Loop Dropout: Regularizing Shared Updates in Looped Language Models

Zirui Zhu; Hailun Xu; Xuanlei Zhao; Yong Liu; Yingxuan Ren; Kanchan Sarkar; Kun Xu; Yang You

cs.LGSep 28, 2026

Loop Dropout: Regularizing Shared Updates in Looped Language Models

Authors: Zirui Zhu, Hailun Xu, Xuanlei Zhao, Yong Liu, Yingxuan Ren, Kanchan Sarkar, Kun Xu, Yang You

Abstract

Looped language models separate computational depth from parameter count by repeatedly applying the same transformer block. Adapting these models requires a shared update that remains effective as hidden states evolve throughout the recurrent computation. Our empirical analysis reveals a pronounced late-loop bias in standard low-rank adaptation (LoRA): the shared update provides limited adaptation at early loop positions. This imbalance motivates training shared updates under varying combinations of their applications. Randomly omitting adapter applications alone, however, does not improve task performance; it reduces expected update strength during training while leaving inference unchanged. We introduce Loop Dropout, which couples stochastic masking of adapter applications with inverse-survival rescaling to preserve expected update strength and promote effective adaptation across loops. Extensive experiments demonstrate improved mathematical reasoning across model sizes, adapter ranks and training recipes, with benefits extending to general instruction tuning and code generation. Loop Dropout outperforms existing LoRA variants and adapter regularizers, while further analysis shows stronger early-loop adaptation. Every backbone loop remains active, and inference applies the adapter at all loops using standard LoRA without additional trainable parameters or inference computation. Code is available at https://github.com/NUS-HPC-AI-Lab/loop-dropout .

Figures & tables

Panel of Figure 1 — Figure 1: Loop Dropout reduces the late-loop bias of shared adaptation. Ouro ( Zhu et al., 2025 ) uses four loops through a shared transformer block by default. To probe adaptation across loops after GSM8K fine-tuning, we activate the adapter in one loop at a time while retaining all four backbone loops. LoRA’s loss reduction falls from 32% at the fourth loop to 12% at the first; Loop Dropout maintains 31–36% across positions. All reductions are relative to the frozen model and use final-loop teacher-forced test loss.

Panel of Figure 2 — Figure 2: Loop Dropout regularizes shared adaptation across loops. The shared block $F$ reuses the same $W$ and $\Delta$ over $K$ loops. Following Eq. ( 3 ), training uses $g_{t}=b_{t}/q$ , where $q=1-p$ and $b_{t}\sim\operatorname{Bernoulli}(q)$ independently for each example and loop. The training row illustrates one sampled pattern: retained updates have gate $1/q$ , and dropped updates have gate $0$ . LoRA and inference use $g_{t}=1$ at every loop. A zero gate removes only $\Delta$ ; every backbone loop remains active.

	Ouro-1.4B			Ouro-2.6B
Benchmark	LoRA	LoRA+	Loop Dropout	LoRA	LoRA+	Loop Dropout
GSM8K	85.34 $\pm$ 0.64	85.54 $\pm$ 0.23	86.53 $\pm$ 0.44	87.62 $\pm$ 0.52	87.21 $\pm$ 0.38	88.73 $\pm$ 0.31
MATH-500	38.87 $\pm$ 1.67	38.13 $\pm$ 0.42	47.53 $\pm$ 2.61	42.87 $\pm$ 1.17	43.40 $\pm$ 0.53	47.20 $\pm$ 0.53

Table 1: Mathematical reasoning performance. Accuracy in %, mean

\pm

SD over three training seeds per method. Bold marks the highest mean within each model and benchmark.

Method	GSM8K	MATH-500	Train (min)	Memory (GB)
LoRA	85.34 $\pm$ 0.64	38.87 $\pm$ 1.67	110.6	45.36
LoRA+	85.54 $\pm$ 0.23	38.13 $\pm$ 0.42	112.8	45.40
CoTo-on-Ouro	85.60 $\pm$ 0.76	37.80 $\pm$ 1.06	94.1	45.28
LoRA Dropout	85.77 $\pm$ 0.70	42.13 $\pm$ 0.31	444.9	45.37
Loop Dropout	86.53 $\pm$ 0.44	47.53 $\pm$ 2.61	128.9	45.36

Table 3: Comparison with existing adaptation methods. Ouro-1.4B, rank 16, mathematical recipe. Accuracy is mean

\pm

SD over three training seeds; training time and peak memory are their means. Displayed costs use an H100 80GB and exclude learning-rate search and evaluation. All methods use 15.1M adapter parameters.

