Training-Free Transformer Merging via Sequential Local Operator Alignment

Akansh Maurya; Ya-Wei Eileen Lin; Stefanie Jegelka; Sebastian U Stich; Rotem Mulayoff

cs.LGOct 5, 2026

Training-Free Transformer Merging via Sequential Local Operator Alignment

Authors: Akansh Maurya, Ya-Wei Eileen Lin, Stefanie Jegelka, Sebastian U Stich, Rotem Mulayoff

Organizations: CISPA Helmholtz Center for Information Security · Munich Center for Machine Learning · Technical University of Munich, School of Computation, Information and Technology · Massachusetts Institute of Technology, Department of EECS, CSAIL

Abstract

Training-free model merging aims to combine multiple fine-tuned models into a single model without further optimization on labeled data. Yet, in transformers, independently merging individual layers can affect a shared attention computation because the query-key and value-output operators depend on composed matrices, overlooking the functional structure. Moreover, when merging earlier components, downstream components receive different activations than they do in the original model, thus, the merged and original execution paths no longer match. In this paper, we introduce Sequential Local Operator Alignment, a training-free method that merges transformers along the execution path of the partially merged model. Our method uses calibration data to estimate the local behavior of each functional component, aligns operators sequentially under the intermediate activation of the partially merged model, and subsequently factorizes the merged operators back into valid transformer parameters. We empirically show that this sequential step reduces error accumulation across layers. Furthermore, the proposed operator factorization step enables rank expansion, providing a principled mechanism for increasing multi-task capacity. We demonstrate that our approach generalizes across modalities, model scales, and varying numbers of tasks, from CLIP and RoBERTa to billion-parameter LLMs, and further extends naturally to the merging of LoRA-fine-tuned models. The results indicate improvements over strong merging baselines without requiring rank expansion, while optional expansion provides a further accuracy-inference-cost trade-off. Project link: https://akansh12.github.io/SLOA-Merge/

Figures & tables

Figure 1 : Illustration of Sequential Local Operator Alignment . Our method aligns the effective local functional operators, merges them sequentially, and then factorizes the aligned operators. This sequential operator-level view respects coupled attention computation, reduces error accumulation across layers, and enables rank expansion to have additional multi-task capacity.

Figure 2 : Illustration of the non-invariance of independent merging in attn. operator.

\begin{array}[]{c|c|c|c}\text{Component}&\text{Local operator

\bm{\theta}

}&\text{Behavior map

\mathcal{A}({\bm{Z}};\bm{\theta})

}&\text{Metric}\\ \hline\cr\text{LayerNorm}&(\bm{\gamma},\bm{\beta})&\mathcal{A}({\bm{X}};\bm{\gamma},\bm{\beta})=\nu({\bm{X}})\operatorname{diag}(\bm{\gamma})+\mathbf{1}\bm{\beta}^{\top}&\operatorname{Cov}(\nu(\chi))\\[4.2679pt] \text{Linear map}&{\bm{W}}&\mathcal{A}({\bm{X}};{\bm{W}})={\bm{X}}{\bm{W}}&{\bm{\Lambda}}\\[4.2679pt] \text{QK component}&{\bm{W}}_{Q}{\bm{W}}_{K}^{\top}/\sqrt{d_{k}}&\mathcal{A}({\bm{X}};\bm{\theta})={\bm{X}}\bm{\theta}{\bm{X}}^{\top}&{\bm{\Lambda}}\otimes{\bm{\Lambda}}\\[4.2679pt] \text{VO component}&{\bm{W}}_{V}{\bm{W}}_{O}&\mathcal{A}({\bm{A}}{\bm{X}};\bm{\theta})={\bm{A}}{\bm{X}}\bm{\theta}&{\bm{\Lambda}}\end{array}

Table 1 : Local functional operators and their data-dependent metrics. Here,

\nu({\bm{X}})

denotes the normalized input before the learned LayerNorm’s scale and shift, and

{\bm{A}}

denotes the attention matrix

Panel of Figure 3 — Figure 3 : Per-task comparison of SLOA ( $r{=}128$ ) with data-free and training-free baselines on ViT-B/32 with 20 vision tasks, Panels 3(a) , 3(b) and RoBERTa with 7 GLUE tasks, Panels 3(c) , 3(d) . SLOA consistently outperforms both data-free and training-free baselines.

