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LoRA vs Full Fine-tuning: An Illusion of Equivalence

Reece Shuttleworth, Jacob Andreas, Antonio Torralba, Pratyusha Sharma

arXiv:2410.21228v3cs.LGcs.CL

TL;DR

The paper asks whether LoRA and full fine-tuning learn equivalent solutions, given LoRA’s parameter efficiency and prior findings on forgetting. It analyzes weight-matrix spectra and intervenes on intruder dimensions, finding distinct spectral structures and a link between those dimensions and forgetting, including in continual learning.

  • Problem

    Although LoRA can match full fine-tuning performance, it is unclear whether the two methods learn similar solutions or inherently different ones.

    Source: §1 · pp. 1–2

  • Method

    The paper compares fine-tuned weight matrices through their singular-value structure and intervenes on intruder dimensions by scaling their singular values.

    Source: §1 · p. 2, §3 · p. 5, §5 · p. 8

  • Results

    LoRA introduces intruder dimensions absent from full fine-tuning, and scaling their singular values reduces pretraining-distribution loss with minimal test-performance drop; LoRA also tends to forget more in continual learning.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    Intruder dimensions are a target for reducing forgetting while retaining downstream test performance, and their accumulation is relevant to LoRA’s continual-learning behavior.

    Source: §1 · pp. 2–3

Abstract

from arXiv · show

Fine-tuning is a crucial paradigm for adapting pre-trained large language models to downstream tasks. Recently, methods like Low-Rank Adaptation (LoRA) have been shown to effectively fine-tune LLMs with an extreme reduction in trainable parameters. But, \emph{are their learned solutions really equivalent?} We study how LoRA and full-finetuning change pre-trained models by analyzing the model's weight matrices through the lens of their spectral properties. We find that LoRA and full fine-tuning yield weight matrices whose singular value decompositions exhibit very different structure: weight matrices trained with LoRA have new, high-ranking singular vectors, which we call \emph{intruder dimensions}, while those trained with full fine-tuning do not. Further, we extend the finding that LoRA forgets less than full fine-tuning and find its forgetting is vastly localized to the intruder dimension -- by causally intervening on the intruder dimensions by changing their associated singular values post-fine-tuning, we show that they cause forgetting. Moreover, scaling them down significantly improves modeling of the pre-training distribution with a minimal drop in downstream task performance. Given this, we should expect accumulating intruder dimensions to be harmful and lead to more forgetting. This will be amplified during continual learning because of sequentially fine-tuning, and we show that LoRA models do accumulate intruder dimensions here tend to perform worse in this setting, emphasizing the practicality of our findings.

1 Introduction

The paper asks whether LoRA and full fine-tuning learn similar solutions despite LoRA’s parameter efficiency. It compares their spectral structure and reports distinct updates, links intruder dimensions to forgetting, and examines continual learning.

1 Introduction

  • LoRA introduces high-ranking singular vectors dissimilar to pretrained vectors, whereas full fine-tuning remains spectrally similar and has no intruder dimensions.The paper calls these vectors intruder dimensions.Source: §1 · p. 2
  • At nearly identical fine-tuning accuracy, LoRA’s forgetting varies with alpha and learning rate and can exceed full fine-tuning; better-generalizing models have fewer intruder dimensions.This qualifies the finding that LoRA forgets less than full fine-tuning.Source: §1 · p. 2
  • Scaling down intruder-dimension singular values reduces pretraining-distribution loss with only a minimal drop in downstream test performance.The paper attributes forgetting to these dimensions through post-fine-tuning interventions.Source: §1 · p. 2
  • LoRA tends to forget more on previously learned tasks during sequential fine-tuning, consistent with concern that intruder dimensions accumulate and increase forgetting.The introduction says this continual-learning result provides additional support for the findings.Source: §1 · pp. 2–3

2 Background & Related Work

This section introduces full fine-tuning, LoRA, and the spectral lens used to compare their learned solutions. It situates the study against prior work on LoRA’s performance, forgetting, and intrinsic rank.

