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Most important LLM weights can't be fine-tuned alone

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2026-07-10 02:08 UTC pith:F7RFWX6U

load-bearing objection Clean experimental design with a real gap between claims and controls the 1 major comments →

arxiv 2607.08733 v1 pith:F7RFWX6U submitted 2026-07-09 cs.LG

Super Weights in LLMs and the Failure of Selective Training

Shreyas Subramanian , Adewale Akinfaderin , Akarsha Sehwag This is my paper
classification cs.LG
keywords Super Weightsparameter-efficient fine-tuningLoRAsparse trainingintrinsic dimensionalitymodel pruningOLMogradient amplification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper investigates a natural hypothesis: if certain individual parameters in a language model are so important that removing them destroys performance, then selectively training those parameters should be an efficient fine-tuning strategy. The authors show the opposite. Training only Super Weights — the highest-impact individual parameters identified by their role in producing activation spikes — collapses model accuracy to random-guessing levels, regardless of whether 100 or 36,864 parameters are included. This failure is specific to Super Weight coordinates: training an equal number of randomly chosen positions in the same layers improves over the pretrained baseline, and applying a low-rank update to those same layers also succeeds. Meanwhile, vanilla LoRA succeeds by updating every position in attention weight matrices through rank-8 decomposition, using only 0.16% of parameters, and constraining LoRA updates at Super Weight-corresponding positions has no measurable effect across a 10-seed ablation. The paper establishes that parameter importance and parameter trainability in isolation are fundamentally different properties, and that effective fine-tuning requires coordinated, structured updates across entire layers rather than targeting individually important weights.

Core claim

The central discovery is a dissociation: the parameters most critical for a model's function (as measured by pruning sensitivity) are precisely the ones that cannot be trained in isolation. This is not a general failure of sparse training — random positions at matched sparsity in the same layers succeed — but a specific failure of targeting Super Weight coordinates, attributable to gradient amplification at outlier positions and the destruction of relational structure that those weights depend on. The finding inverts the natural intuition that important parameters should be the best targets for efficient fine-tuning, and shows that LoRA succeeds not because it is sparse but because its low-r

What carries the argument

Super Weights

Load-bearing premise

The claim that the failure is specific to Super Weight coordinates (rather than to sparse training in general) rests primarily on a single random-position control experiment using one seed at one parameter budget on one model scale, which the authors themselves flag as a limitation.

What would settle it

If random positions at matched sparsity also collapse to random guessing under different seeds or parameter budgets, the specificity claim weakens — the failure would then be attributable to sparse training itself rather than to Super Weight coordinates.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Where Pith is reading between the lines

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

  • If the importance/trainability dissociation holds more broadly, methods that use saliency or magnitude to select fine-tuning targets may be systematically choosing the worst possible parameters to train.
  • The gradient amplification mechanism suggests a general principle: parameters in high-curvature regions of the loss landscape are both more important for model function and harder to train, creating an inverse relationship between pruning criticality and trainability.
  • The result implies that the intrinsic dimensionality framework for fine-tuning needs refinement: not all low-dimensional subspaces are equal, and coordinate-aligned subspaces are qualitatively inferior to low-rank subspaces even at comparable dimensionality.
  • The architecture-specific nature of Super Weight identification suggests that the interaction between parameter importance and trainability may depend on pretraining dynamics in ways that differ across model families.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. This paper investigates whether Super Weights—individual parameters whose removal catastrophically degrades LLM performance—can be fine-tuned in isolation. The authors find that training Super Weights and their local neighborhoods (100–36,864 parameters) collapses accuracy to random-guessing levels on OLMo-1B and OLMo-7B, while full-layer low-rank updates (LoRA) succeed. A random-position control on the same down_proj layers improves over baseline, which the authors use to argue the failure is specific to Super Weight coordinates rather than to sparsity or module choice. The paper also shows that constraining LoRA updates at positions corresponding to Super Weight coordinates has no measurable effect across a 10-seed ablation. Theoretical analysis via intrinsic dimensionality and optimizer dynamics is provided in Appendices H–I.

