Strength Controls Change How Interventions Compare Across Sparse Autoencoder Codes

Jinchen Zhang; Zheyuan Liu; Yihan Zhu; Meng Jiang

cs.LGOct 19, 2026

Strength Controls Change How Interventions Compare Across Sparse Autoencoder Codes

Authors: Jinchen Zhang, Zheyuan Liu, Yihan Zhu, Meng Jiang

Organizations: University of Notre Dame

Abstract

Changing sparse autoencoder (SAE) codes can alter both intervention direction and magnitude, so comparing intervention effects requires a strength control specifying what is held equal. Norm matching and output calibration provide different controls, but whether interventions yield different effects across codes after strength matching remains unclear. We compare intervention effects under these controls for fixed feature groups in Gemma and Llama and observe: (1) Whole-patch scaling. With one coefficient per input, full-output Kullback-Leibler (KL) calibration leaves smaller mean effect gaps from the encoder-based reference intervention than total-norm matching for nonnegative least-squares (NNLS) refits. On new Llama inputs, total-norm matching loses 11 reference transfers, whereas full-output calibration preserves every transfer decision. (2) Positionwise scaling. On a small matched gender-agreement subset, positionwise norm matching leaves smaller observed mean gaps than full-output calibration. (3) Answer exclusion. Compared with unscaled NNLS, answer-excluded calibration reduces gaps in the measured subject-verb agreement comparisons but increases them for gender refits. Full-output KL calibration also reduces larger gaps for norm-matched constructed candidates, although matching coverage is incomplete in several settings. Since calibration uses the evaluated input's outputs, it serves as a diagnostic of residual disagreement.

Figures & tables

Figure 1: Strength control can change a transfer decision. A Llama-128K example selected post hoc. (1) Prompts with task answers appended. (2)

E

is the reference intervention built from the released encoder’s codes. Separate patches on the same native recipient. Rankings (

>

) compare only the two answer logits. Ticks indicate donor-answer transfer, crosses its absence. (3) Transfers lost or gained relative to

E

across all 304 inputs. Agreement with

E

is not a faithfulness standard.

Figure 2: Four-step comparison with a fixed model, dictionary, feature group, and sentence pair. The reference intervention

E

, built from the released encoder’s codes, supplies the reference targets (gray) for rescaling the original alternative patch (color). Norm segments show squared patch norms, matched in sum or per position. KL controls use trial model outputs to choose one scaling coefficient (knob) per input for matching scalar KL. Q2 excludes both answer tokens (crossed) and renormalizes before matching conditional KL. Step 4 keeps beginning-of-sequence (BOS) and last prompt tokens, comparing native-to-patched KL (nats), donor-oriented margin change (logits) and transfer with

E

. Matching can fail.

Configuration	Width	Layer	$\|G\|$	NNLS	Cand.
Gemma-9B-it SVA	131,072	20	6	166	504
Gemma-2B SVA	131,072	12	6	62	64
Llama-8B SVA	131,072	15	12	188	124
New material	131,072	15	12	304	—
Llama-8B SVA	32,768	15	12	112	106
Gemma-2B gender	131,072	12	12	42	40

Table 1: Configurations and eligible inputs before matching. Width denotes dictionary size, Layer is zero-indexed, and

|G|

is group size. SVA denotes subject–verb agreement. Llama-128K/32K indicate dictionary widths. The 9B NNLS and candidate panels are separate. New Llama material has no candidate comparison.

Panel of Figure 3 — Figure 3: NNLS gap reductions relative to total-norm matching. Full-output KL calibration (Q1, panel a) reduces mean gaps in every cohort. Positionwise matching (b) improves on total-norm matching in most observed means. Points show mean paired differences in absolute effect gaps from the encoder-based reference intervention $E$ (total-norm minus alternative), so positive values favor the alternative. Each comparison uses its $n$ common matched inputs. Means are case/block-balanced (pooled for 9B). Bars show pointwise 95% paired block-bootstrap intervals, and “CI n/a” marks an unavailable interval. Transfer counts report lost/gained relative to $E$ .