	Method	# Samples	Avg. Accuracy $\uparrow$
	Method	# Samples	8 Tasks	14 Tasks	20 Tasks
	Fine-tuned	–	90.39	89.26	89.78
Data-Free	Weight Averaging	–	66.29	65.37	61.08
	Task Arithmetic	–	69.49	66.47	60.60
	TIES-Merging	–	71.70	67.58	62.76
	TSV-M	–	83.04	79.11	73.66

Table 2 : Performance comparison of model merging methods on ViT-B/32 across 8, 14, and 20 vision classification tasks. For SLOA,

r

denotes the retained factorization rank used for both QK and VO local operators. The setting

r=64

matches the original ViT-B/32 attention-head dimension.

Data-Free
Method	Avg. Metric $\uparrow$
Individual (oracle)	0.8483
Weight Averaging	0.6165
Task Arithmetic	0.6724
TIES-Merging	0.6448
TSV-M	0.7120

Table 3 : Avg. GLUE metric for merged RoBERTa base models. Full per-task results are in Table 16 .

	Method	Benchmark			Average
	Method	GSM8K	HumanEval	IFEval	Avg. $\uparrow$
Experts	Llama2-7B Pretrained	4.32	0.00	20.89	8.40
	Llama2-7B IF-Tulu	12.59	1.22	66.17	26.66
	Llama2-7B Code (Magic-Coder)	9.70	32.93	26.99	23.21
	Llama2-7B Math (Meta-Math)	65.50	0.00	15.53	27.01
Data-Free	Weight Averaging	40.11	18.29	40.67	33.02

Table 4 : LLaMA-2-7B LoRA-expert merge results (math, code, instruction-following). Avg. is the mean score over GSM8K (math), HumanEval (code, pass@1), and IFEval (instruction-following). Bold/underline marks the best/second-best merging method per column.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Task	Data Source	# Classes	Train	Test	Calib.	Calib. (%)
8-Tasks
SUN397	tanganke/sun397	397	76,128	10,875	256	0.34
GTSRB	tanganke/gtsrb	43	39,209	12,630	256	0.65
Cars	tanganke/stanford_cars	196	8,144	8,041	256	3.14
EuroSAT	tanganke/eurosat	10	21,600	5,400	256	1.19
RESISC45	tanganke/resisc45	45	25,200	6,300	256	1.02

Appendix

Table 5 : Data statistics of the 20 image classification datasets used for model merging, grouped by task set: the first 8 form 8-tasks the next 6 extend it to 14-tasks, and the final 6 extend it to 20-tasks.

Task	# Labels	Primary Metric	Train	Val.	Calib.	Calib. (%)
CoLA	2	Matthews correlation	8,551	1,043	256	2.99
MNLI	3	Accuracy	392,702	9,815	256	0.07
MRPC	2	Avg. F1 & Acc.	3,668	408	256	6.98
QNLI	2	Accuracy	104,743	5,463	256	0.24
QQP	2	Avg. F1 & Acc.	363,849	40,430	256	0.07
RTE	2	Accuracy	2,490	277	256	10.28

Appendix

Table 6 : Summary of GLUE tasks, Metric used, and dataset statistics

Task	Train Corpus	Benchmark	Primary Metric	Train	Eval	Calib.	Calib. (%)
Math	MetaMathQA	GSM8K	Exact match (flexible extract)	395,000	1,319	256	0.065
Code	Magicoder-Evol-Instruct-110K	HumanEval	pass@1	111,183	164	256	0.230
Instruction-following	Tulu-3 Persona-IF	IFEval	Prompt-level strict accuracy	29,980	541	256	0.854

Appendix

Table 7 : Summary of the LLM fine-tuning corpora, evaluation benchmarks, and dataset statistics. Each row is one LoRA expert (rank 32, all-linear modules) fine-tuned from LLaMA-2-7B.

Figure 4 : Prompt templates applied uniformly at training, calibration, and evaluation time for each LoRA expert. The math and instruction-following experts share the Alpaca-style template 4(a) ; the code expert uses Magicoder’s own template 4(b) .