Significance. The paper addresses a well-posed and timely question: whether parameter importance (as defined by pruning sensitivity) implies parameter trainability in isolation. The experimental design is a strength: six experiments with appropriate controls, a 10-seed ablation for the key LoRA comparison, and two model scales. The random-position control isolating the failure to specific coordinates is a strong design choice. The theoretical framework (Proposition 1, Theorem 1) provides falsifiable predictions mapped to experimental results. The finding that LoRA's full-layer coordination drives success while isolated training of important weights fails is a useful contribution to the PEFT literature. However, the central specificity claim has a confound that needs to be addressed before the contribution is fully established.

major comments (1)
  1. The paper's central specificity claim is: 'The failure is specific to Super Weight coordinates' (Abstract, §4.8, Table 6). However, the training experiments use 'top-k Super Weights by magnitude' (Table 2 caption), meaning the k highest-magnitude weights in down_proj. Only 2–9 of these are confirmed activation-spike Super Weights (Table 1); at k=8,192, the vast majority of trained positions are simply high-magnitude non-Super-Weight positions. Yet all k values collapse to chance. The random-position control (k=4,096, seed 42) trains random down_proj positions, which are overwhelmingly low-magnitude. The paper therefore cannot distinguish between two hypotheses: (H1) the failure is specific to Super Weight coordinates, versus (H2) the failure is specific to high-magnitude coordinates. The paper's own gradient amplification theory (Appendix H, Eq. 5) predicts H2: large weights create a '→'
minor comments (5)
  1. Table 2 caption: 'top-k Super Weights by magnitude' is ambiguous about whether these are confirmed activation-spike Super Weights or simply high-magnitude positions. Clarify which positions are being trained.
  2. §3.1 states 'We identify Super Weights by magnitude ranking,' but Appendix B shows magnitude-based identification does not generalize to all architectures. Clarify the identification method used for training experiments vs. pruning replication.
  3. Table 6 labels the random-position control with a single seed (seed 42, OLMo-1B only). This should be noted in the table for clarity.
  4. Appendix I, Experiment 2: the mean Hessian ratio (2.48) falls short of the predicted M² scaling. The paper acknowledges finite-difference noise, but the discrepancy is large enough to warrant discussion of whether the theory's assumptions hold in practice.
  5. Figure 4: the y-axis label 'Ratio (log scale)' is unclear; specify what ratio is being plotted (e.g., 'Super Weight update / typical position update').

Circularity Check

0 steps flagged

No significant circularity: theoretical derivations are parameter-free from standard results, and experimental claims rest on independent measurements rather than self-definitional chains.

full rationale

The paper's derivation chain is largely self-contained. Proposition 1 (intrinsic dimensionality) is a parameter-free derivation from standard random matrix theory (Haar measure on the Grassmannian), yielding k/N subspace coverage without fitting any parameter to the data it predicts. Theorem 1 (implicit suppression, 1/M scaling) is derived from first principles of gradient and Hessian computation under stated assumptions, not from a self-cited empirical fit. The experimental claims rest on independent measurements: pruning replication uses coordinates from Yu et al. (2025) (external authors), and the random-position control provides an external comparison group. The gradient amplification explanation (§H.1) is framed as a mechanistic hypothesis, not a fitted prediction. The only minor self-referential element is that the paper's own empirical weight-change measurements (Table 10) are used to validate Proposition 4's prediction about relative perturbation scaling, but this is standard empirical validation of a theoretical prediction, not circular derivation—the prediction (O(1/|W_ij|) scaling) is derived independently before the measurement is compared to it. No step in the paper's argument reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 4 axioms · 0 invented entities

The paper introduces no new entities or postulated objects. All free parameters are standard hyperparameters (LoRA rank, learning rate, etc.) rather than theory-derived constants. The axioms are drawn from prior literature (Aghajanyan et al., Yu et al., An et al.) with one ad-hoc modeling assumption (diagonal Hessian update rule) that the paper itself flags as approximate. The theoretical framework recombines existing results (intrinsic dimensionality, activation outliers, nuclear norm regularization) rather than postulating new mechanisms.

