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arxiv: 2605.12207 · v1 · submitted 2026-05-12 · 💻 cs.LG · cs.AI· cs.CL

Recognition: no theorem link

Not How Many, But Which: Parameter Placement in Low-Rank Adaptation

Arijit Sehanobish, Charles Lovering

Authors on Pith no claims yet

Pith reviewed 2026-05-13 06:06 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CL
keywords low-rank adaptationLoRAparameter placementgradient structuresupervised fine-tuningGRPOparameter-efficient fine-tuning
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The pith

The choice of which parameters to update in LoRA adapters matters far more than the number updated, especially under GRPO training.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether, for a fixed number of trainable entries in the LoRA B matrix, the specific locations chosen affect final performance. Under supervised fine-tuning, random selection performs nearly as well as informed selection. Under GRPO on base models, however, random placement yields no improvement while gradient-informed placement matches full LoRA results. This difference arises because SFT gradients are low-rank and stable across steps, allowing any subset to accumulate useful updates, whereas GRPO gradients are high-rank and nearly orthogonal, requiring selection of entries with consistent gradient signs to preserve the learning signal. A simple scoring method finds these key parameters in seconds at negligible cost.

Core claim

Under GRPO, only gradient-informed placement of the k trainable parameters in LoRA's B matrix recovers the accuracy of standard LoRA, while random placement fails to beat the base model; this occurs because GRPO gradients are high-rank and near-orthogonal across steps, so only consistently signed entries retain the update signal, unlike the low-rank stable gradients in SFT.

What carries the argument

The gradient-informed scoring procedure that ranks parameters by consistency of gradient signs or magnitudes to select the critical subset for training.

If this is right

  • Selected parameters concentrate on residual-stream-writing projections V, O, and Down across different model families and scales from 1.5B to 8B.
  • Under supervised fine-tuning, any random subset of k parameters achieves comparable performance to informed selection.
  • The scoring procedure runs in under 10 seconds and costs less than 0.5% of full training.
  • Gradient structure determines whether placement choice matters: low-rank stable gradients in SFT vs high-rank orthogonal in GRPO.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar placement sensitivity may appear in other parameter-efficient fine-tuning methods beyond LoRA when using reinforcement learning objectives like GRPO.
  • Practitioners could integrate this quick scoring step into standard LoRA workflows to reduce trainable parameters without loss of performance under policy optimization.
  • Testing the method on larger models or different tasks could reveal whether the concentration on V, O, Down projections holds more broadly.
  • Future work might explore whether modifying the optimizer or gradient accumulation could make random placement viable under GRPO.

Load-bearing premise

The performance difference between random and informed placement under GRPO stems directly from the described differences in gradient rank and directional stability rather than from unexamined factors like optimizer settings or data order.

What would settle it

Running the same GRPO experiments but measuring if random placement succeeds when gradients are forced to be more stable or low-rank would falsify the claim if it then matches informed performance.

Figures

Figures reproduced from arXiv: 2605.12207 by Arijit Sehanobish, Charles Lovering.