Panel of Figure 4 — Figure 4: NNLS control orderings differ across settings. Q1 denotes full-output KL, Q2 answer-excluded KL. (a) Case/block-balanced mean gaps from the encoder-based reference intervention $E$ on inputs matched by all three controls in each row, shown descriptively without intervals. (b) Case/block-balanced mean paired gap differences (Q2 minus unscaled NNLS), with pointwise 95% block-bootstrap intervals. Parentheses give input counts. Gender subsets differ between panels. Llama rows use the new cohort, where Q2 is a follow-up after inspection.

Panel of Figure 5 — Figure 5: Constructed candidates under full-output KL calibration (Q1). (a) Pooled gaps from the encoder-based reference intervention $E$ before (open circles) and after (filled circles) search within $\alpha\leq 2$ . All-attempt rows retain failed searches at their best measured point. Diamonds show post hoc continuation to $\alpha\leq 4$ (Appendix C.5 ). (b) Paired reductions on original matched inputs, case-balanced for 9B and case/block-balanced otherwise. Pointwise 95% intervals use lexical-block Student- $t$ for 9B and paired block bootstrap elsewhere. The 9B primary interval’s half-width exceeds the planned 0.25 logit. Gender lacks a balanced estimate and interval because one case has no matches. Coverage follows the original per-case rule. Llama intervals describe selected subsets.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

			Mean gap (logit)		Paired difference (logit)		Lost/gained
Cohort	$n$	Weighting	Total norm	Q1	Estimate	95% interval	Total norm	Q1
Earlier panels (follow-up controls)
9B SVA	165	pooled	0.056	0.015	$0.041$	unavailable	1/2	1/0
2B SVA	62	balanced	0.033	0.008	$0.024$	$[0.011,0.042]$	0/0	0/0
Llama-128K	188	balanced	0.160	0.017	$0.142$	$[0.117,0.168]$	10/1	0/1
Llama-32K	112	balanced	0.127	0.012	$0.115$	$[0.077,0.159]$	3/1	0/1

Appendix

Table 3: Coefficient-count matched NNLS controls. Q1 leaves smaller mean effect gaps on inputs matched by both controls. Differences are total norm minus Q1 (logits). Balanced means equally weight cases and blocks within case. Pointwise 95% paired block-bootstrap intervals assume independent blocks; the 9B interval is unavailable. Lost/gained transfers are unweighted counts relative to

E

. The prospective cohort uses new material in the same Llama-128K configuration and is estimated separately from the earlier panel. These are the full values underlying Figure 3 a.

Cohort	$n$	Estimate	95% interval
9B SVA	150	$-0.024$	unavailable
2B SVA	62	$0.001$	$[-0.004,0.006]$
Llama-128K	188	$-0.048$	$[-0.067,-0.030]$
Llama-32K	109	$-0.029$	$[-0.047,-0.012]$
2B gender	24	$-0.073$	unavailable
Prospective confirmation

Appendix

Table 4: Positionwise minus total-norm effect gap. Negative values favor positionwise norm matching. Each row uses its own common matched inputs; Figure 3 b plots the negative of these values. Values are in logits, with case/block balancing except for pooled 9B. Intervals use the assumptions in Table 3 . The gender interval is unavailable because too few bootstrap draws preserve the required cells.

Cohort	$n$	Median $\|e_{i}(E)\|$	Encoder transfers
9B SVA	165	3.92	85
2B SVA	62	4.61	55
Llama-128K	188	3.33	126
Llama-32K	112	4.05	96
2B gender	42	4.47	42

Appendix

Table 5: Effect magnitude of the encoder-based reference intervention

E

on the five earlier total-norm/Q1 common-input cohorts. Absolute-effect medians are in logits; transfer counts are unweighted. These background quantities describe the same inputs as the paired comparison, separately for each cohort.

Code	Inputs	Q2 matches	Before	Final	Reduction and 95% interval
NNLS	40	40	0.089	0.139	$-0.050$ $[-0.091,-0.008]$
Candidate	40	13	0.754	0.401	$0.353$ $[0.265,0.447]$

Appendix

Table 6: Answer exclusion is a boundary, not a uniform robustness result. Q2 worsens NNLS effect agreement on the gender common-input cohort despite complete matching. Mean candidate gaps decrease across all attempts, with failed coverage. Means and reductions are case/block-balanced logits; final means retain unmatched attempts. Intervals are pointwise conditional 95% block-bootstrap intervals. A negative reduction denotes deterioration.

