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CrispEdit: Low-Curvature Projections for Scalable Non-Destructive LLM Editing

Zarif Ikram, Arad Firouzkouhi, Stephen Tu, Mahdi Soltanolkotabi, Paria Rashidinejad

arXiv:2602.15823v2cs.LGcs.AI

TL;DR

LLM editing must change targeted behavior without quietly damaging general capabilities. CrispEdit enforces this preservation goal through low-curvature second-order projections, using Bregman-Gauss-Newton curvature and scalable K-FAC-based computation. Across evaluated settings, it reports strong edit success with substantially reduced capability degradation and modest compute and storage requirements.

  • Problem

    Existing editing methods can appear successful while degrading broader capabilities, motivating direct capability preservation without full retraining.

  • Method

    CrispEdit formulates editing as constrained optimization and projects updates onto low-curvature capability-loss directions using Bregman-Gauss-Newton curvature, K-FAC, and a matrix-free projector.

  • Results

    CrispEdit consistently improves the edit–capability trade-off across evaluated settings, achieving strong edit success while substantially reducing capability degradation.

  • Takeaways & Limitations

    The approach makes constraint-aware second-order editing feasible for billion-parameter LLMs and supports reusable curvature statistics for batch and sequential edits.

  • Takeaways & Limitations

    Prior soft-penalty optimization can be prohibitively expensive when the capability reference set is much larger than the edit set, motivating CrispEdit’s constrained approximation.

Abstract

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A central challenge in large language model (LLM) editing is capability preservation: methods that successfully change targeted behavior can quietly game the editing proxy and corrupt general capabilities, producing degenerate behaviors reminiscent of proxy/reward hacking. We present CrispEdit, a scalable and principled second-order editing algorithm that treats capability preservation as an explicit constraint, unifying and generalizing several existing editing approaches. CrispEdit formulates editing as constrained optimization and enforces the constraint by projecting edit updates onto the low-curvature subspace of the capability-loss landscape. At the crux of CrispEdit is expressing capability constraint via Bregman divergence, whose quadratic form yields the Gauss-Newton Hessian exactly and even when the base model is not trained to convergence. We make this second-order procedure efficient at the LLM scale using Kronecker-factored approximate curvature (K-FAC) and a novel matrix-free projector that exploits Kronecker structure to avoid constructing massive projection matrices. Across standard model-editing benchmarks, CrispEdit achieves high edit success while keeping capability degradation below 1% on average across datasets, significantly improving over prior editors.

1 Introduction

CrispEdit treats capability preservation as an explicit editing constraint and projects updates into low-curvature directions, enabling scalable edits that retain general capabilities. It combines a Bregman-Gauss-Newton formulation with K-FAC and a matrix-free projector, improving the edit–capability trade-off across settings.

  • Motivation: Model editing offers a faster alternative to retraining for targeted updates, but successful edits can quietly degrade broader capabilities.Reported degradation includes brittle reasoning, weaker instruction-following, and broken fluency.
  • Core formulation: CrispEdit formulates editing as constrained optimization, minimizing edit loss while requiring negligible change in capability loss.The constraint is enforced directly rather than replaced by a soft penalty.
  • Core formulation: CrispEdit projects updates into low-curvature Hessian subspaces, where parameter movement minimally affects capabilities.This exploits the observed anisotropy of pretrained loss landscapes, with sharp directions concentrated in a small subspace.
  • Second-order constraint: Bregman divergence yields a Gauss-Newton Hessian quadratic form exactly, without requiring the base model to be trained to convergence.This avoids stationarity assumptions that accompany the standard Hessian formulation.
  • Scalability: K-FAC approximates curvature, while a matrix-free Kronecker projector masks high-curvature components without constructing massive projection matrices.Cached curvature can be reused across future edits, supporting batch and sequential editing.
  • Evaluation: Across small- and large-scale experiments, CrispEdit improves the edit–capability trade-off with strong edit success and substantially reduced capability degradation.Evaluations include LLaMA-3-8B-Instruct and Qwen-2.5-1.5B-Instruct, alongside batch and sequential editing.

2 The Model editing problem

The paper formalizes model editing as minimizing edit loss subject to preserving a reference set of capabilities. It then motivates low-curvature projections as a practical approximation to this hard constraint at LLM scale.

  • Problem setup: Model editing updates a pretrained model using an edit target pair while preserving the base model’s existing capabilities.The model maps inputs x to outputs y through parameters θ, with θ0 denoting the pretrained parameters.
  • Capability constraint: The capability dataset Dcap is a reference set representing domains on which the edited model should continue performing well.Capability preservation is measured through empirical loss on this dataset.
  • Constrained optimization: The constrained objective minimizes edit loss while keeping capability loss within a small tolerance of its pre-edit value.The distance measure may be a loss difference or Bregman divergence, and the formulation unifies several editing frameworks.
  • Problem setup: The edit dataset Dedit contains desired edit pairs, and Ledit measures the loss associated with producing their target outputs.One example is the negative log-likelihood of edit outputs.
  • Computational challenge: At LLM scale, solving the hard constraint directly is challenging because the reference capability set typically far exceeds the edit set, making soft-penalty optimization costly.CrispEdit addresses this complexity by approximating the constrained problem with low-curvature projections.