Panel of Figure 5 — Figure 5 : Average performance of SLOA as the number of calibration data increases for 5(a) ViT-B/32 and 5(b) ViT-B/16 on 20 vision tasks and 5(c) RoBERTa on 7 NLU tasks.

	Method	# Samples	Avg. Accuracy $\uparrow$
	Method	# Samples	8 Tasks	14 Tasks	20 Tasks
	Fine-tuned	–	92.34	91.31	91.61
Data-Free	Weight Averaging	–	72.28	69.72	64.81
	Task Arithmetic	–	77.17	70.92	64.23
	TIES-Merging	–	80.21	71.48	66.31
	TSV-M	–	86.70	82.09	78.31

Appendix

Table 8 : Performance comparison of model merging methods on ViT-B/16 across 8, 14, and 20 vision classification tasks. For SLOA,

r

denotes the retained factorization rank used for both QK and VO local operators. The setting

r=64

matches the original ViT-B/16 attention-head dimension.

	Method	# Samples	Norm. Acc. (%) $\uparrow$
	Fine-tuned (LoRA, oracle)	–	100.00
Data-Free	Simple Avg	–	63.76
	TA	–	63.70
	TIES	–	63.70
	DARE-TIES	–	63.70
	KnOTS-TIES	–	68.00

Appendix

Table 9 : Normalized per-task accuracy of model merging methods on ViT-B/32 LoRA (KnOTS,

r{=}16

[ Stoica et al., 2025 ] ) experts across 8 vision tasks. Normalized accuracy is the mean over tasks of (merged/fine-tuned) accuracy

\times

100; the fine-tuned LoRA oracle’s raw average accuracy is 84.28%.

Panel of Figure 6 — Figure 6 : Diagnosing the source of SLOA’s gains : 6(a) error accumulation across network depth on ViT-B/16, and 6(b) merging accuracy versus the number of calibration samples on ViT-B/32, comparing SLOA at increasing rank ( $r=64,128,256$ ) against sequential correction in isolation (no operator composition), RegMean, and RegMean++.

Method	Data requirement	Merge time (s)	Accuracy (%)
Task Arithmetic	Data-free	14.63	64.23
TIES-Merging	Data-free	39.64	66.31
TSV-M	Data-free	153.34	78.31
Iso-C	Data-free	17.09	75.82
Iso-CTS	Data-free	167.24	78.50
Fisher Merging	Calibration-based ( $N=256$ )	112.70	66.50

Appendix

Table 10 : Wall-clock merging cost and accuracy when merging 20 task-specific ViT-B/16 checkpoints. Calibration-based methods use

N=256

samples per task; data-free methods use none.

	ViT-B/32		ViT-B/16
# Tasks	Accuracy (%)	Merge time (s)	Accuracy (%)	Merge time (s)
8	85.50 $\pm$ 0.05	74.2 $\pm$ 0.5	88.04 $\pm$ 0.24	107.50 $\pm$ 1.76
14	82.61 $\pm$ 0.04	103.3 $\pm$ 3.1	85.16 $\pm$ 0.09	136.50 $\pm$ 0.49
20	81.03 $\pm$ 0.12	130.6 $\pm$ 1.3	83.43 $\pm$ 0.09	180.49 $\pm$ 1.88

Appendix

Table 11 : Accuracy and merge time versus the number of merged tasks, at

N=256

calibration samples per task and

r=128

	ViT-B/32		ViT-B/16
$N$	Accuracy (%)	Merge time (s)	Accuracy (%)	Merge time (s)
128	79.77 $\pm$ 0.38	99.31 $\pm$ 2.13	82.67 $\pm$ 0.12	124.96 $\pm$ 0.92
256	81.03 $\pm$ 0.12	130.64 $\pm$ 1.26	83.43 $\pm$ 0.09	180.49 $\pm$ 1.88
512	81.63 $\pm$ 0.13	147.59 $\pm$ 3.28	83.86 $\pm$ 0.08	245.69 $\pm$ 0.58
1024	81.92 $\pm$ 0.08	177.78 $\pm$ 1.89	84.12 $\pm$ 0.11	376.18 $\pm$ 0.57
2048	82.01 $\pm$ 0.07	252.40 $\pm$ 2.37	84.24 $\pm$ 0.03	635.90 $\pm$ 2.24

Appendix

Table 12 : Accuracy and merge time versus calibration samples per task, at 20 tasks and

r=128

Backbone	8 tasks	14 tasks	20 tasks
ViT-B/32	2.1	2.5	2.9
ViT-B/16	7.0	8.0	9.0

Appendix

Table 13 : Peak GPU memory (GB) vs. the number of merged tasks, at

N=256

calib. samples.