Configuration	Code	Quantity	$n$	Estimate	Lower	Upper
9B SVA	Candidate	Q1 reduction	478	0.7239	0.3241	1.1236
9B SVA	Candidate	Q1 gap	478	0.0284	0.0223	0.0344
9B SVA	Candidate	Q2 gap $-$ Q1 gap	504	0.0071	0.0042	0.0100
2B SVA	Candidate	Q1 reduction	64	0.5459	0.3637	0.7485
2B SVA	Candidate	Q1 gap	64	0.0149	0.0106	0.0201
2B SVA	Candidate	Q2 reduction	64	0.5410	0.3590	0.7446

Appendix

Table 7: Primary and supporting calibration estimates with pointwise 95% intervals. All quantities are in logits. A reduction is the original gap minus the selected gap, so a negative reduction denotes deterioration. The 9B rows use case-balanced means and block-based Student-

t

intervals. Other rows use case/block-balanced means and lexical-block bootstrap intervals. Candidate Q1 rows use matched inputs, including the prespecified 9B primary reduction. The 9B Q2-minus-Q1 row retains all attempts, and the gender row uses the complete 42-input NNLS cohort, distinct from the common-40 comparison in Table 6 . Each row preserves its own input set and inferential assumptions.

Balanced mean absolute gap (logit)
Panel	$n$	Before	After
9B: all attempts	504	0.816	0.076
9B: matched	478	0.752	0.028
2B: all, matched	64	0.561	0.015

Appendix

Table 8: Candidate absolute effect gaps from the encoder-based reference intervention

E

, in logits, before and after Q1. Means use each setting’s specified balancing;

n

counts inputs. The 9B all-attempt row includes 26 failed matches. The 9B matched row uses the inputs for the prespecified primary comparison, whose estimand is the paired reduction. All 2B attempts match.

Subject–attractor	Matched	Before	After
Plural–plural	118/124	0.643	0.020
Plural–singular	117/124	0.702	0.036
Singular–plural	130/136	0.740	0.025
Singular–singular	113/120	0.923	0.033

Appendix

Table 9: Matching coverage and mean absolute effect gaps (logit) before and after Q1, by original subject and attractor number. Counts are matched/attempted inputs. Both means exclude unmatched inputs; the case label is retained when donor and recipient are reversed.

		Matched inputs		All-attempt gap		Matched gap		Rescued	Lost
Setting	Sel.	$\alpha{\leq}2$	$\alpha{\leq}4$	$\alpha{\leq}2$	$\alpha{\leq}4$	original	rescued	median $\alpha$	$\alpha{\leq}2{\to}4$
9B SVA	Q1	478/504	498/504	0.076	0.047	0.028	0.392	3.03	11 $\to$ 6
Llama-128K	Q1	92/124 †	116/124	0.270	0.193	0.130	0.241	2.53	13 $\to$ 7
Llama-128K	Q2	91/124 †	117/124	0.302	0.234	0.174	0.274	2.39	14 $\to$ 9
Llama-32K	Q1	75/106 †	92/106 †	0.197	0.144	0.041	0.087	2.38	10 $\to$ 10
Llama-32K	Q2	76/106 †	94/106	0.205	0.173	0.051	0.112	2.47	10 $\to$ 10

Appendix

Table 10: Post hoc search-range sensitivity for unmatched candidate searches. Matched counts use the original denominators; † marks matched subsets that do not meet the original per-case coverage rule. Gaps are pooled mean absolute effect gaps from the encoder-based reference intervention

E

(logits). All-attempt gaps retain each unmatched search at its best attempt. The matched-gap columns compare inputs matched within

\alpha\leq 2

with inputs first matched by the continuation. Lost counts are encoder transfers lost over the full cohort. Search records overlap across Q1 and Q2 and are not independent inputs.

Panel of Figure 6 — Figure 6: Candidate residual gaps across answer-excluded KL budgets. Small mean gaps persist on both changing and fixed matched subsets. $E$ is the encoder-based reference intervention. Both reference and candidate patches are rescaled to each budget. Filled points use each budget’s joint $E$ /candidate matches (counts out of 346 in 9B, 64 in 2B). Open rings and dashed lines use the all-six-budget intersection; superposed symbols indicate overlapping means. Both axes are shared, but weighting differs: pooled in 9B, case/block-balanced in 2B. The 2B coverage failures are labeled; 9B counts are descriptive. Lines connect measured means, without uncertainty bands or an equivalence claim. Exact values and coverage are in Appendix D.3 .