3 CrispEdit: Curvature-Restricted In-Situ Parameter Editing

CrispEdit formulates model editing as constrained optimization and preserves capabilities by restricting updates to low-curvature directions. Bregman divergence yields a Gauss–Newton constraint without convergence assumptions, while K-FAC enables scalable projections and generalizes prior restrictive editing methods.

  • 3.1 Preserving capabilities with low-curvature updates: The quadratic capability constraint uses the Hessian of the capability loss when the base parameters are a local minimum.The second-order approximation is 1/2(θ − θ0)⊤Hcap(θ − θ0), with the linear term vanishing under the local-minimum assumption.
  • 3.1 Preserving capabilities with low-curvature updates: Low-rank capability Hessians provide many low-curvature directions where capability loss remains approximately invariant while edit loss decreases.The ellipsoidal constraint can be enforced with projected gradient or trust-region methods to control tolerated capability degradation.
  • 3.1 Preserving capabilities with low-curvature updates: CrispEdit enforces capability preservation by projecting edit gradients into approximate nullspaces of the capability-loss curvature.The projection removes gradient components in directions where capability loss is sensitive.
  • 3.2 Gauss-Newton constraint via Bregman divergence: Bregman divergence removes the need for base-model convergence and produces a Gauss–Newton Hessian that remains well-behaved for partially trained, overparameterized networks.This formulation is first-order flat at the base parameters and supports reliable scalable approximations.
  • 3.2 Gauss-Newton constraint via Bregman divergence: Representation constraints used by AlphaEdit and related methods are restrictive special cases whose feasible update subspaces are strict subsets of CrispEdit’s curvature-based subspace.This restrictiveness limits accessible parameter space and worsens the editing-efficacy–capability-preservation trade-off.
  • 3.3 K-FAC for scalable, matrix-free projections: K-FAC approximates the Gauss–Newton Hessian with Kronecker-factored blocks, enabling scalable matrix-free curvature projection and sequential editing.The approach exploits Kronecker structure and supports maintaining the statistics needed for approximate nullspace projections without retaining all edits.

4 Experiments

Experiments evaluate CrispEdit from tractable low-curvature projections to large-scale LLM editing, measuring edit reliability, generalization, runtime, scalability, and capability preservation. Across these settings, CrispEdit maintains a strong edit–capability trade-off, remains robust to dataset size and edit scale, and supports sequential editing.

  • 4.1 Comparison of various second-order constraints: Hessian low-curvature projections improve Fashion-MNIST fine-tuning while maintaining MNIST performance, and K-FAC and EK-FAC reasonably approximate Gauss-Newton behavior.The comparison uses LeNet-5 and evaluates Hessian, Gauss-Newton, K-FAC, EK-FAC, and activation-covariance nullspaces.
  • 4.2 Large-scale LLM evaluations: CrispEdit evaluates 3,000 edits on LLaMA-3-8B-Instruct across ZsRE, CounterFact, and WikiBigEdit using reliability, generalization, WILD evaluation, and five base-capability benchmarks.WILD combines context-guided autoregressive decoding with LLM-as-a-judge evaluation, including prompts with and without QA context.
  • 4.2 Large-scale LLM evaluations: CrispEdit achieves superior editing performance while preserving base-model capabilities, with computational efficiency from projected-gradient-descent fine-tuning.Prior methods can trade capability preservation for edit quality, especially under autoregressive decoding.
  • 4.2 Large-scale LLM evaluations: Capability preservation remains robust across energy thresholds, and CrispEdit stays effective with capability datasets as small as 100 samples.Removing the projection mechanism harms capability preservation, while preserving capabilities can improve autoregressive edit performance by maintaining fluency and instruction-following.
  • 4.2 Large-scale LLM evaluations: CrispEdit scales robustly from 3,000 to 10,000 edits, whereas baselines degrade or plateau at larger scales; it also generalizes across model architectures.CrispEdit maintains performance up to 10,000 edits and remains effective on Qwen-2.5-1.5B-Instruct.
  • 4.2 Large-scale LLM evaluations: CrispEdit-Seq retains previous edits, matches CrispEdit’s sequential-editing strength, operates 8× faster than AlphaEdit, and preserves base capabilities nearly intact.This result challenges the assumption that depth-first fine-tuning is ill-suited to sequential model editing.

5 Related work

Prior model-editing methods use memory, localization, hypernetworks, or constrained fine-tuning to modify models while mitigating capability loss. CrispEdit differs by avoiding auxiliary memory and prediction networks while deriving less restrictive projections from capability-loss curvature.