Backbone	$N=128$	$N=256$	$N=512$	$N=1024$	$N=2048$
ViT-B/32	1.4	2.1	2.4	3.2	5.5
ViT-B/16	3.8	7.0	8.1	10.9	20.2

Appendix

Table 14 : Peak GPU memory (GB) versus calib. samples per task, at 8 tasks.

Deployed model	QK/VO rank	Accuracy (%)	Throughput (img/s) $\uparrow$	Latency (ms/img) $\downarrow$	Peak memory (GB) $\downarrow$
All baseline merges (shared original architecture)	64 / 64	64.23–78.50	$\approx$ 3257	0.307	0.60
SLOA (original rank, $r=64$ )	64 / 64	82.37	3261.1	0.307	0.60
SLOA (expanded rank, $r=128$ )	128 / 128	83.53	2711.5	0.369	0.65
SLOA (expanded rank, $r=256$ )	256 / 256	83.74	1999.1	0.500	0.83

Appendix

Table 15 : Inference cost by deployed rank on ViT-B/16 (20-task merge). Inference measured with batch size 64, FP16, on a single A100 GPU, averaged over 100 iterations.

	Method	# Samples	Accuracy							Average
	Method	# Samples	CoLA	SST2	MRPC	QQP	MNLI	QNLI	RTE	Avg. $\uparrow$
	Individual	–	0.6018	0.9404	0.8922	0.9141	0.8720	0.9271	0.7906	0.8483
Data-Free	Weight Averaging	–	0.1808	0.8188	0.8116	0.7381	0.4383	0.7106	0.6173	0.6165
	Task Arithmetic	–	0.2909	0.8463	0.8188	0.7869	0.5591	0.7584	0.6462	0.6724
	TIES-Merging	–	0.1951	0.8429	0.8333	0.8351	0.6501	0.7309	0.4260	0.6448
	TSV-M	–	0.3477	0.8739	0.8446	0.8286	0.6163	0.8124	0.6606	0.7120

Appendix

Table 16 : Multi-task performance on seven GLUE tasks with merged RoBERTa models. Training-free methods use 256 calibration samples. SLOA denotes rank-preserving operator alignment, while SLOA (

r=128

) expands the factorization rank of the merged QK and VO operators.

Method	SUN397	GTSRB	Cars	EuroSAT	RESISC45	DTD	MNIST	SVHN	Avg.
Fine-tuned (individual)	74.90	98.91	78.47	99.07	95.14	79.79	99.58	97.27	90.39
Simple Avg.	65.39	54.92	62.42	75.67	70.63	50.53	86.28	64.49	66.29
Task Arithmetic	64.30	62.71	61.42	78.30	70.57	51.76	93.01	73.88	69.49
TIES-Merging	61.40	78.68	60.15	69.04	69.62	51.06	97.72	85.92	71.70
TSV-M	68.59	91.27	72.01	93.41	85.14	64.36	98.76	90.79	83.04
Iso-C	67.73	93.51	71.61	90.00	84.98	65.05	98.82	89.39	82.64

Appendix

Table 17 : Per-task accuracy (%) of merged ViT-B/32 models on 8 vision tasks. Training-free methods use

N{=}256

calibration samples per task; their averages are mean

\pm

std over three seeds. For SLOA,

r

is the retained rank of the QK and VO operators (

r{=}64

preserves the original architecture). Bold/underline marks the best/second-best merging method per column. Data-free methods are deterministic given the checkpoints, so standard deviations are reported only for calibration-based methods, computed over three random draws of the calibration set.