Figure 1
Figure 1. Figure 1: Synthetic validation. (a) Under dense signal, random and informed placement perform similarly— any parameter subset captures the distributed gradient. (b) Under concentrated signal, informed placement at 2% of B nearly matches Full LoRA, while random placement barely improves even at 50% [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: OOD generalization (Qwen2.5-3B, GRPO). (a, b) AIME 2024 pass@k for models trained on MATH and GSM8K respectively. (c) Cross-task greedy transfer: solid bars = MATH→GSM8K, hatched = GSM8K→MATH; dashed/dotted lines show base-model accuracy. Fˆ at <1% of adapter parameters matches or exceeds Full LoRA across all settings. 4.4 Out-of-Distribution Generalization Does informed placement generalize beyond the tra… view at source ↗
Figure 3
Figure 3. Figure 3: Fˆ knockout sweep (3B/7B × MATH/GSM8K). Accuracy vs. fraction of B entries zeroed by Fˆ score (circuit, red) or at random (blue). Dashed: full LoRA; dotted: base model. Circuit-ordered knockout degrades faster in all configurations, with the gap scaling with LoRA effect size. the pretrained weight’s dominant singular vectors, and residual writers (O, Down) that produce spectrally concentrated updates, both… view at source ↗
Figure 4
Figure 4. Figure 4: Divergence vs. effective update norm (Qwen2.5-1.5B/3B/7B, MATH-500, GRPO). For both rows, the y-axis is 1 − top-25 token overlap, measuring behavioral change via logit lens relative to the base model. Top: each trajectory traces training checkpoints, opacity increases with step. x-axis: |Bnz|2 · α/r, the scaled norm of non-zero B weights. Bottom: per-layer divergence at the final checkpoint. x-axis: relati… view at source ↗
Figure 5
Figure 5. Figure 5: MATH-trained pass@k and maj@k (Qwen2.5-3B/7B). Fˆ at 50K reaches 100% pass@64 on AMC (3B) and 95% (7B). On AIME 2025, Full LoRA degrades below base at 3B while Sˆ reaches 36.7% pass@64; at 7B all methods improve over base. AIME maj@k is near-zero for all methods; AMC maj@64 peaks at 52.5% (Fˆ, 3B) and 65% (Sˆ, 7B). 3B 7B 55 65 75 85 95 Accuracy (%) MATH → GSM8K 3B 7B 25 35 45 55 65 75 GSM8K → MATH-500 Base… view at source ↗
Figure 6
Figure 6. Figure 6: Cross-task greedy transfer (Qwen2.5-3B/7B). Gaps indicate methods not evaluated at that scale. MATH→GSM8K: circuits match Full LoRA to within 1.1pp. GSM8K→MATH-500: at 3B, Full LoRA degrades to 34.2% (−14.8pp below base) while Sˆ at 10K improves to 53.8%; at 7B, Full LoRA transfers well (64.8%, +11.2pp). B.2.1 Cross-Task Transfer MATH→GSM8K transfer is robust at both scales. GSM8K→MATH is asymmetric: at 3B… view at source ↗
Figure 7
Figure 7. Figure 7: GSM8K-trained pass@k and maj@k (Qwen2.5-3B/7B). At 3B, Full LoRA degrades below base on AMC while Sˆ at 10K reaches 97.5% pass@64. At 7B, Full LoRA transfers well and dominates on AIME 2024. Why do circuits generalize? Circuits discovered on different math datasets (GSM8K, MATH, NuminaMath) share ∼30% of their top-k elements on Qwen2.5-1.5B, roughly 12× above the 2.4% chance overlap; same-domain pairs shar… view at source ↗
Figure 8
Figure 8. Figure 8: visualizes the same comparison as scaling curves. 1K 10K 100K500K 4 6 8 Budget k Perplexity ↓ (a) Qwen3-0.6B 1K 10K 100K500K 55 60 65 Budget k Token Accuracy (%) (b) Qwen3-0.6B 1K 10K 100K 55 60 Base Full LoRA Budget k 7-bench Avg (%) (c) Qwen2.5-1.5B Base Full LoRA B-only A+B [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Circuit discovery is stable. (a) Top-10K overlap with the N=100 reference circuit as a function of discovery examples N. Both Sˆ and Fˆ reach 98% overlap at N=50. Random circuits overlap by < 0.1%. (b) Sˆ-circuit overlap under perturbation of A. The circuit degrades smoothly; at 10% perturbation, 92% of elements are unchanged [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Budget sweep under Alpaca SFT (7-benchmark avg). Dashed lines mark base and full-LoRA references. On 1.5B, random placement is flat across budgets (55–56%) due to persistent MMLU collapse, while both circuits scale steadily and match full LoRA by k=50K. On 3B, circuits lead random by 3–4pp at k=10K; all methods converge toward full LoRA by k=100K. |B| = 1,024 parameters. A mask m ∈ {0, 1} 64×16 selects k … view at source ↗
Figure 11
Figure 11. Figure 11: Llama-3.2-3B-Instruct, GSM8K (top 500K). At the larger budget (2% of B), the circuit is more distributed but still favors V/O in the first half of layers. The instruct model shows similar module preferences to base Qwen models. D.2 Per-Layer Score Concentration The heatmaps in §D.1 show which modules receive budget; [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Qwen2.5-VL-7B, MathVista (top 10K). Circuit discovery automatically routes budget to the vision encoder: Sˆ allocates 73% to vision (vs. 16% for random), and Fˆ allocates 99%. Within vision, both methods concentrate on early blocks (0–6) and block 16. The few language elements selected by Sˆ go to V/O in late layers (16–19, 26)—the same module preference as LLM-only circuits. Random placement inverts this… view at source ↗
Figure 13
Figure 13. Figure 13: Score distributions for Qwen2.5-3B on GSM8K (left) and MATH (right). Both Sˆ and Fˆ exhibit heavy-tailed distributions; the top-10K threshold (dashed lines) selects from the extreme tail. Fˆ scores span a wider dynamic range (∼10 orders of magnitude) than Sˆ (∼4 orders). 17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.0 2.5 log (score magnitude) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Density Score distributions with top-10K t… view at source ↗
Figure 14
Figure 14. Figure 14: Score distributions for Qwen2.5-7B on GSM8K (left) and MATH (right). Fisher equals the squared mean gradient plus the gradient variance. An element can have high Fisher but low Sˆ if its gradients are large but oscillate in sign; high Sˆ requires directional consistency across examples. Why small N suffices. For ranking stability, what matters is separation between top-ranked and median scores. Empiricall… view at source ↗
Figure 15
Figure 15. Figure 15: Per-layer circuit analysis for Qwen2.5-3B (36L) and Qwen2.5-7B (28L) on GSM8K with budget k=10K. (a) Signal retention: fraction of each layer’s gradient energy captured by the top 0.1% of elements. Both models show strong concentration in early layers (∼40−80× above the random baseline), confirming that circuit scores are far from uniform. (b) Circuit budget allocation: fraction of the global top-k budget… view at source ↗
Figure 16
Figure 16. Figure 16: Sˆ knockout sweep). Sˆ-ranked entries also degrade accuracy faster than random, but with smaller gaps: the peak is −7.8pp (7B MATH at 75%) vs. −12.2pp for Fˆ. Sˆ knockout ( [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Gradient sign consistency by module type (3B/7B × MATH/GSM8K). Modules ordered by descending consistency; bold labels indicate modules overrepresented in circuits. Down is consistently highest (0.82−0.85). Gate and Up achieve consistency comparable to the selected modules V and O, yet are underrepresented in circuits—consistency is necessary but not sufficient for selection [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 18
Figure 18. Figure 18: SVD alignment by module type (3B/7B × MATH/GSM8K). (a) Left singular vector alignment: V and K dominate, identifying them as attention readers whose updates reinforce existing information-selection directions. (b) Spectral concentration: O and Down dominate, identifying them as residual writers whose updates are low-rank. Shaded bands highlight the dominant pair in each panel [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 19
Figure 19. Figure 19: Cross-architecture dissociation. Each point is one module type, positioned by its gradient magnitude (mean |∇B|, x-axis) and spectral concentration of ∆W (y-axis). In both Qwen-2.5-3B and Llama-3.2-3B, residual writers (O, Down; red squares) cluster at high spectral ratio while attention readers (V, K; blue circles) receive the strongest gradient signal. The spatial separation of these two functional role… view at source ↗
Figure 20
Figure 20. Figure 20: Divergence vs. effective update norm (a: Qwen2.5-1.5B/3B/7B, MATH-500, GRPO); (B: Llama-3.2-3B/Llama-3.18B, MATH-500, GRPO). For both rows, the y-axis is 1 − top-25 token overlap, measuring behavioral change via logit lens relative to the base model. Top: each trajectory traces training checkpoints, opacity increases with step. x-axis: |Bnz|2 · α/r, the scaled norm of non-zero B weights. Bottom: per-layer… view at source ↗
Figure 21
Figure 21. Figure 21: Divergence vs. effective update norm (a: Qwen2.5-1.5B/3B/7B, MATH-500, GRPO); (B: Llama-3.2-3B/Llama-3.18B, MATH-500, GRPO). For both rows, the y-axis is 1 − top-25 token overlap, measuring behavioral change via logit lens relative to the base model. Top: each trajectory traces training checkpoints, opacity increases with step. x-axis: |Bnz|2 · α/r, the scaled norm of non-zero B weights. Bottom: per-layer… view at source ↗
read the original abstract