Methods for fine-tuning

  • Prior work found LoRA can match full fine-tuning on classification, instruction tuning, and chat, but reported gaps on harder code-generation tasks.The cited studies include [Hu et al., 2021] for sequence classification and [Biderman et al., 2024, Zhuo et al., 2024] for code generation.Source: §2 · p. 3
  • LoRA represents a weight update as the product of two low-rank matrices, reducing trainable parameters from mn to mr + rn per matrix.The rank r is generally much smaller than min(m, n).Source: §2 · p. 3

Analysis of Solutions

  • Prior intrinsic-rank findings motivated LoRA, but earlier work had not compared LoRA and full fine-tuning weight-matrix rank or other properties at matched performance.The intrinsic dimension measure [Li et al., 2018] was used by Aghajanyan et al. [2021] to argue fine-tuning updates have low intrinsic rank.Source: §2 · p. 3

Singular Value Decomposition

  • The SVD separates a matrix into singular vectors and singular values, with singular vectors ordered by singular value revealing its most important transformation axes.The factorization is M = UΣV^T; U and V represent rotations, while Σ represents scaling.Source: §2 · p. 3

3 Structural Differences

LoRA and full fine-tuning produce structurally different weight updates: LoRA introduces high-ranking singular vectors unlike those in pre-trained weights, while full fine-tuning largely preserves the pre-trained spectral structure. Intruder dimensions appear across LoRA variants and change as training progresses.

3 Structural Differences

  • Among the top 10 singular vectors, LoRA contains intruder dimensions while full fine-tuning almost never does across the tested similarity thresholds.For LLaMA2-7B, intruders occur at ranks up to r ≤ 256; for RoBERTa, they occur at ranks up to r ≤ 16.Source: Figure 4 · p. 5, §3 · p. 5
  • The detection procedure counts a fine-tuned singular vector as an intruder when its maximum cosine similarity to any pre-trained singular vector falls below ε.It examines the top k fine-tuned singular vectors and sums counts across the model's weight matrices.Source: §3 · p. 5
  • As LoRA rank increases beyond a threshold, intruder dimensions decline and can disappear, as in LLaMA2-7B fine-tuned on math at r = 2048.At higher ranks, LoRA begins to resemble a high-rank update and its intruder-dimension curve resembles full fine-tuning.Source: §3 · p. 6
  • AdaLoRA, LoRA+, PiSSA, and VeRA also exhibit intruder dimensions with characteristics similar to vanilla LoRA.The authors report this as evidence that the finding extends to these four LoRA variants.Source: §3 · p. 6
  • LoRA has more intruder dimensions than full fine-tuning across the examined fractions of singular values, with intruders distributed across high and low singular values.The analysis holds ε fixed while varying the proportion of fine-tuned singular vectors examined.Source: §3 · p. 6
  • During LoRA fine-tuning, intruder dimensions gradually rise in rank and singular value while changing direction.Figure 5 tracks their rank, associated singular values, and test accuracy across training steps.Source: Figure 5 · p. 5, §3 · p. 6
  • Intruder counts increase with fine-tuning dataset size up to saturation, while random adapter-initialization seeds do not affect the resulting structure.Effective rank did not explain the behavioral differences, and its global nature prevents the precise examinations used later.Source: §3 · p. 6

4 Model Differences: Forgetting and Out-of-Distribution Generalization

Across the evaluated tasks, LoRA generally forgets less than full fine-tuning, including when downstream fit is equivalent. However, LoRA’s forgetting varies with its settings and is associated with intruder dimensions; scaling down an intruder dimension can reduce forgetting with little performance cost.

4 Model Differences: Forgetting and Out-of-Distribution Generalization

  • LoRA forgets less than full fine-tuning across tasks, including when RoBERTa models have equivalent downstream accuracy.For LLaMA2-7B, forgetting is measured on unrelated tasks; for RoBERTa, it is measured with pre-training pseudo-loss.Source: §4 · p. 6, Figure 6 · p. 7
  • With α = 8, LoRA exhibits intruder dimensions and worse generalization than with α = 2r, despite equivalent target-task performance.The α = 8 models show more forgetting; the α = 2r models have fewer intruder dimensions and generalize better.Source: §4 · p. 7
  • More intruder dimensions track increased forgetting without a corresponding test-accuracy relation, and high-learning-rate LoRA can forget more than full fine-tuning.The reported Spearman correlation between intruder count and forgetting is ρ = 0.971, p-value ≪0.001 for RoBERTa; the relationship is not statistically significant for test accuracy.Source: §4 · p. 7, Figure 7 · p. 7
  • Scaling down the top intruder dimension in each matrix reduces forgetting with minimal impact on fine-tuning performance.Figure 8 plots pre-training loss as forgetting and test accuracy as performance.Source: Figure 8 · p. 8, §4 · p. 8

