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Procedural Knowledge Is Not Low-Rank: Why LoRA Fails to Internalize Multi-Step Procedures

This paper demonstrates that LoRA fails to effectively internalize complex multi-step procedural knowledge compared to full fine-tuning because the necessary weight updates exhibit a high effective rank that far exceeds LoRA's low-rank capacity, leading to significant performance gaps even at high ranks.

Original authors: Simon Dennis, Kevin Shabahang, Hao Guo, Rivaan Patil

Published 2026-07-27

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Simon Dennis, Kevin Shabahang, Hao Guo, Rivaan Patil

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a super-smart robot how to do a complex job, like booking a flight or fixing a broken computer. In the world of artificial intelligence, there's a popular trick called "fine-tuning." Think of a giant AI model as a massive library containing almost every book ever written. To teach it a specific job, you usually have to rewrite the whole library, which is expensive and slow. To save time, scientists invented a shortcut called LoRA (Low-Rank Adaptation). Instead of rewriting the whole library, LoRA tries to teach the robot by adding a tiny, thin notebook of sticky notes to the shelves. The big idea was that these sticky notes could capture everything the robot needed to know about a new job without touching the heavy books underneath. It's like thinking you can teach someone to drive a race car just by giving them a small cheat sheet, rather than letting them practice on the track for hours.

But here is the catch: this paper asks if that cheat sheet is actually enough when the job isn't just about remembering facts or writing in a cool style, but about following a strict, multi-step procedure. Procedural knowledge is like a recipe with lots of "if this happens, then do that" steps, where you have to keep track of your progress, handle mistakes, and make sure you don't skip a step. The researchers wanted to know: Can a tiny notebook (LoRA) teach a robot to follow a complex, branching recipe, or does it need to rewrite the whole library (Full Fine-Tuning) to get it right?

The Great Cheat Sheet Failure

The researchers set up a series of tests to see if the "sticky note" method could handle the messy reality of following instructions. They didn't just ask the robot to write a poem; they made it act as a travel agent, a Zoom tech support specialist, and an insurance claims adjuster. These aren't simple tasks; they are like navigating a giant, twisting maze where every turn depends on what the customer said five minutes ago. If the customer gets angry, you have to change your path. If they forget a detail, you have to ask for it again. The goal was to see if the robot could successfully guide the conversation from the start to a happy ending without getting lost.

The results were a bit of a shock. When the researchers used the full library rewrite method (Full Fine-Tuning), the robots became excellent agents. They followed the rules, remembered the details, and solved the problems, scoring a solid 4.11 out of 5 on their travel booking test. But when they tried the "cheat sheet" method (LoRA), even with the biggest, most expensive cheat sheets they could make, the robots completely failed the procedure.

Here is the weird part: the LoRA robots didn't just get slightly worse; they got stuck in a weird loop. They could still chat! They could keep the conversation going for 14 turns without crashing. They sounded polite and natural. But they were failing the actual job. On the travel test, the LoRA robots scored between 2.10 and 2.54—barely half as good as the full-trained robots. They would skip important steps, forget to ask for the destination, or give up too early. It was like a tour guide who knows how to say "Hello" and "Goodbye" perfectly but leads the tourists into a wall because they forgot the map.

The Paradox of Bigger Notebooks

You might think, "Okay, maybe the cheat sheet was just too small. Let's make it bigger!" The researchers tried this. They increased the size of the LoRA notebook from a tiny rank 16 up to a huge rank 128. This meant they were changing more of the robot's brain, using up to 3.4% of its total parameters. You would expect that a bigger notebook would help, right?

Surprisingly, it made things worse. As the notebook got bigger, the robots actually got worse at following the procedure. The best score was at the smallest size, and by the time they hit the biggest size, the task success score dropped to 2.10. It turns out that making the cheat sheet bigger didn't help the robot learn the logic of the maze; it just helped the robot memorize the words of the maze better. The robot became really good at sounding like it knew what to do, but it was still lost. The researchers found that the robots were "overfitting," which is a fancy way of saying they were memorizing the training examples like a parrot without understanding the rules.

Why the Cheat Sheet Wasn't Enough

So, why did the tiny notebook fail so miserably? The authors dug deep into the robot's brain to find the answer. They used a mathematical tool called SVD (Singular Value Decomposition) to look at exactly how much the robot's brain changed when it learned the procedure.