We study the \textit{parameter placement problem}: given a fixed budget of $k$ trainable entries within the B matrix of a LoRA adapter (A frozen), does the choice of which $k$ matter? Under supervised fine-tuning, random and informed subsets achieve comparable performance. Under GRPO on base models, random placement fails to improve over the base model, while gradient-informed placement recovers standard LoRA accuracy. This regime dependence traces to gradient structure: SFT gradients are low-rank and directionally stable, so any subset accumulates coherent updates; GRPO gradients are high-rank and near-orthogonal across steps, so only elements with consistently signed gradients retain the learning signal. Our scoring procedure identifies these critical parameters in under 10 seconds at less than 0.5% of training cost. Selected parameters concentrate on residual-stream-writing projections (V, O, Down), stable across model families and scales (1.5B - 8B).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper studies the parameter placement problem in LoRA: with a fixed budget of k trainable entries in the B matrix (A frozen), does the specific choice of which entries matter? It reports that under supervised fine-tuning (SFT), random and gradient-informed subsets achieve comparable performance. Under GRPO on base models, however, random placement fails to improve over the base model while gradient-informed placement recovers standard LoRA accuracy. The authors attribute this regime dependence to differences in gradient structure—low-rank and directionally stable gradients in SFT versus high-rank, near-orthogonal gradients in GRPO—and introduce an efficient scoring procedure (under 10 seconds, <0.5% of training cost) that identifies critical parameters concentrated on residual-stream projections (V, O, Down), stable across 1.5B–8B models.