5 Intruder Dimensions Cause Forgetting

Scaling down the highest-ranked intruder dimensions reduces pre-training forgetting with little change in downstream performance. During sequential training, LoRA accumulates intruder dimensions and degrades faster than full fine-tuning; the paper connects this accumulation to forgetting.

5 Intruder Dimensions Cause Forgetting

  • Scaling down top intruder dimensions substantially reduces pre-training forgetting while having little impact on test accuracy.For LLaMA2-7B fine-tuned on MetaMath, λ = 0.3 yielded a 0.1% test-accuracy drop and a 33.3% reduction in forgetting; for RoBERTa-base on QQP, λ = 0.7 preserved test accuracy and reduced forgetting by 33.2%.Source: §5 · p. 8
  • This effect is specific to intruder dimensions: scaling down similarly weighted pre-trained singular vectors instead increases forgetting.Increasing intruder contributions (λ > 1) also increases forgetting.Source: §5 · p. 8
  • Full fine-tuning retains pre-trained structure and performance better than LoRA across six sequential tasks, while LoRA accumulates intruder dimensions.Figure 9 compares performance over the sequence and shows the accumulation of intruder dimensions in LoRA but not full fine-tuning.Source: Figure 9 · p. 9, §5 · p. 9
  • In continual learning, all LoRA ranks degrade faster than full fine-tuning, with low ranks degrading most and each task adding intruder dimensions.The six-task sequence is MNLI, QQP, SST-2, SIQA, WinoGrande, and FEVER; the authors attribute the divergence to accumulated intruder dimensions driving forgetting.Source: §5 · pp. 8–9
  • For equally performing downstream models, the authors propose selecting the one with fewer intruder dimensions as a data-free way to assess fine-tuning overfit.They also recommend avoiding intruder dimensions where possible and note that using separate adapters without combining them may mitigate continual-learning risks.Source: §5 · p. 9

6 Conclusion

The paper concludes that LoRA often introduces high-ranking singular vectors dissimilar to those in pre-trained weights, and that reducing their magnitude can reduce forgetting with minimal test-performance changes. Its appendix discusses possible causes and how learning rate and LoRA parameterization relate to intruder dimensions.

6 Conclusion

  • LoRA often introduces high-ranking singular vectors dissimilar to pre-trained ones; reducing their magnitude yields little test-performance change but substantially reduces pre-training forgetting.During continual training, LoRA accumulates intruder dimensions and forgets more than full fine-tuning.Source: §6 · p. 9

A.1 What do intruder dimensions do?

  • The authors conjecture that intruder dimensions are task-specific: they are learned from smaller fine-tuning datasets and can overpower pre-trained vectors, while full fine-tuning retains pre-trained spectral properties.Scaling an intruder dimension upward (λ > 1) substantially increases pre-training loss with little change in adaptation performance.Source: §A.1 · p. 15

A.2 Adding a random vector

  • A rank-one update λvvT is expected to introduce an intruder dimension when λ exceeds the largest singular value of the pre-trained matrix, because v is likely nearly orthogonal to existing singular vectors.The passage presents this as mathematical intuition for adding new singular vectors.Source: §A.2 · p. 15

A.3 Differences in update rule

  • LoRA projects gradients into a low-rank space, producing conditions similar to the paper’s random-vector toy example.The paper notes that LoRA and full fine-tuning have different update rules even on the same training examples.Source: §A.3 · p. 15

A.4 Impact of learning rate on intruder dimensions.

