Rank Measures of Fine-Tuning Updates Need a Noise Baseline

Muhammad Usman Altaf

cs.LGOct 15, 2026

Rank Measures of Fine-Tuning Updates Need a Noise Baseline

Authors: Muhammad Usman Altaf

Organizations: Independent Researcher

Abstract

Studies of what fine-tuning does to a language model often summarise the weight update $ΔW$ by its rank. We compare the rank measures used for this against the same measures on a random matrix of the same shape, and against two harder references: the pretrained weights, and matched fine-tunes without alignment content. Measures that count directions put our fine-tuning updates only a little below noise; stable rank, computed from the same singular values, puts them far below. Both are right: an update has a few strong directions on top of a long tail. The pattern holds for thirteen released pairs up to 8B parameters, and it is not a mark of alignment: with training held fixed, instruction data does not stand out from Wikipedia or code, DPO gives the same update whether its preference labels are real or half swapped at random (one model, modest preference strength), and scrambled text gives the most concentrated update of all. Applied to a recent claim that post-training is superficial because the effective rank of the weights barely changes, the baselines show that a random update of the same size leaves that rank almost unchanged while wrecking the model, and that an edit built on the effective rank of the update has no measurable effect in our tests. Rank measures of weight updates need a noise baseline, and claims about alignment need a control without alignment content.

Figures & tables

model	matrix	threshold	eRank	90% energy	particip.	stable	top-1 energy
Qwen2.5-0.5B	SFT update	0.973	0.816	0.541	0.381	0.097	4.2%
	wikitext update	0.972	0.810	0.531	0.365	0.093	4.0%
	code update	0.975	0.817	0.543	0.374	0.094	3.8%
	shuffled-words update	0.941	0.745	0.440	0.202	0.042	9.0%
	DPO update	0.966	0.810	0.536	0.295	0.053	5.8%
	swapped-DPO update	0.967	0.811	0.537	0.304	0.055	5.7%

Table 1: Rank measures as a fraction of the maximum rank, and the share of energy in the top direction, averaged over all attention and MLP matrices. “Noise” is the random-init pair; “pretrained

W

” is the base model’s own weights. The DPO updates are measured from the SFT model they start from.

Figure 1: How far below noise each measure puts each update,

1-\text{update}/\text{noise}

, with 95% layer-bootstrap intervals. Measures are ordered by how much weight they give to the top directions. The gap grows from a few percent for the threshold count to 78–94% for stable rank, and to 90–97% for the shuffled-words updates.

Figure 2: Where the energy sits, averaged over the 168 matrices of Qwen2.5-0.5B. Left: share of energy in the top fraction of directions. Right: singular values relative to the largest. Updates put much more energy in their top directions than noise does, then decay slowly over a long tail. The pretrained weights (dashed) sit between the updates and noise.

difference	Qwen2.5-0.5B	SmolLM2-360M
SFT $-$ wikitext	$+0.004$ ( $-0.001$ , $+0.010$ )	$+0.005$ ( $+0.003$ , $+0.006$ )
SFT $-$ code	$+0.003$ ( $-0.000$ , $+0.006$ )	$-0.009$ ( $-0.011$ , $-0.008$ )
code $-$ wikitext	$+0.002$ ( $-0.004$ , $+0.007$ )	$+0.014$ ( $+0.012$ , $+0.016$ )
shuffled words $-$ SFT	$-0.054$ ( $-0.062$ , $-0.047$ )	$-0.016$ ( $-0.018$ , $-0.014$ )
code, steps matched $-$ code	$+0.020$ ( $+0.018$ , $+0.023$ )	$+0.007$ ( $+0.007$ , $+0.008$ )
swapped DPO $-$ DPO	$+0.002$ ( $+0.000$ , $+0.004$ )	–

Table 2: Differences in stable rank between runs on the same model, paired matrix by matrix, with 95% layer-bootstrap intervals. The intervals describe the spread over layers within one run, not between training runs; “second seed” repeats a run with a different seed and data sample. “Steps matched” trains on code for as many optimizer steps as SFT, which is 2.5–2.7 times more code tokens.