Imagine the robot's brain is a giant grid of numbers. When the robot learns a new skill, the numbers in this grid shift. The "Low-Rank" idea assumes that these shifts are simple, like drawing a straight line or a flat sheet on the grid. But the researchers found that for procedural tasks, the shifts are incredibly complex and messy. They are like a 3D sculpture made of thousands of tiny, twisting wires.

The math showed that to capture the changes needed for these tasks, the robot needed to use a "rank" of about 761 to 1,026. But the LoRA method was only allowed to use a rank of 128 at its maximum. This means the LoRA notebook was trying to describe a complex 3D sculpture using only a flat, 2D drawing. It simply didn't have enough dimensions to hold the information. The researchers calculated that even the biggest LoRA notebook could only capture about 43% to 51% of the necessary changes. The rest of the "knowledge" was just left out, like trying to fit a whole ocean into a teacup.

The Verdict: You Can't Cheat the Maze

The study concludes that for tasks requiring a robot to follow a strict, multi-step procedure with lots of "if-then" decisions, the shortcut method (LoRA) just doesn't work. It's not that the robot wasn't trained hard enough; it's that the method itself is too limited. The knowledge needed to navigate these complex mazes is spread out across the robot's entire brain in a way that a small, low-rank notebook cannot capture.

The researchers are very sure about this because they tested it in three different areas (travel, tech support, and insurance) and with two different sizes of robot brains (3 billion and 8 billion parameters). In every case, the full library rewrite won, and the cheat sheet lost. They also ruled out the idea that the robot just needed more training time, because the LoRA robots actually learned the words faster than the full-trained ones—they just failed to learn the rules.

So, if you want an AI agent that can reliably book your flight, fix your Zoom call, or process your insurance claim without getting confused, you can't just stick a few sticky notes on its brain. You have to let it rewrite the whole library. The efficiency of the shortcut comes at the cost of the robot's ability to actually do the job.

Technical Summary: Procedural Knowledge Is Not Low-Rank

Problem Statement

While parameter-efficient fine-tuning (PEFT) methods like Low-Rank Adaptation (LoRA) have become the standard for adapting Large Language Models (LLMs) due to their memory efficiency and success in instruction following, style transfer, and factual adaptation, their efficacy on procedural knowledge remains unverified. Procedural knowledge is defined here as the ability to follow multi-step procedures involving conditional branching, implicit state tracking across turns, and navigation to specific terminal states (e.g., customer support workflows, booking systems, diagnostic interviews).

The core assumption underpinning LoRA is that task-specific adaptations reside in a low-dimensional subspace of the parameter space. This paper challenges that assumption, positing that procedural knowledge is distributed across the full parameter space in a manner that low-rank approximations cannot capture at practical ranks. If LoRA fails to internalize procedural knowledge, the efficiency gains it offers are unavailable for the emerging class of agentic applications that rely on structured workflows.

Methodology

The study employs a systematic ablation and cross-domain replication approach using synthetic data generated from flowchart-based procedure definitions.

1. Task Definition and Data Generation

Procedures are formalized as directed graphs $F = (N, E, n_0, T)$ representing multi-turn conversations with conditional branching. Three domains of varying complexity were used:

Travel Booking (3B model): 14 nodes, 3 decision hubs, 86 unique acyclic paths.
Zoom Support (8B model): 14 nodes, 3 decision hubs, 60 unique acyclic paths, incorporating domain-specific UI and error codes.
Insurance Claims (8B model): 55 nodes, 6 decision hubs, 2,381 unique acyclic paths, featuring nested loops and cross-phase dependencies.

Synthetic conversations were generated by sampling paths through these flowcharts and using an LLM (Claude Sonnet 4.5) to generate turn-by-turn dialogue, varying scenario variables to ensure diversity. The inference model sees only natural dialogue without explicit procedural annotations.

2. Fine-Tuning Configurations

The study compares Full Parameter Fine-Tuning (Full FT) against LoRA at four ranks ( $r = 16, 32, 64, 128$ ).

Full FT: Updates all parameters. Training utilized AdamW with cosine decay, running for 20 epochs (3B) or 10–20 epochs (8B) until convergence.
LoRA: Targets all linear layers (attention $q, k, v, o$ and MLP gate, up, down). The scaling factor $\alpha = 2r$ . LoRA learning rates were set higher ( $2 \times 10^{-4}$ ) than Full FT ( $2 \times 10^{-5}$ ) to provide LoRA the best chance of performance. Adapters were merged prior to evaluation to ensure identical inference costs.