Method	Only loop 1	Only loop 2	Only loop 3	Only loop 4
LoRA	0.03 $\pm$ 0.04	0.30 $\pm$ 0.46	2.63 $\pm$ 3.07	6.07 $\pm$ 4.42
Loop Dropout	1.31 $\pm$ 0.69	64.72 $\pm$ 13.56	86.23 $\pm$ 0.84	86.18 $\pm$ 0.78

Table 4: GSM8K generation with one adapter application. Ouro-1.4B after MetaMath-GSM fine-tuning; accuracy in %, mean

\pm

SD over three seeds. Only the indicated adapter application is enabled, with all four backbone loops retained.

Method	GSM8K	MATH-500
LoRA	85.32 $\pm$ 0.92	40.13 $\pm$ 1.62
Unscaled	84.71 $\pm$ 0.19	39.33 $\pm$ 1.70
Dose control	85.57 $\pm$ 0.88	38.60 $\pm$ 1.71
Module-wise	85.52 $\pm$ 0.62	41.13 $\pm$ 0.76
Low-rank weight noise	85.70 $\pm$ 0.54	39.13 $\pm$ 0.83
Parallel noise	85.62 $\pm$ 0.64	37.93 $\pm$ 1.33

Table 5: Ablating loop-level masking and inverse-survival rescaling. The controls separate update scaling, gate variation across loops, masking granularity and matched noise. All rows use shared rank-16 adapters on Ouro-1.4B, retain four backbone loops and apply the full adapter update at inference. MetaMath-GSM recipe, learning rate

10^{-4}

; accuracy in %, mean

\pm

SD over three seeds. Noise strength

c=1

matches the expected perturbation energy of Loop Dropout. Per-seed results are in Tables 10 and 15 .

			Without loop mask		With loop mask
Adapters	Rank	Params	GSM8K	MATH-500	GSM8K	MATH-500
Shared	16	15.1M	84.69	37.00	87.04	48.40
Independent	4	15.1M	84.08	39.60	87.57	44.40
Independent	16	60.6M	84.53	39.40	86.43	43.80

Table 6: Loop Dropout with shared and independent adapters. Shared adapters reuse their parameters across all four loops; Independent adapters learn separate parameters for each loop. Rank is per loop, and Params counts all trainable adapter parameters. With loop mask uses complete-application masking and inverse-survival rescaling during training; all adapters are active at inference. MetaMath-GSM recipe with development selection over five learning rates per method; accuracy in % for training seed 101.

Panel of Figure 3 — Figure 3: Mathematical accuracy across adapter ranks. Ouro-1.4B after MetaMath-GSM fine-tuning. Markers and horizontal bars show mean $\pm$ SD over three seeds. Both methods use the same five-rate search budget at each rank, with $\alpha/r=2$ ; Table 20 gives the values.

Panel of Figure 4 — Figure 4: Depth transfer on Ouro. Loop Dropout gains over LoRA on GSM8K (percentage points). Outlined cells match training and evaluation depth.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

	MetaMath-GSM-100k	GSM8K	Tülu 2-100k
Training examples	100,000	6,973	98,415
Epochs / optimizer steps	1 / 3,125	2 / $\approx$ 1,744	1 / 769
Effective batch	$8\times 4=32$	$8\times 1=8$	$8\times 16=128$
Maximum sequence length	1,024	512	2,048
Warm-up	3%	5%	3%
Learning rates	See below	3e-5 to 6e-4	1e-4

Appendix

Table 7: Fine-tuning recipes. Effective batch = micro-batch

\times

gradient accumulation. The adapter recipes use AdamW ( Loshchilov and Hutter, 2019 ) (

\beta_{1}=0.9

\beta_{2}=0.999

), zero weight decay, gradient clipping at 1.0, a cosine schedule decaying to 10% of the peak learning rate, and the final checkpoint.