Figure 3: Released updates (blue) against a Gaussian matrix of the same shape (grey), in three groups: base/instruct releases, the OLMo 2 post-training stages, and code/math specialists. Left: the threshold count (axis starts at 0.85). Right: stable rank.

update	threshold	noise	above floor	stable rank	noise	stable w/o top
SmolLM2-135M instruct	0.971	0.996	96%	0.086	0.354	0.119
SmolLM2-360M instruct	0.964	0.996	95%	0.039	0.356	0.059
Qwen2.5-0.5B instruct	0.990	0.996	97%	0.280	0.434	0.314
Qwen3-0.6B	0.989	0.996	99%	0.124	0.342	0.145
Qwen2.5-1.5B instruct	0.969	0.996	78%	0.051	0.432	0.073
SmolLM2-1.7B instruct	0.959	0.993	91%	0.075	0.336	0.090

Table 3: Released pairs, means over matrices. “Above floor” is the share of the update’s singular values larger than the largest singular value of the simulated bf16 rounding error: directions that rounding alone cannot produce (§ 2 ). It is not computed (–) for OLMo 2 base to SFT, whose base, a mid-training checkpoint, is stored in fp32. The last column drops the top direction, to check that one outlier does not drive the result.

Figure 4: Left: size of the change in the effective rank of the full weight matrix

W

|\Delta\,\mathrm{eRank}(W)|/\mathrm{eRank}(W)

averaged over matrices, for the real base-to-instruct update (blue) and a random update of the same Frobenius norm (grey). Right: perplexity on held-out wikitext of Qwen3-0.6B base, unchanged, after a random update of the same size as its real post-training update (Gaussian, or with the real update’s singular values), and after the real update, which gives the instruct model. The numbers above the bars give the size of the change in eRank(

W

) for each update, as in the left panel.

model	accuracy	difference	$\|$ change $\|$ in eRank( $W$ )
base	59.5	–	–
base + Gaussian update, draw 1	4.1	$-55.4$ ( $-58.2$ , $-52.7$ )	0.50%
base + Gaussian update, draw 2	4.8	$-54.7$ ( $-57.5$ , $-52.2$ )	–
base + real singular values, random directions	5.8	$-53.8$ ( $-56.6$ , $-50.9$ )	0.48%
instruct (= base + real update)	64.5	–	0.13%
instruct + Gaussian update	11.9	$-52.6$ ( $-55.3$ , $-49.9$ )	–

Table 4: GSM8K accuracy, in percent (greedy, all 1,319 test problems), of Qwen3-0.6B with a random update of the same size as its real post-training update, added to every attention and MLP matrix. Each pair uses the same prompt and token limit: the base model an 8-shot plain-text prompt, the instruct model its chat template. Differences in points with 95% paired bootstrap intervals.

Figure 5: GSM8K pass@

k

for Qwen3-0.6B (non-thinking mode) with tail edits and with uniform scaling of the whole update. Eight samples per problem, the same sampling seed per problem for every variant. Left: first 300 test problems, main variants (every variant is in Table 15 ). Right: all 1,319 (Table 5 ).

edit	unscaled	pass@1	pass@8	answered	$\Delta$ pass@1	$\Delta$ pass@8
none	100%	62.3	85.1	97.4%	–	–
tail $\times 0.2$ , eRank cutoff (Timber)	98.9%	62.3	85.3	97.1%	$-0.0$ ( $-0.8$ , $+0.8$ )	$+0.2$ ( $-1.4$ , $+1.8$ )
tail $\times 0.5$ , eRank cutoff (Timber)	98.9%	63.1	85.9	97.1%	$+0.7$ ( $-0.0$ , $+1.6$ )	$+0.8$ ( $-0.6$ , $+2.3$ )
tail $\times 0.5$ , stable-rank cutoff	41.4%	62.9	86.9	96.0%	$+0.6$ ( $-0.5$ , $+1.7$ )	$+1.8$ ( $+0.2$ , $+3.5$ )
whole update $\times 0.75$ (same size)	0%	65.8	87.8	96.4%	$+3.4$ ( $+2.4$ , $+4.5$ )	$+2.7$ ( $+1.1$ , $+4.2$ )