3. Evaluation Protocol

Performance was evaluated on $n=200$ scenarios per condition using dynamic user simulation and an LLM-as-a-judge (Claude Sonnet 4.5, verified by GPT-4.1). Metrics included:

Task Success: Correct execution of the procedure to a terminal state.
Information Accuracy, Consistency, Graceful Handling, and Naturalness.
Statistical significance was assessed using Wilcoxon signed-rank and Mann–Whitney U tests with Holm–Bonferroni correction.

4. Structural Analysis (SVD)

To explain behavioral failures, the authors performed Singular Value Decomposition (SVD) on the weight differences ( $\Delta W = W_{ft} - W_{base}$ ) across all weight matrices. They measured the "Energy at $r$ " (fraction of squared Frobenius norm captured by the top $r$ singular vectors) to determine the theoretical upper bound of what a rank- $r$ LoRA adapter could capture, regardless of optimization quality.

Key Results

1. Uniform Failure Across Ranks

On the Travel Booking task (3B model), all LoRA configurations failed to match Full FT, regardless of rank.

Task Success: LoRA scores ranged from 2.10 to 2.54, compared to 4.11 for Full FT ( $p < 0.001$ ).
Rank Insensitivity: Increasing the rank from 16 to 128 (an 8x increase in trainable parameters) yielded no improvement. In fact, scores degraded at higher ranks (e.g., Task Success dropped from 2.54 at $r=32$ to 2.10 at $r=128$ ).
Completion vs. Correctness: LoRA models maintained high conversation completion rates (95.5%–99.0%), indicating they did not collapse mid-conversation. However, they failed to follow the correct procedural paths, often skipping decision hubs or reaching terminal states prematurely.

2. Cross-Domain Generalization

The failure generalized to the 8B model across Zoom Support and Insurance Claims.

LoRA underperformed Full FT by an average of 0.8–2.2 points across all criteria at both $r=32$ and $r=128$ .
The performance gap was largest on the most complex procedure (Insurance Claims), where LoRA $r=128$ achieved a Task Success of 2.10 vs. 4.47 for Full FT.
Increasing rank from 32 to 128 provided only marginal improvements, failing to close the gap to Full FT.

3. High-Rank Nature of Weight Updates

SVD analysis revealed that the weight changes required for procedural internalization are genuinely high-rank.

Effective Rank: The mean effective rank of the Full FT update across domains ranged from 761 to 1,026.
Energy Capture: A rank-128 LoRA adapter captures only 43% to 51% of the squared Frobenius norm of the Full FT update.
MLP Bottleneck: The MLP layers (gate, up, down projections) exhibited the highest effective ranks (up to 1,872 in Insurance Claims). Rank 128 covers only 7–13% of the required rank for these layers, capturing only 26–44% of the update energy. Attention K/V projections were better covered due to smaller matrix sizes (GQA), but MLPs remain the primary bottleneck.

4. Training Dynamics and Overfitting

Analysis of training dynamics showed that LoRA does not suffer from underfitting.

Lower Loss, Worse Behavior: At convergence, LoRA ( $r=32$ ) achieved a lower held-out per-token loss (0.81) than Full FT (0.89).
Divergence: Despite lower loss, LoRA performed significantly worse on behavioral criteria. This indicates LoRA fits the surface token distribution (memorizing local patterns) more precisely but fails to acquire the state-to-action mapping required for procedural correctness.
Position-Dependent Overfitting: LoRA's loss on early turns (formulaic) remained stable, but loss on late turns (requiring state tracking) increased after epoch 3, signaling overfitting to surface correlations rather than procedural structure.

Significance and Claims

The paper establishes that procedural knowledge is not low-rank. The findings suggest a fundamental limitation of parameter-efficient methods for agentic applications:

Structural Limitation: The inability of LoRA to capture procedural knowledge is not an optimization failure but a representational one. The required weight updates are distributed across hundreds of dimensions, exceeding the capacity of standard low-rank approximations (even at $r=128$ ).
Efficiency Trade-off: For tasks requiring reliable multi-step procedure following, the efficiency gains of LoRA come at the cost of procedural competence. Full fine-tuning remains necessary to compile such procedures into model weights with near-frontier quality.
Paradoxical Degradation: Increasing LoRA rank within the practical range does not linearly improve performance; instead, it may increase the capacity for surface-pattern overfitting without capturing the deeper procedural structure.

The authors conclude that while LoRA succeeds in instruction following and factual adaptation, it falls well short of Full Fine-Tuning for procedural tasks, a critical distinction for the deployment of LLM-based agents. The paper notes that heterogeneous-rank LoRA (allocating higher ranks specifically to MLP layers) is a plausible avenue for future research but was not tested in this study.