Method	Per-seed time	Mean $\pm$ SD	Memory (GB)
LoRA	111.93 / 106.74 / 113.28	110.65 $\pm$ 3.45	45.36
LoRA+	111.20 / 107.74 / 119.40	112.78 $\pm$ 5.99	45.40
CoTo-on-Ouro	94.95 / 93.32 / 94.06	94.11 $\pm$ 0.82	45.28
LoRA Dropout	442.39 / 438.34 / 453.98	444.91 $\pm$ 8.12	45.37
Loop Dropout	130.86 / 128.16 / 127.65	128.89 $\pm$ 1.72	45.36

Appendix

Table 8: Training time across seeds on H100 80GB. Times are in minutes for seeds 101 / 102 / 103, with their mean and sample SD. Peak memory is the mean over the same runs.

	LoopUS-Qwen3-4B		Huginn
Benchmark	LoRA	Loop Dropout	LoRA	Loop Dropout
GSM8K	83.93	85.52	59.89	60.11
MATH-500	34.60	40.80	13.00	13.80

Appendix

Table 9: Mathematical evaluation on other looped transformers. Accuracy in %. Bold marks the higher value within each model and benchmark.

Method	LR	GSM8K seeds 101 / 102 / 103	MATH-500 seeds 101 / 102 / 103
LoRA	1e-4	84.6 / 85.0 / 86.4	39.2 / 42.0 / 39.2
Loop Dropout	1e-4	87.7 / 87.0 / 88.0	44.0 / 45.4 / 42.6
Unscaled	1e-4	84.5 / 84.7 / 84.9	40.6 / 40.0 / 37.4
Unscaled	2e-4	83.8 / 83.8 / 84.8	35.6 / 38.0 / 37.4
Dose control	1e-4	85.4 / 84.8 / 86.5	36.8 / 40.2 / 38.8
Dose control	2e-4	85.5 / 84.9 / 85.9	37.8 / 42.4 / 37.8

Appendix

Table 10: Per-seed results of the mask-control comparison on Ouro-1.4B under the MetaMath protocol, at the learning rates shown; accuracy in %.

Metric	LoRA	Loop Dropout
MMLU (0-shot)	68.64 $\pm$ 0.40	68.91 $\pm$ 0.13
BBH (3-shot CoT)	71.20 $\pm$ 0.14	70.78 $\pm$ 0.11
TruthfulQA MC2	47.32 $\pm$ 1.02	48.19 $\pm$ 0.68
HumanEval	71.54 $\pm$ 2.14	72.56 $\pm$ 0.61
HumanEval+	67.68 $\pm$ 3.05	69.92 $\pm$ 0.35
MBPP	75.57 $\pm$ 0.40	76.72 $\pm$ 0.53

Appendix

Table 11: All downstream metrics after Tülu 2 instruction tuning of Ouro-1.4B. Rank 16; accuracy in %, mean

\pm

SD over three matched seeds. HumanEval+ and MBPP+ add the expanded EvalPlus tests to the original ones; IFEval reports prompt-level accuracy. The last three rows average the four code metrics, the six-benchmark set, and all nine metrics per seed; bold marks the highest average in each summary row.

Variant	Mask	LR 1e-4	LR 3e-4
LoRA	none	80.6 $\pm$ 0.3	78.1 $\pm$ 1.0
Loop Dropout	uniform $p=0.5$	81.7 $\pm$ 0.2	81.3 $\pm$ 0.7
Early-heavy profile	$p=(.75,.75,.25,.25)$	82.1 $\pm$ 0.2	79.4 $\pm$ 2.6
Late-heavy profile	$p=(.25,.25,.75,.75)$	80.0 $\pm$ 0.5	77.9 $\pm$ 1.0
Ramp down	$p=(.8,.6,.4,.2)$	80.7 $\pm$ 0.8	79.3 $\pm$ 0.4
Ramp up	$p=(.2,.4,.6,.8)$	80.0 $\pm$ 0.8	79.0 $\pm$ 2.6

Appendix

Table 12: Structured, scheduled and adaptive masks on the GSM8K recipe (Ouro-1.4B, strict exact match, %; two seeds, mean

\pm

SD). Masks control adapter applications while all four backbone loops run; all variants keep the

1/P(\text{keep})

rescaling. Bottom: five-seed follow-up of the two closest variants. No variant exceeds the uniform rule at both learning rates.