Panel of Figure 7 — Figure 7: Candidate effect-gap reductions under shared-coefficient and per-input calibration. Gaps are measured from the encoder-based reference intervention $E$ . Shared coefficients improve means while worsening many individual comparisons. Negative reductions denote larger gaps. Dots at zero mark the shared-coefficient worsening fraction. The curves weight inputs equally, retaining 11 failed per-input Q1 attempts in 9B. Table 13 reports balanced means. Beyond the displayed 3-logit limit lie 2 shared-coefficient and 40 per-input values in 9B, none in 2B (full curves: Appendix D.4 , Figure 8 ). Selection protocols differ between panels. Shared-coefficient selection holds out evaluation inputs, which were exposed in earlier analyses. Per-input Q1 uses each evaluation input’s outputs. Curves are descriptive, without confidence intervals.

Panel of Figure 8 — Figure 8: Full-range counterpart to Figure 7 . The wider axis reveals the positive tail clipped in that figure, with unchanged cohorts and equal input weights. Reductions are in logits; negative values denote larger gaps from the encoder-based reference intervention $E$ , and dots at zero mark the shared-coefficient worsening fraction. All 11 failed 9B Q1 searches remain included. The panels retain their distinct selection protocols, and Q1 uses additional evaluation-input information. Curves are descriptive, without confidence intervals.

	9B: pooled		2B: balanced
$q$	Matched	Gap	Matched	Gap
0.25	339/346	0.074	64/64	0.019
0.50	343/346	0.056	64/64	0.017
0.75	343/346	0.043	64/64	0.019
1.00	334/346	0.037	64/64	0.020
1.25	303/346	0.038	56/64	0.027

Appendix

Table 11: Candidate joint matching coverage and absolute effect gaps (logit) at each answer-excluded KL budget. Means condition on both patches matching: pooled inputs in 9B and case/block-balanced in 2B. The 9B panel is the global evaluation subset, not all 504 lexical inputs. High-budget coverage failures prevent extending these subset means to the full panels.

$q$	NNLS matched	Gap	Candidate
0.25	62/62	0.00782	0.01927
0.50	62/62	0.00798	0.01679
0.75	62/62	0.00986	0.01902
1.00	62/62	0.00821	0.01977
1.25	62/62	0.01217	0.04455
1.50	50/62	0.01225	0.13646

Appendix

Table 12: Additional 2B budget accounting. NNLS gaps condition on joint matching. The candidate column retains all 64 inputs, using nearest measured attempts for failed matches; its larger-budget entries are not matched comparisons. All gaps are balanced mean absolute differences in logits.

Balanced mean absolute gap (logit)
Setting	Before	Shared	Per-input
9B ( $n=346$ )	0.876	0.704	0.046
2B ( $n=64$ )	0.561	0.313	0.015

Appendix

Table 13: Balanced candidate effect gaps (logit) on the same inputs within each setting. Both selectors return

\alpha=1.25

; every 2B fold selects that value. Per-input Q1 uses each evaluation input’s reference and separate search, including 11 failed attempts in 9B. It is an information-richer diagnostic, not a same-budget comparator.

Pooled mean absolute gap (logit)
Config. / code	Selector	Matched / attempted	Matched only		All attempts		Coverage
			Before	After	Before	Final
128K / NNLS	Q1	188/188	0.166	0.016	0.166	0.016	Pass
	Q2	188/188	0.166	0.023	0.166	0.023	Pass
32K / NNLS	Q1	112/112	0.159	0.012	0.159	0.012	Pass
	Q2	112/112	0.159	0.014	0.159	0.014	Pass

Appendix

Table 14: Earlier Llama cohorts: pooled mean absolute effect gaps from the encoder-based reference intervention

E

in logits. Matching counts use each code’s own qualified cohort. Before/after columns compare identical inputs within a row; matched membership can differ between selectors. All attempts retain failed searches at their best measured point, without calling them calibrated. Coverage requires the prespecified per-case retention and support conditions. These descriptive means are distinct from the original case/block-balanced estimates.