  • Increasing LoRA learning rate increases intruder-dimension counts, trading faster convergence and potentially better test accuracy against greater drift in language-modeling performance.In the MNLI sweep, models had similar test accuracies after five epochs, while smaller learning rates converged more slowly and sometimes struggled to reach larger-rate performance.Source: §A.4 · p. 15
  • The authors use prior-literature and common-library hyperparameters and externally trained models to argue that their findings reflect common practice rather than their own hyperparameter selection.They note that prior learning rates may have been selected for convergence speed and test accuracy, while coincidentally producing intruder dimensions.Source: §A.4 · p. 15

A.5 Matrix product parameterization of LoRA

  • Freezing a singular-value-1 initialization of A and training only B sharply reduces high-ranking intruder dimensions relative to standard LoRA, implicating the BA product parameterization.The authors suggest matrix multiplication matters because it amplifies spectral differences between A and B.Source: §A.5 · p. 16, Figure 10 · p. 16

B Implementation Details

The appendix describes evaluation datasets and procedures, compute resources, and RoBERTa fine-tuning settings. It also identifies the main accuracy tables and explains why a vector can have low cosine similarity to every vector in a spanning orthonormal basis in high dimensions.

B.1 Evaluation details

  • LLaMA2-7B code and math fine-tuning are evaluated on HumanEval and GSM8K, with forgetting measured on Hellaswag, WinoGrande, and Arc-Challenge.The stated evaluation harnesses are bigcode-eval-harness for HumanEval and lm-eval-harness for GSM8K and the forgetting evaluations.Source: §B.1 · p. 16

B.2 Compute Resources

  • Experiments used a shared 8×A100-SXM4-80GB machine; RoBERTa-base runs and evaluations used one A100, and LLaMA2-7B models were publicly released fine-tuned models due to compute constraints.A RoBERTa-base fine-tuning run took at most six hours; evaluating a LLaMA2-7B model for accuracy and forgetting took about 45 minutes on one GPU.Source: §B.2 · p. 16

B.3 RoBERTa fine-tuning details

  • RoBERTa models used Adam without weight decay, maximum sequence length 512, and all linear layers except embeddings; LoRA ranks ranged from 1 to 64 with α = 2r and α ∗ η fixed at 2.4e-3.Full fine-tuning used learning rate 1e-5, and the models were fine-tuned to equivalent downstream accuracy.Source: §B.3 · p. 17

C Model Accuracies

D Cosine Similarity with Orthogonal Vectors that Span a Space

  • In a high-dimensional space, the minimum cosine similarity between an arbitrary vector and the vectors of a spanning orthonormal basis can be small.The appendix states that for an N-dimensional space, Z = 1/√n.Source: D Cosine Similarity with Orthogonal… · p. 17

E Derivation of LoRA Adapter’s Gradients

The derivation shows that LoRA and full fine-tuning follow different update rules despite identical initial layer outputs on the first training step. LoRA’s update is factorized through its adapter matrices.

E Derivation of LoRA Adapter’s Gradients

F.1 For RoBERTa

Across the reported RoBERTa and LLaMA2-7B experiments, scaling down intruder dimensions reduces forgetting with comparatively small changes to downstream accuracy. Intruder counts also correlate with forgetting, but not significantly with test accuracy in the reported RoBERTa and LLaMA2-7B analyses.

F.1 For RoBERTa

  • ρ = 0.971 (p-value ≪0.001) for RoBERTa intruder counts versus forgetting, while their correlation with test accuracy was not significant.For RoBERTa, intruder count versus test accuracy had ρ = −0.3381 and p-value = 0.218.Source: §F.1 · p. 18

F.2 LLaMA2-7B

  • ρ = 0.59 (p-value = 0.0006) for LLaMA2-7B intruder counts versus forgetting across training epochs; the correlation with test accuracy was not significant.For LLaMA2-7B, intruder count versus test accuracy had ρ = −0.3178 and p-value = 0.0869.Source: §F.2 · p. 19

G.1.1 RoBERTa

  • 0.0% drop in adaptation performance and -33.2% drop in forgetting were reported for QQP with LoRA r=8 when scaling intruders by λ=0.7.Across three datasets and LoRA ranks r=1 and r=8, scaling down intruders reduced forgetting while producing a much smaller test-accuracy drop.Source: §G.1.1 · p. 19, Table 3 · p. 20

G.1.2 LLaMA2-7B

  • -25.2% forgetting and +1.8% test accuracy were reported for LLaMA2-7B fine-tuned on MetaMath with r=256 and λ=0.5.The reported pattern of less forgetting with similar test accuracy also holds for LLaMA2-7B.Source: §G.1.2 · p. 19, Table 4 · p. 20

G.2 Intruder Dimensions Cause Worse OOD Performance

Scaling intruder dimensions down reduces forgetting with little adaptation-performance loss, while scaling them up increases forgetting; scaling neighboring pretrained vectors has different effects. The effective rank of LoRA updates also depends on how α is set relative to rank.