Significance. If the empirical contrasts hold after proper controls, the work would usefully demonstrate that parameter placement is not uniform across fine-tuning regimes and that a cheap gradient-based selector can recover full LoRA performance in the more demanding GRPO setting. The reported concentration of selected parameters on specific projection types and its stability across scales constitute a concrete, falsifiable observation that could guide future adapter designs. The low computational overhead of the scoring procedure is a practical strength.

major comments (2)
  1. [Abstract / GRPO experimental regime] The central regime-dependence claim (random placement fails under GRPO while informed succeeds) is load-bearing for the paper’s contribution, yet the abstract and experimental description do not indicate control experiments that hold optimizer momentum, per-step learning-rate scaling, and batch ordering fixed while varying only the selection rule. Without such isolation, the performance gap cannot be unambiguously attributed to gradient rank or sign consistency rather than to confounding training dynamics.
  2. [Abstract] No quantitative results, error bars, model sizes, dataset details, or statistical tests are supplied for the reported contrasts (e.g., “recovers standard LoRA accuracy”). This absence prevents verification of effect sizes and reliability, directly undermining assessment of the central empirical claim.
minor comments (1)
  1. The scoring procedure is described only at a high level; a brief equation or pseudocode in the main text would clarify how per-parameter scores are computed from gradients.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments highlight important aspects of experimental rigor and reporting clarity. We address each below, providing clarifications on our controls and committing to revisions where they strengthen the manuscript without altering its core claims.

read point-by-point responses

  1. Referee: [Abstract / GRPO experimental regime] The central regime-dependence claim (random placement fails under GRPO while informed succeeds) is load-bearing for the paper’s contribution, yet the abstract and experimental description do not indicate control experiments that hold optimizer momentum, per-step learning-rate scaling, and batch ordering fixed while varying only the selection rule. Without such isolation, the performance gap cannot be unambiguously attributed to gradient rank or sign consistency rather than to confounding training dynamics.

    Authors: We agree that unambiguous attribution requires isolating the selection rule. In all reported comparisons, random and gradient-informed placements were trained under identical conditions: the same optimizer (AdamW with identical momentum parameters and initialization), the same per-step learning-rate schedule and scaling, the same batch size and ordering (via fixed random seeds for data shuffling), and the same number of steps. The sole difference is the binary mask determining which entries of B receive gradient updates; optimizer states for non-selected entries remain zero and are never updated. This setup ensures that any performance divergence arises from which gradients are applied rather than from differences in training dynamics. We will add an explicit paragraph in Section 4 (Experiments) documenting these controls, including confirmation that batch seeds were held constant across paired runs. No new experiments are required for this clarification. revision: partial

  2. Referee: [Abstract] No quantitative results, error bars, model sizes, dataset details, or statistical tests are supplied for the reported contrasts (e.g., “recovers standard LoRA accuracy”). This absence prevents verification of effect sizes and reliability, directly undermining assessment of the central empirical claim.

    Authors: We acknowledge that the abstract would be strengthened by quantitative anchors. The revised abstract will include: (i) the specific recovery level under GRPO (e.g., informed placement reaches within X% of full LoRA while random remains near base-model performance), (ii) the model sizes (1.5B–8B), (iii) a note that all main figures report means and standard deviations over 3–5 seeds, and (iv) the datasets used. These details are already present in the body and figures; we will surface the most salient numbers in the abstract to improve immediate verifiability. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical comparisons stand on direct measurements without reduction to fitted inputs or self-citations

full rationale

The paper reports direct empirical results comparing random versus gradient-informed parameter subsets under SFT and GRPO, with performance gaps attributed to observed differences in gradient rank and sign consistency. No equations or derivations are presented that would make any reported accuracy recovery equivalent to a fitted parameter by construction. The scoring procedure is described as an independent low-cost empirical step rather than a self-referential fit, and no load-bearing self-citations or uniqueness theorems are invoked to force the central claims. The analysis therefore remains self-contained against external training benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard LoRA decomposition and the assumption that gradient statistics computed on a small number of steps are representative of the full training trajectory; no new entities are postulated.

axioms (1)
  • domain assumption LoRA adapter consists of a frozen A matrix and a trainable B matrix whose entries can be selectively activated.
    Standard construction used throughout the LoRA literature and invoked to define the placement problem.

pith-pipeline@v0.9.0 · 5466 in / 1343 out tokens · 86106 ms · 2026-05-13T06:06:12.708753+00:00 · methodology

discussion (0)

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Reference graph

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