Rank	Benchmark	LoRA	Loop Dropout	Difference
4	GSM8K	85.82 $\pm$ 0.20	86.38 $\pm$ 0.50	$+0.56$
4	MATH-500	39.00 $\pm$ 1.25	46.07 $\pm$ 1.17	$+7.07$
64	GSM8K	85.44 $\pm$ 1.12	87.01 $\pm$ 0.49	$+1.57$
64	MATH-500	37.87 $\pm$ 1.70	43.33 $\pm$ 1.85	$+5.47$
128	GSM8K	85.06 $\pm$ 0.62	86.48 $\pm$ 0.22	$+1.42$
128	MATH-500	34.80 $\pm$ 1.00	41.60 $\pm$ 1.39	$+6.80$

Appendix

Table 13: Paired comparisons at different shared adapter ranks. Three seeds, LR

10^{-4}

and

\alpha/r=2

. Accuracy is in %; differences are in percentage points.

	Missing answers (%)		Net gain (points)
Rank	LoRA	Loop Dropout	Both extractable	Missing in either	Total
4	8.80	9.60	6.80	0.27	7.07
64	7.60	5.60	4.80	0.67	5.47
128	7.80	5.67	6.00	0.80	6.80

Appendix

Table 14: Decomposing the net MATH-500 accuracy difference. Missing-answer rates are percentages. Both net-difference components use all 500 problems as denominator and sum to the overall gain in percentage points. This decomposition does not change the official scorer.

Method	Rank	LR	$c$	GSM8K	MATH-500
LoRA	4	1e-4	–	85.90 / 85.60 / 85.97	38.0 / 38.6 / 40.4
Loop Dropout	4	1e-4	–	86.13 / 86.05 / 86.96	45.6 / 47.4 / 45.2
LoRA	64	1e-4	–	84.15 / 86.20 / 85.97	36.2 / 39.6 / 37.8
Loop Dropout	64	1e-4	–	87.57 / 86.66 / 86.81	44.4 / 41.2 / 44.4
LoRA	128	1e-4	–	84.91 / 84.53 / 85.75	35.8 / 33.8 / 34.8
Loop Dropout	128	1e-4	–	86.35 / 86.35 / 86.73	43.2 / 40.8 / 40.8

Appendix

Table 15: Per-seed results of the additional suites. Seeds are ordered 101 / 102 / 103. Accuracy is in %;

c

is the noise strength and is inapplicable to LoRA and Loop Dropout.

Seed	GSM8K	MATH-500
101	84.84	37.40
102	85.60	39.00
103	86.35	37.00
Mean $\pm$ SD	85.60 $\pm$ 0.76	37.80 $\pm$ 1.06

Appendix

Table 16: Per-seed CoTo-on-Ouro results. Full GSM8K and MATH-500 test sets under the MetaMath protocol; accuracy in %. The summary uses the sample standard deviation.

	LoRA+		LoRA Dropout
Seed	GSM8K	MATH-500	GSM8K	MATH-500
101	85.29	37.80	85.44	42.20
102	85.60	38.60	85.29	41.80
103	85.75	38.00	86.58	42.40
Mean $\pm$ SD	85.54 $\pm$ 0.23	38.13 $\pm$ 0.42	85.77 $\pm$ 0.70	42.13 $\pm$ 0.31

Appendix

Table 17: LoRA+ and LoRA Dropout across three training seeds. Ouro-1.4B, rank 16, MetaMath-GSM recipe; accuracy in %.

Method	Params	GSM8K	MATH-500
Shared LoRA, rank 16	30.3M	87.62 $\pm$ 0.52	42.87 $\pm$ 1.17
LoRA+, rank 16	30.3M	87.21 $\pm$ 0.38	43.40 $\pm$ 0.53
Independent, rank 4	30.3M	85.95 $\pm$ 2.59	43.60 $\pm$ 0.40
Independent, rank 16	121.1M	85.87 $\pm$ 0.87	41.40 $\pm$ 1.71
Loop Dropout, rank 16	30.3M	88.73 $\pm$ 0.31	47.20 $\pm$ 0.53

Appendix

Table 18: Shared and independent adapters on Ouro-2.6B. MetaMath-GSM recipe, mean

\pm

SD over three training seeds, in %. All backbone loops and all trained adapter applications are active at inference.