G.2 Intruder Dimensions Cause Worse OOD Performance

  • Scaling neighboring pretrained singular vectors down sharply increased forgetting and reduced adaptation performance more than scaling intruders.Scaling those neighboring vectors up also produced drops in both forgetting and adaptation performance.Source: §G.2 · pp. 20–21
  • Across six RoBERTa models, scaling intruders down reduced forgetting with negligible test-accuracy loss, while scaling them up increased forgetting.For λ=1.5, adaptation performance remained relatively flat despite a large increase in forgetting.Source: §G.2 · p. 20, Figure 12 · p. 21
  • Removing intruders with λ=0 does not restore the pretrained model because some matrices have no intruders, others may have multiple, and the vectors are not perfectly orthogonal.SVD orthogonality constraints can slightly change all singular vectors when an intruder is removed.Source: §G.2 · pp. 20–21

H The Effective Rank of the Update Matrix Depends on Alpha

  • With α=2r, LoRA ranks 64 and 768 had higher effective rank than ranks 8 and 16; at α=8, rank 768 stayed below 50, while full fine-tuning stayed above 400.The authors report that holding α fixed as rank increases can result in high-rank LoRA converging to low-rank solutions.Source: H The Effective Rank of the Update… · p. 22, Figure 14 · p. 23

I Impact of Matrix Percentage on Number of Intruder Dimensions

LoRA generally has more intruder dimensions than full fine-tuning across the examined fractions of fine-tuned singular vectors. The pattern has exceptions for rank-1 LoRA and code fine-tuning.

I Impact of Matrix Percentage on Number of Intruder Dimensions

  • For some datasets, full fine-tuning exceeds LoRA r=1 in the last 20% of singular vectors; for code fine-tuning, their intruder-dimension distributions are similar.The authors suggest limited expressivity for rank-1 updates and a large language-to-code domain shift as possible explanations, respectively.Source: I Impact of Matrix Percentage on… · p. 23
  • LoRA generally has more intruder dimensions than full fine-tuning regardless of the fraction of fine-tuned singular vectors examined.The comparison fixes ϵ at 0.5 while varying k, the proportion of examined singular vectors.Source: I Impact of Matrix Percentage on… · p. 23, Figure 15 · p. 24

J Impact of Dataset Size On Intruder Dimensions

Across continual learning, LoRA accumulates intruder dimensions and lower-rank models forget more; results also show dataset size and scaling choices affect intruder dimensions and forgetting. The appendix examines checkpoint coverage and random-seed sensitivity.

J Impact of Dataset Size On Intruder Dimensions

K LLaMA/LLaMA-2 Instruction Tuned Models

L.1 Performance during continual learning

  • At matched accuracy, lower-rank LoRA forgets more previously learned tasks than higher-rank LoRA and full fine-tuning during six-task continual learning.The comparison measures performance across all six tasks as training proceeds.Source: §L.1 · p. 25, Figure 17 · p. 26

L.2 Similarity matrices during continual learning

  • Across sequential tasks, LoRA accumulates intruder dimensions while full fine-tuning retains pretrained structure, a pattern the authors suggest explains faster degradation.Source: §L.2 · p. 25, Figure 18 · p. 26

M Case Study: Setting Alpha=8 instead of Alpha=2r

N Impact of Random Seeds

  • Across five RoBERTa-base MNLI runs, random initialization has a negligible role in the number of observed intruder dimensions.Source: N Impact of Random Seeds · p. 27

O LoRA Variants

The examined LoRA variants also exhibit intruder dimensions, with differences across variants and ranks. The authors connect these patterns to variant design choices such as modifying singular vectors or learning rates.

O LoRA Variants

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