Method	SUN397	GTSRB	Cars	EuroSAT	RESISC45	DTD	MNIST	SVHN	Flowers	PCAM	FER2013	Pets	STL10	CIFAR100	Avg.
Fine-tuned (individual)	74.90	98.91	78.47	99.07	95.14	79.79	99.58	97.27	88.62	87.96	71.62	92.37	97.55	88.38	89.26
Simple Avg.	64.73	45.63	60.40	66.96	67.06	47.13	76.57	50.66	67.25	65.27	51.62	84.25	97.20	70.39	65.37
Task Arithmetic	64.41	50.00	59.63	67.74	67.29	47.77	80.75	53.93	66.17	69.80	53.05	84.25	96.62	69.14	66.47
TIES-Merging	62.10	63.95	54.58	62.93	65.29	50.21	92.56	65.70	58.22	77.10	54.95	81.30	94.84	62.44	67.58
TSV-M	67.30	87.80	63.85	91.04	81.22	61.17	98.35	85.23	68.01	82.03	65.53	87.82	97.18	71.06	79.11
Iso-C	70.02	81.45	63.66	83.85	80.10	60.96	96.36	75.98	70.34	83.29	65.30	87.87	97.08	74.99	77.94

Appendix

Table 18 : Per-task accuracy (%) of merged ViT-B/32 models on 14 vision tasks. Setup and notation as in Table 17 .

Method	SUN397	GTSRB	Cars	EuroSAT	RESISC45	DTD	MNIST	SVHN	Flowers	PCAM	FER2013	Pets	STL10	CIFAR100	CIFAR10	Food101	FMNIST	EMNIST	KMNIST	SST2	Avg.
Fine-tuned (individual)	74.90	98.91	78.47	99.07	95.14	79.79	99.58	97.27	88.62	87.96	71.62	92.37	97.55	88.38	97.60	88.40	94.75	95.62	98.23	71.28	89.78
Simple Avg.	64.22	43.14	59.56	60.85	64.76	46.44	71.79	47.22	66.29	63.95	50.14	83.92	96.99	69.80	92.66	80.38	71.31	14.98	11.41	61.78	61.08
Task Arithmetic	62.03	48.95	53.74	58.15	60.87	45.96	79.45	48.43	60.95	73.45	51.32	82.23	94.89	64.64	91.38	71.88	73.86	17.81	12.14	59.86	60.60
TIES-Merging	65.02	49.22	59.77	60.78	66.22	48.51	79.27	52.37	66.55	66.75	51.66	84.06	97.05	70.40	93.33	80.45	72.43	17.32	12.16	61.94	62.76
TSV-M	65.50	83.79	55.80	87.11	75.76	58.62	97.61	80.28	65.00	80.57	64.29	86.29	96.55	68.92	93.89	76.37	84.27	35.91	46.48	70.18	73.66
Iso-C	67.13	80.72	47.76	78.19	73.16	57.07	96.82	75.47	63.16	79.79	62.75	83.73	96.05	71.96	94.36	73.38	82.60	33.99	35.11	70.02	71.16

Appendix

Table 19 : Per-task accuracy (%) of merged ViT-B/32 models on 20 vision tasks. Setup and notation as in Table 17 .

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Model merging integrates task-specific fine-tuned models into a single multi-task model, but often suffers from parameter interference caused by conflicting task-vector updates. Existing methods typically mitigate conflicts by pruning task vectors based on weight magnitude or random heuristics, treating Transformers as unstructured ``bags of parameters'' and overlooking their inherent modularity. In this paper, we propose \textbf{C}ontribution-\textbf{A}ware \textbf{S}tructured \textbf{S}parsity (CASS), a unified framework that reduces parameter interference by identifying and preserving task-specific components. At the core of CASS is a contribution-aware structured mask that identifies task-relevant attention heads and FFN neurons. We instantiate this mask in two settings: CASS-Merging, the primary post-hoc setting where masks serve as a plug-and-play denoising filter for existing merging operators, and CASS-Tuning, an extension for scenarios with fine-tuning access where masks constrain gradients to reduce structural overlap between task vectors. Our analysis shows that task-relevant components are sparse and partially disjoint, supporting structured component-level filtering as an effective way to reduce merging interference. Extensive experiments across vision (ViT, 20 tasks) and language (RoBERTa, 8 tasks; Qwen2.5, 4 tasks) benchmarks demonstrate that CASS improves a range of representative merging baselines.

Abstract

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