Method	LR	GSM8K	MATH-500
Shared LoRA, rank 16	$2\times 10^{-4}$	88.02 / 87.04 / 87.79	42.00 / 44.20 / 42.40
LoRA+, rank 16	$10^{-4}$	87.04 / 86.95 / 87.64	43.80 / 42.80 / 43.60
Independent, rank 4	$2\times 10^{-4}$	86.88 / 87.95 / 83.02	43.20 / 44.00 / 43.60
Independent, rank 16	$2\times 10^{-4}$	85.90 / 84.99 / 86.73	39.80 / 41.20 / 43.20
Loop Dropout, rank 16	$10^{-4}$	89.01 / 88.78 / 88.40	47.80 / 47.00 / 46.80

Appendix

Table 19: Per-seed Ouro-2.6B results after independent tuning. Seeds are ordered 101 / 102 / 103; accuracy in %.

Rank	Method	GSM8K	MATH-500
4	LoRA	85.82 $\pm$ 0.20	39.00 $\pm$ 1.25
4	Loop Dropout	86.38 $\pm$ 0.50	46.07 $\pm$ 1.17
16	LoRA	85.34 $\pm$ 0.64	38.87 $\pm$ 1.67
16	Loop Dropout	86.53 $\pm$ 0.44	47.53 $\pm$ 2.61
64	LoRA	85.44 $\pm$ 1.12	37.87 $\pm$ 1.70
64	Loop Dropout	85.87 $\pm$ 0.48	40.73 $\pm$ 1.42

Appendix

Table 20: Independent tuning at each adapter rank. Ouro-1.4B, three seeds per rank and method; full GSM8K and MATH-500 evaluation, mean

\pm

SD in %. Both methods have five learning-rate candidates at each rank.

Rank	Method	LR	GSM8K	MATH-500
4	LoRA	$10^{-4}$	85.90 / 85.60 / 85.97	38.00 / 38.60 / 40.40
4	Loop Dropout	$10^{-4}$	86.13 / 86.05 / 86.96	45.60 / 47.40 / 45.20
16	LoRA	$2\times 10^{-4}$	84.69 / 85.37 / 85.97	37.00 / 40.20 / 39.40
16	Loop Dropout	$5\times 10^{-5}$	87.04 / 86.35 / 86.20	48.40 / 49.60 / 44.60
64	LoRA	$10^{-4}$	84.15 / 86.20 / 85.97	36.20 / 39.60 / 37.80
64	Loop Dropout	$2\times 10^{-4}$	85.60 / 85.60 / 86.43	42.00 / 39.20 / 41.00

Appendix

Table 21: Per-seed results after independent rank-wise tuning. Seeds are ordered 101 / 102 / 103; accuracy in %. The same learning rate is used for all three seeds in a row.

Group	$n$	LoRA	Loop Dropout	Difference
Algebra	124	58.33 $\pm$ 3.05	72.04 $\pm$ 3.05	$+13.71$
Counting & Probability	38	29.82 $\pm$ 9.24	39.47 $\pm$ 2.63	$+9.65$
Geometry	41	32.52 $\pm$ 6.14	38.21 $\pm$ 3.73	$+5.69$
Intermediate Algebra	97	19.59 $\pm$ 1.03	25.43 $\pm$ 6.21	$+5.84$
Number Theory	62	40.32 $\pm$ 5.59	44.09 $\pm$ 4.06	$+3.76$
Prealgebra	82	54.88 $\pm$ 4.40	62.60 $\pm$ 1.86	$+7.72$

Appendix

Table 22: MATH-500 breakdown for independently tuned rank-16 adapters. Mean

\pm

SD over three seeds, in %; differences in percentage points. Each partition covers all 500 problems, with

n

denoting the number of problems per group. Checkpoints are identical to those in Table 20 .

Panel of Figure 5 — Figure 5: Zero-shot generalization beyond the training depth. Ouro-1.4B after direct GSM8K fine-tuning at four loops, evaluated without further training; accuracy is mean $\pm$ SD over three seeds. Annotations show the gain of Loop Dropout over LoRA in percentage points.