free parameters (6)
  • LoRA rank r = 8
    Chosen hyperparameter for all LoRA experiments; standard value but not derived from theory.
  • LoRA alpha = 16
    Scaling factor for LoRA; standard choice, not derived.
  • Learning rate = 1e-4
    AdamW learning rate for all training experiments; not tuned per-method.
  • Neighborhood radius = 1
    Fixed radius for neighborhood training; defines 3x3 patches. Not justified against alternatives.
  • Scale factor s values = {0.0, 0.1, 0.2, 0.5, 0.8, 1.0}
    Discrete set chosen for LoRA-dproj-SW-freeze sweep; not derived from theory.
  • Top-k for direct training = {100, 1000, 4096, 8192}
    Chosen sparsity levels; not justified by intrinsic dimensionality theory.
axioms (4)
  • domain assumption Fine-tuning operates in a low-dimensional subspace S of dimension d << N (Aghajanyan et al., 2021)
    Invoked in §H.1 to derive Proposition 1; the entire intrinsic dimensionality framework depends on this empirical result from prior work.
  • domain assumption Super Weights are identifiable via activation spikes in down_proj layers (Yu et al., 2025)
    All training experiments use coordinates identified by this method; the paper validates criticality via pruning replication but does not independently derive the identification method.
  • domain assumption Weight outliers and activation outliers co-occur with 100% consistency (An et al., 2025)
    Invoked in §I to establish M >> 1 at Super Weight positions; the curvature suppression theorem depends on this coupling.
  • ad hoc to paper Diagonal second-order update rule approximates optimizer dynamics
    Theorem 1 assumes Δw = -ηg/H; the paper acknowledges Adam does not fully replicate this (§I.1) but uses the diagonal approximation as the formal framework.

pith-pipeline@v1.1.0-glm · 25783 in / 3260 out tokens · 475958 ms · 2026-07-10T02:08:21.080541+00:00 · methodology

0 comments
read the original abstract

Recent work identified Super Weights, individual parameters whose removal degrades model performance by orders of magnitude. We show that this degradation due to pruning Super Weights does not universally apply to all LLMs. Furthermore, if these parameters are so important, Super Weight-aware training should be effective. We show the opposite. Training Super Weights in isolation (100 to 8,192 parameters) drops accuracy to random-guessing levels on both OLMo-1B and OLMo-7B, and expanding to local neighborhoods of up to 36K parameters provides no improvement. The failure is specific to Super Weight coordinates: training an equal number of randomly chosen positions in the same down_proj layers instead improves over the baseline, so the collapse comes from targeting Super Weights, not from sparsity itself. Vanilla LoRA, updating every position in attention weight matrices through low-rank structure, succeeds with only 0.16% of parameters, and applying the same low-rank update to down_proj succeeds as well. A 10-seed ablation confirms that constraining LoRA updates at positions corresponding to Super Weight coordinates yields statistically indistinguishable results. These findings establish that parameter importance does not imply parameter trainability in isolation, and that effective fine-tuning relies on structured decompositions over entire layers rather than targeting individually important weights.

Figures

Figures reproduced from arXiv: 2607.08733 by Adewale Akinfaderin, Akarsha Sehwag, Shreyas Subramanian.

Figure 1
Figure 1. Figure 1: The three LoRA variants. All operate on attention projections ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Expanding from isolated Super Weights to neighborhoods of 100, 1,000, or 36,864 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Per-seed accuracy for vanilla LoRA and LoRA-dproj-SW-freeze (OLMo-1B, seeds [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Theorem 1 mechanism: Under second-order optimization, the 1 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiment 1: Activation amplification M = |xc|/µx measured across selected transformer layers. Mean (solid bars) and maximum (transparent bars) values show signif￾icant outliers in mid-to-late layers (layers 2, 3, 21, 23), validating the assumption M ≫ 1 underlying Theorem 1. Typical layers exhibit M ∈ [5, 10]. Interpretation. While the mean ratio (2.48) falls short of the predicted M2 ≈ 100 (given M ≈ 5–… view at source ↗
Figure 6
Figure 6. Figure 6: Experiment 2: Distribution of Hessian ratios [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Experiment 3: Loss sensitivity to parameter perturbations, showing Super Weight [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗

discussion (0)

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Reference graph

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