresponds to different retained percentages at different sites); the viable codecs crowd into a span a linear axis would collapse to a point. Dotted lines are each site’s seed floor. Pythia’s fp16 point is omitted: the bf16 master makes it bit-lossless.

Activation Compression

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Scaling Laws for SAE Training Data
Nov 5, 2026Nikita Koriagin, Nikita Balagansky, Daniil GavrilovSparse Autoencoders Memory-Efficient Training
Yandex · T-Tech
Perplexity Can Miss SAE Feature Damage Under Quantization
Jun 2, 2026Evan DuanLLM Quantization LLM Interpretability
University of Michigan
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Aug 22, 2025David Chanin, Adrià Garriga-AlonsoActivation Sparsity LLM Interpretability
University College London · MATS Research

Nov 5, 2026J/K move · Enter open · S save

Scaling Laws for SAE Training Data

Nikita Koriagin, Nikita Balagansky, Daniil Gavrilov

Yandex · T-Tech

Sparse Autoencoders Memory-Efficient Training Data-Efficient Learning

Open paper AskPDF arXiv

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.

Abstract

Activation Compression

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