Method	Train $K$	Eval $K=4$	Eval $K=6$	Eval $K=8$
LoRA	4	79.9 $\pm$ 1.0	75.0 $\pm$ 1.0	70.8 $\pm$ 1.9
LoRA	6	82.4 $\pm$ 1.1	82.3 $\pm$ 0.7	77.8 $\pm$ 0.7
LoRA	8	80.3 $\pm$ 1.4	81.8 $\pm$ 0.6	81.6 $\pm$ 0.9
Loop Dropout	4	81.8 $\pm$ 0.5	78.0 $\pm$ 1.7	75.6 $\pm$ 1.3
Loop Dropout	6	83.0 $\pm$ 0.3	83.3 $\pm$ 0.7	80.5 $\pm$ 0.8
Loop Dropout	8	83.4 $\pm$ 0.5	83.6 $\pm$ 0.5	82.1 $\pm$ 0.8

Appendix

Table 23: Accuracy across training and evaluation depths. GSM8K recipe, Ouro-1.4B, LR

10^{-4}

(

A

-factor rate for LoRA+), seeds 101–103; mean

\pm

SD of strict exact match, %. Loop Dropout has higher mean accuracy than LoRA in all nine depth pairs. Comparisons within a training depth share the training budget; different training depths use different numbers of recurrent passes per optimizer step.

Model	Method	LR	GSM8K
Ouro-1.4B	LoRA	1e-4	79.9 $\pm$ 1.0
Ouro-1.4B	LoRA+	1e-4	79.8 $\pm$ 0.5
Ouro-1.4B	rsLoRA	1e-4	79.8 $\pm$ 0.9
Ouro-1.4B	Loop Dropout	1e-4	81.8 $\pm$ 0.5
Ouro-1.4B	LoRA	1e-4	80.4 $\pm$ 0.6
Ouro-1.4B	Loop Dropout	1e-4	81.9 $\pm$ 0.4

Appendix

Table 24: The gain persists across learning rates and model sizes under the GSM8K recipe. Zero-shot strict exact match (%) after fine-tuning on the GSM8K training split, mean

\pm

SD. The first and last blocks use three seeds; the middle block uses seeds 0–4. LR denotes the

A

-factor rate for LoRA+. Bold marks the highest mean within each model, seed set and learning rate.

LR	LoRA	Loop Dropout	$\Delta$
3e-5	80.4 $\pm$ 0.4	80.9 $\pm$ 0.3	$+0.56$
1e-4	80.4 $\pm$ 0.6	81.9 $\pm$ 0.4	$+1.52$
2e-4	79.8 $\pm$ 1.5	82.2 $\pm$ 0.6	$+2.38$
3e-4	78.6 $\pm$ 1.1	81.2 $\pm$ 0.6	$+2.62$
6e-4	72.4 $\pm$ 0.7	73.8 $\pm$ 0.8	$+1.36$

Appendix

Table 25: Learning-rate curve on the GSM8K recipe (Ouro-1.4B,

K=4

; lm-eval zero-shot strict exact match, %). Five seeds at

10^{-4}

and

3\times 10^{-4}

, three seeds elsewhere.

		GSM8K-Platinum		GSM-Plus mini
Method	LR	Accuracy	$\Delta$ vs. LoRA	Accuracy	$\Delta$ vs. LoRA
LoRA	1e-4	82.4 $\pm$ 0.0	–	58.9 $\pm$ 0.7	–
Input dropout	1e-4	81.7 $\pm$ 0.4	$-0.66$	59.1 $\pm$ 0.4	$+0.21$
Loop Dropout	1e-4	83.8 $\pm$ 0.4	$+1.38$	60.0 $\pm$ 0.2	$+1.14$
LoRA	3e-4	80.3 $\pm$ 0.4	–	56.0 $\pm$ 1.0	–
Input dropout	3e-4	80.7 $\pm$ 1.7	$+0.33$	57.0 $\pm$ 1.1	$+1.04$

Appendix

Table 26: Transfer of the saved GSM8K-recipe adapters (Ouro-1.4B, three seeds) to GSM8K-Platinum (1,209 relabelled problems) and GSM-Plus mini (2,400 perturbed problems); zero-shot strict exact match, %. “Input dropout” applies standard dropout of 0.1 to the LoRA input.