Table 5: Edits of the Qwen3-0.6B update on all 1,319 GSM8K test problems; pass@

k

in percent. “Unscaled” is the share of the update’s energy the edit leaves at full strength (uniform scaling shrinks every direction by the same factor); “answered” is the share of samples with a final boxed answer; “same size” means scaled to the size of the stable-rank edit. Differences from the unedited model in points, with 95% paired bootstrap intervals over problems, not corrected for multiple comparisons.

edit	accuracy	$\Delta$ accuracy	pass@4	$\Delta$ pass@4
none	25.9	–	46.6	–
tail $\times 0.2$ , eRank cutoff (Timber)	26.9	$+0.9$ ( $-2.1$ , $+4.0$ )	46.0	$-0.6$ ( $-5.9$ , $+4.8$ )
tail $\times 0.5$ , eRank cutoff (Timber)	26.4	$+0.4$ ( $-2.5$ , $+3.5$ )	47.5	$+0.9$ ( $-4.2$ , $+6.4$ )
tail $\times 0.8$ , eRank cutoff (Timber)	26.9	$+0.9$ ( $-2.0$ , $+3.9$ )	47.0	$+0.4$ ( $-4.8$ , $+5.6$ )
tail $\times 0.2$ , stable-rank cutoff	21.3	$-4.6$ ( $-9.3$ , $+0.2$ )	48.0	$+1.4$ ( $-6.4$ , $+9.4$ )
whole update, scaled to the stable-rank edit’s size	22.9	$-3.1$ ( $-7.1$ , $+0.9$ )	54.0	$+7.5$ ( $+0.5$ , $+14.3$ )

Table 6: The same edits on GPQA-Diamond ( Rein et al., 2024 ) , Wu et al.’s benchmark, 198 questions, non-thinking mode with their sampling settings. “Accuracy” is the mean over samples (their Mean@

k

), in percent, as is pass@4; eight samples per question for the unedited model, four for each edit. Differences in points with 95% paired bootstrap intervals over questions.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

setting	value
weights / compute precision	fp32 master weights, bf16 autocast
optimizer	8-bit AdamW, learning rate $2\times 10^{-5}$ , no weight decay
schedule	cosine, 3% warmup, 3 epochs, gradient clipping at 1.0
batch	1 sequence $\times$ 16 accumulation steps
sequence length	512 tokens
embeddings	frozen (tied with the output head; not analysed)

Appendix

Table 7: Settings shared by all our fine-tunes. Each language-model run trains on the same number of tokens as the SFT run of the same model, counted after truncation. DPO uses the same precision, memory settings and batch, with the changes listed at the bottom.

model	run	steps	loss	$\\|\Delta W\\|_{F}$	stable rank	stable w/o top	hours
Qwen2.5-0.5B	SFT	1,840	1.00	10.4	0.097	0.126	2.8
	SFT, seed 2	1,840	–	10.5	0.096	0.126	–
	wikitext	1,910	2.23	12.1	0.093	0.122	2.8
	wikitext, seed 2	1,910	2.24	12.0	0.092	0.121	2.7
	code	680	1.11	6.3	0.094	0.123	1.3
	code, seed 2	670	1.12	6.3	0.094	0.122	1.3

Appendix

Table 8: Every fine-tune. Steps are optimizer steps (rounded to 10); loss is the mean training loss;

\|\Delta W\|_{F}

is over all attention and MLP matrices. The second SFT seed was interrupted and resumed from step 1,600 with its optimizer and scheduler state, so its logged mean loss covers only the last steps and is not shown.

model	matrix	$0.01\,\sigma_{1}$	$0.05\,\sigma_{1}$	$0.1\,\sigma_{1}$
Qwen2.5-0.5B	SFT update	0.973	0.889	0.787
	wikitext update	0.972	0.886	0.775
	code update	0.975	0.895	0.799
	shuffled-words update	0.941	0.752	0.530
	noise	0.996	0.982	0.964
	pretrained $W$	0.976	0.913	0.851

Appendix

Table 9: The threshold count at three cutoffs. At

0.01\,\sigma_{1}

every update looks almost like noise; at

0.1\,\sigma_{1}

the gap is large. The verdict depends on an arbitrary choice.

matrix	shape	predicted	random-init pair
q , o	$896\times 896$	0.250	0.252, 0.251
k , v	$128\times 896$	0.527	0.536, 0.535
gate , up	$4864\times 896$	0.490	0.492, 0.492
down	$896\times 4864$	0.490	0.493

Appendix

Table 10: Stable rank of the random-init pair against the prediction

mn/(\sqrt{m}+\sqrt{n})^{2}/r

for a Gaussian matrix of the same shape, Qwen2.5-0.5B, mean over layers.