Intervention	Inputs	Balanced gap	Pooled gap	Transfers	Lost	Gained
Reference $E$	304	0.000	0.000	175	0	0
Unscaled NNLS	304	0.213	0.207	181	3	9
Total-norm matching	304	0.181	0.176	165	11	1
Positionwise norm matching	304	0.134	0.130	167	10	2
Full-output KL (Q1)	304	0.021	0.020	175	0	0
Follow-up after inspection of the confirmation results

Appendix

Table 15: New-material Llama-128K controls on common inputs. Q1 preserves the transfer decisions of the encoder-based reference intervention

E

and leaves the smallest observed mean gap among these controls. Every measured matching control matches all 304 inputs. Gaps are in logits; balanced values equally weight cases and blocks within case, whereas pooled values weight inputs equally. Transfers and losses/gains relative to

E

are unweighted counts. The separated Q2 row is a follow-up after inspection of the confirmation, on the same inputs. This cohort remains separate from the earlier 188-input panel.

Case	$n$	Contrast
Singular / singular	65	0.195
Singular / plural	87	0.122
Plural / singular	65	0.185
Plural / plural	87	0.138

Appendix

Table 16: New-material confirmation by recipient subject/attractor number;

n

counts inputs, all matched by both primary controls. The mean total-norm minus Q1 gap is positive in every case. Contrasts are in logits, with equal block weights within each case.

Effect-gap contrast	Mean	95% interval
Unscaled NNLS $-$ Q2	$0.184$	$[0.149,0.223]$
Q2 $-$ Q1	$0.008$	$[0.005,0.011]$
Total norm $-$ Q2	$0.152$	$[0.124,0.183]$

Appendix

Table 17: Q2 follow-up contrasts on common Llama inputs. Answer exclusion retains a reduction but leaves a larger residual gap than Q1. All 304 inputs match the compared controls. Estimates and pointwise 95% block-bootstrap intervals are in logits, with equal case weights and equal block weights within case.

Comparison	Evidence status	Inferential scope
9B lexical candidate	Prespecified primary; effects unmeasured when specified	Coverage passes; planned precision unmet
2B and Llama SVA extensions	Specified after group qualification, before alternative effects	2B matches fully; Llama candidate coverage fails
Gender	Selected group validated before code comparison	Partial extension; Q2 reverses the NNLS pattern
Single-scale controls	Prespecified follow-ups on exposed panels	Paired comparisons, with unavailable intervals reported
New Llama-128K NNLS panel	Materials and analysis specified before tokenization or group effects	Prospective material confirmation; all support, coverage and precision requirements pass
Q2 on the new Llama panel	Specified after inspection of the original confirmation results	Follow-up reduction; larger residual gap and transfer losses than Q1 on the same inputs

Appendix

Table 18: Evidence status and scope. Exposition starts with practical NNLS comparisons without replacing the prespecified primary endpoint. The new Llama panel provides prospective material confirmation separately from exposed-panel follow-ups.

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Sparse autoencoder (SAE) features are increasingly used to steer language models, but feature steering is rarely clean: the same intervention can behave inconsistently across contexts and perturb unrelated features. We introduce a pre-intervention screening framework for forecasting SAE steering side effects from feature statistics computed before steering. We operationalize side effects along two axes of steering modularity, effect stability and collateral spread, and evaluate GPT-2-small, Pythia-70M-deduped, Gemma-2-2B, and Llama-3.1-8B across ReLU, JumpReLU, and TopK SAE dictionaries. Across these settings, decoder geometry, activation statistics, co-activation structure, and direct-logit footprint predict steering modularity better than frequency-only and activation-magnitude baselines. The signal is strongest in GPT-2-small, Pythia-70M, and Llama-3.1-8B, where it survives residualization against magnitude-related confounds, and weaker in Gemma-2-2B. Held-out screening shows that ranking unseen features by predicted cleanliness can select features that steer more cleanly on fresh contexts, but the successful axis varies by setting: GPT-2 improves most cleanly, Pythia improves mainly on stability, Llama mainly on collateral, and Gemma only partially. A controlled Llama Scope width comparison shows that the predictive signal persists under a 32K-to-128K dictionary-width change, although the screening payoff becomes less stable. Overall, SAE steering side effects are predictable in advance, but the useful predictor signature and transferred modularity axis are model- and dictionary-setting dependent.

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