$K$	Seeds	LoRA	Loop Dropout	$\Delta$
2	0–2	68.6 $\pm$ 0.8	70.7 $\pm$ 0.3	$+2.17$
4	0–4	80.4 $\pm$ 0.6	81.9 $\pm$ 0.4	$+1.52$
6	0–2	81.6 $\pm$ 0.6	83.2 $\pm$ 1.2	$+1.59$
4	101–103	79.9 $\pm$ 1.0	81.8 $\pm$ 0.5	$+1.90$
6	101–103	82.3 $\pm$ 0.7	83.3 $\pm$ 0.7	$+1.06$
8	101–103	81.6 $\pm$ 0.9	82.1 $\pm$ 0.8	$+0.51$

Appendix

Table 27: Training and evaluating at the same recurrence depth on the GSM8K recipe (Ouro-1.4B, LR

10^{-4}

; strict exact match, %). Seeds 0–2 (five at

K=4

) use independent initializations; seeds 101–103 share initialization across methods.

		Final-loop loss				All-on
Method	LR	Only 1	Only 2	Only 3	Only 4	accuracy
LoRA	$10^{-4}$	0.7437	0.6834	0.6208	0.5709	80.59
Unscaled	$10^{-4}$	0.5309	0.5082	0.4984	0.5037	77.03
Loop Dropout	$10^{-4}$	0.5847	0.5475	0.5409	0.5432	81.73
LoRA	$3\times 10^{-4}$	0.7493	0.6835	0.6159	0.5317	77.86
Unscaled	$3\times 10^{-4}$	0.5152	0.4907	0.4824	0.4895	73.84

Appendix

Table 28: Single-loop activation of a trained update. Ouro-1.4B, direct GSM8K recipe. The four loss columns are final-loop teacher-forced cross entropy (lower is better). The last column separately reports standard all-on generation accuracy on all 1,319 test questions, in %.

Readout	LoRA	Loop Dropout
Loop 1	1.240 $\pm$ 0.041	1.047 $\pm$ 0.010
Loop 2	0.836 $\pm$ 0.006	0.765 $\pm$ 0.005
Loop 3	0.742 $\pm$ 0.009	0.717 $\pm$ 0.002
Loop 4	0.737 $\pm$ 0.010	0.700 $\pm$ 0.005

Appendix

Table 29: MATH-500 readouts along the fully adapted recurrence. Ouro-1.4B after MetaMath-GSM fine-tuning. Token-weighted cross entropy, mean

\pm

SD over three training seeds; lower is better. Every adapter application and all four backbone loops are enabled.

Panel of Figure 6 — Figure 6: Generation and early readouts after MetaMath-GSM fine-tuning. Ouro-1.4B, rank 16, four backbone loops. Left: GSM8K generation with only the indicated adapter application enabled. Right: MATH-500 teacher-forced cross entropy with all applications enabled, read out at each loop. Points and error bars show mean $\pm$ SD over three training seeds. Loop Dropout supports generation from individual applications at loops two through four and lowers MATH-500 loss at every readout, most strongly at the first loop.

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1 ´Ecole Normale Sup´erieure PSL, Paris, France · 2CNRS · 3Apple, Paris, France.

May 7, 2026, cs.AIJ/K move · Enter open · S save

Rethinking Adapter Placement: A Dominant Adaptation Module Perspective

Suoxin Zhang, Run He, Di Fang, Xiang Tan+2

South China University of Technology, China · Zhejiang University, China

Low-Rank Adaptation Adapters

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Low-rank adaptation (LoRA) is a widely used parameter-efficient fine-tuning method that places trainable low-rank adapters into frozen pre-trained models. Recent studies show that using fewer LoRA adapters may still maintain or even improve performance, but existing methods still distribute adapters broadly, leaving where to place a limited number of adapters to maximize performance largely open. To investigate this, we introduce PAGE (Projected Adapter Gradient Energy), a gradient-based sensitivity probe that estimates the initial trainable gradient energy available to each candidate LoRA adapter. Surprisingly, we find that PAGE is highly concentrated on a single shallow FFN down-projection across two model families and four downstream tasks. We term this module the dominant adaptation module and show that its layer index is architecture-dependent but task-stable. Motivated by this finding, we propose DomLoRA, a placement method that places a single adapter at the dominant adaptation module. With only ~0.7% of vanilla LoRA's trainable parameters, DomLoRA outperforms it on average across various downstream tasks, including instruction following, mathematical reasoning, code generation, and multi-turn conversation. This method also improves other LoRA variants, supporting the dominant adaptation module perspective as a practical placement guideline.

Abstract

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Balanced LoRA: Removing Parameter Invariance to Accelerate Convergence