Figure 6: How far below noise each update is,

1-R_{\alpha}(\text{update})/R_{\alpha}(\text{noise})

as means over matrices, along the Rényi family from

\alpha=1/4

\alpha=\infty

(stable rank, squares). The gap grows with the order and levels off from about

\alpha=8

model	matrix	q	k	v	o	gate	up	down
Qwen2.5-0.5B	SFT update	0.032	0.064	0.251	0.057	0.064	0.072	0.139
	wikitext update	0.031	0.072	0.227	0.057	0.065	0.085	0.112
	code update	0.031	0.070	0.243	0.058	0.057	0.073	0.127
	shuffled-words update	0.011	0.043	0.124	0.021	0.023	0.026	0.048
	noise	0.252	0.536	0.535	0.251	0.492	0.492	0.493
	pretrained $W$	0.058	0.212	0.453	0.078	0.071	0.242	0.135

Appendix

Table 11: Stable rank (fraction of maximum rank) by matrix type.

model	update	first quarter	second	third	last quarter
Qwen2.5-0.5B	SFT	0.094	0.099	0.103	0.091
	wikitext	0.086	0.100	0.104	0.080
	code	0.091	0.097	0.104	0.084
	shuffled words	0.055	0.039	0.034	0.041
SmolLM2-360M	SFT	0.023	0.027	0.029	0.021
	wikitext	0.019	0.024	0.023	0.015

Appendix

Table 12: Stable rank by quarter of the network.

update	particip. ratio	noise	above edge	rounding share	$\\|\Delta W\\|_{F}/\\|W\\|_{F}$	zero entries
SmolLM2-135M instruct	0.321	0.667	30	28%	8.1%	2.5%
SmolLM2-360M instruct	0.239	0.668	46	10%	7.6%	2.7%
Qwen2.5-0.5B instruct	0.657	0.755	22	76%	7.8%	2.2%
Qwen3-0.6B	0.442	0.654	72	15%	12.7%	1.5%
Qwen2.5-1.5B instruct	0.299	0.753	100	94%	1.2%	14.0%
SmolLM2-1.7B instruct	0.375	0.629	140	45%	6.2%	3.1%

Appendix

Table 13: More on the released pairs. “Particip. ratio”: the participation ratio, next to its value for a Gaussian matrix of the same shape. “Above edge”: the number of singular values above the largest one such a Gaussian matrix would have (mean per matrix). “Rounding share”: the number of singular values of the simulated bf16 rounding error above

0.01\,\sigma_{1}

of the update, as a share of the update’s threshold count; a loose upper bound on how much of the count rounding could explain, as a ratio of means over matrices (compare the “above floor” column of Table 3 ).

\|\Delta W\|_{F}/\|W\|_{F}

: the size of the update relative to the base weights. “Zero entries”: the share of entries of

\Delta W

that are exactly zero, the direct trace of rounding.

pair	Qwen2.5-0.5B	SmolLM2-360M
SFT vs SFT, seed 2	0.32 (0.30, 0.33)	0.51 (0.50, 0.52)
wikitext vs wikitext, seed 2	0.30 (0.29, 0.31)	0.49 (0.47, 0.52)
code vs code, seed 2	0.26 (0.25, 0.27)	0.34 (0.33, 0.36)
code vs code, steps matched	0.37 (0.36, 0.39)	0.54 (0.53, 0.55)
SFT vs wikitext	0.22 (0.21, 0.23)	0.13 (0.13, 0.14)
SFT vs code	0.20 (0.19, 0.21)	0.12 (0.12, 0.12)

Appendix

Table 14: Overlap of the top-16 left singular subspaces of two updates, mean over matrices, with 95% layer-bootstrap intervals.

edit	pass@1	pass@8	answered	$\Delta$ pass@1	$\Delta$ pass@8
none	63.4	84.7	96.5%	–	–
tail $\times 0$ , eRank cutoff	63.9	87.7	96.7%	$+0.5$ ( $-1.3$ , $+2.3$ )	$+3.0$ ( $-0.3$ , $+6.3$ )
tail $\times 0.2$ , eRank cutoff	64.0	85.7	96.8%	$+0.5$ ( $-1.3$ , $+2.4$ )	$+1.0$ ( $-2.0$ , $+4.0$ )
tail $\times 0.5$ , eRank cutoff	64.7	88.0	96.8%	$+1.3$ ( $-0.5$ , $+3.0$ )	$+3.3$ ( $+0.7$ , $+6.3$ )
tail $\times 0$ , stable-rank cutoff	59.3	83.0	90.6%	$-4.2$ ( $-6.9$ , $-1.4$ )	$-1.7$ ( $-5.3$ , $+2.0$ )
tail $\times 0.5$ , stable-rank cutoff	64.5	87.3	95.1%	$+1.1$ ( $-1.1$ , $+3.3$ )	$+2.7$ ( $-1.0$ , $+6.3$ )

Appendix

Table 15: All edit variants on the first 300 GSM8K test problems, pass@

k

in percent. Differences from the unedited model in points, with 95% paired bootstrap intervals, not corrected for the fourteen comparisons. “Answered” is the share of samples with a final boxed answer. “Uniform” scales the whole update, per matrix, to the size of the named tail edit.

Fine-Tuning Language Models

See all 34 papers

Aligning Language Model Benchmarks with Pairwise Preferences
Feb 2, 2026Marco Gutierrez, Xinyi Leng, Hannah Cyberey +3LLM Evaluation Learning to Rank
School of Data Science, University of Virginia · Imperial College London · Thomson Reuters Foundational Research +1
IGU-LoRA: Adaptive Rank Allocation via Integrated Gradients and Uncertainty-Aware Scoring
Mar 14, 2026Xuan Cui, Huiyue Li, Run Zeng +5LLM Fine-Tuning Low-Rank Adaptation
Chongqing Technology and Business University · Shanghai Jiao Tong University
Post-Training Science for Supervised Fine-Tuning
Sep 1, 2026Charles O'Neill, Mudith Jayasekara, Harry PartridgeSupervised Fine-Tuning Hyperparameter Optimization
1Baseten

Feb 2, 2026J/K move · Enter open · S save

Aligning Language Model Benchmarks with Pairwise Preferences

Marco Gutierrez, Xinyi Leng, Hannah Cyberey, Jonathan Richard Schwarz+2

School of Data Science, University of Virginia · Imperial College London · Thomson Reuters Foundational Research+1

LLM Evaluation Learning to Rank Pairwise Ranking

Open paper AskPDF arXiv

Language model benchmarks are pervasive and computationally-efficient proxies for real-world downstream performance. However, many recent works find that benchmarks often fail to predict downstream utility. While some works have begun diagnosing sources of misalignment, there remain no ways to systematically update benchmarks to align their scores with downstream usage. Towards bridging this gap, we introduce and study benchmark alignment, where we use information about downstream model performance to automatically update benchmarks, specifically aiming to update static benchmarks so they generalizably rank models according to new pairwise preferences. Our experiments involving 4576 language models and 6 benchmarks show that reweighting benchmark items can successfully rank unseen models, even generalizing across model scales in most cases. And while naive alignment unsurprisingly requires large numbers of models and benchmark questions, an oracle experiment suggests this could be reduced to as few as 20 well-chosen models. Overall, our work takes a step towards efficiently aligning benchmark development with downstream tasks.\footnote{All of our code, models, and data are publicly-available.

Abstract

Fine-Tuning Language Models

Explore similar work

Aligning Language Model Benchmarks with Pairwise Preferences

IGU-LoRA: Adaptive Rank Allocation via Integrated Gradients and Uncertainty-Aware Scoring

Post-Training Science for Supervised Fine-Tuning