  • Memory-based editors store edits in auxiliary models, in-context prompts, low-rank adapters, or retrieval-based alignment rather than directly in model parameters.
  • Locate-then-edit methods identify parameters or neurons associated with undesired behavior, often assuming feed-forward knowledge localization and structured subjects, entities, or relations.
  • Hypernet-based editors learn separate networks to predict parameter shifts, recasting locate-then-edit optimization problems as meta-learning tasks.
  • Constrained fine-tuning adds restrictions such as weight decay, null-space projection, prompt masking, low-rank updates, or strict layer choices.
  • CrispEdit avoids per-step auxiliary loss calculation and additional prediction networks while using curvature-derived projections for scalable LLM editing.

6 Conclusion and future work

CrispEdit formulates sequential billion-parameter LLM editing around Gauss–Newton low-curvature directions and K-FAC projections. The paper also defines the model, datasets, losses, curvature notation, and projection subspaces used by this approach, while identifying broader future applications.

  • 6 Conclusion and future work: CrispEdit and its sequential variant edit billion-parameter LLMs while preserving capabilities through Gauss–Newton Hessian eigenspaces induced by a Bregman-divergence constraint.
  • 6 Conclusion and future work: K-FAC makes projection onto low-curvature capability-loss nullspaces practical at LLM scale.
  • 6 Conclusion and future work: Future work includes applying CrispEdit to safety and personalization, learning interpretable models under capability constraints, and using nonlinear constrained optimization for larger updates.
  • 6 Conclusion and future work: The notation defines θ0 as pretrained parameters, Δθ as parameter updates, and P_S as the orthogonal projector onto subspace S.
  • 6 Conclusion and future work: Capability and edit datasets are distinguished, while capability deviations are measured using distance functions including absolute loss differences and Bregman divergences.
  • 6 Conclusion and future work: The low-curvature subspace is formed from eigenvectors after a threshold index, and K-FAC approximates layerwise Gauss–Newton curvature using activations and pseudo-gradients.

B Proof of Bregman divergence quadratic form

The proof shows that the Bregman divergence has a quadratic second-order approximation governed by the output-loss Hessian and parameter–output Jacobian. This establishes the curvature form used for capability-preserving editing.

  • The proposition defines a Bregman divergence for a twice-differentiable model and convex twice-differentiable loss around base parameters θ0.
  • The chain rule differentiates the divergence with respect to parameters, with the first derivative evaluating to zero at θ0.
  • At θ0, the second derivative’s additional term vanishes because the divergence gradient with respect to model outputs is zero.
  • A second-order Taylor expansion therefore yields the quadratic approximation to the Bregman divergence around θ0.

C Proof of Proposition 1

The proof connects layerwise null directions defined through activation structure to null directions of the capability Gauss–Newton Hessian. Thus, updates invisible to all capability-example Jacobians preserve the quadratic capability curvature.

  • The proof fixes all parameters except one layer’s weight matrix and vectorizes that layer’s weights and updates.
  • Updates in the relevant activation-covariance nullspace produce zero parameter-Jacobian action for every capability example.
  • The layer’s per-example parameter Jacobian is derived by applying the chain rule through the layer pre-activation and downstream network map.
  • Because the capability Gauss–Newton Hessian is positive semidefinite, zero Jacobian action for all examples implies the update lies in its nullspace.

D Proof of matrix-free projection

The proof shows that a Kronecker-structured projector onto a selected eigen-subspace can be applied through matrix operations, without constructing the full projection matrix.

  • Subspace definition: The selected subspace S contains eigenvectors of C whose eigenvalues satisfy the predicate τ.C is the Kronecker product B ⊗ A.
  • Mask construction: The mask M selects factorized eigendirections by setting Mij = τ(λA,i · λB,j).The mask therefore encodes which eigenvalue products belong to the retained subspace.
  • Kronecker eigensystem: The eigen-decompositions of A and B induce eigenvalues λA,i · λB,j for C and eigenvectors uB,j ⊗ uA,i.This Kronecker-product structure enables the projector to be represented using the factor matrices.

E Additional details on LLM experiments

The experiments evaluate edited models on diverse capability and editing benchmarks using standardized protocols, with implementation choices and a judge prompt documented for reproducibility.

  • Evaluation setup: Base capabilities are evaluated with lm-evaluation-harness on IFEval, TruthfulQA, MMLU, GSM8K, and ARC-Challenge.Each task uses 200 examples with chat-template and multi-turn few-shot formatting.
  • Evaluation setup: Editing performance is evaluated with EasyEdit, generally following the WILD protocol except for the teacher-forcing evaluation in Table 3.The QA Context protocol appends a question-answer template and generates up to 40 tokens with predefined stop tokens.
  • Evaluation setup: The LLM judge gives binary correct-or-incorrect assessments from a question, gold target answer, and predicted answer.Figure 8 documents the complete judging prompt.
  • Implementation details: CrispEdit and CrispEdit-Seq use dataset-specific values of n and γ, while other hyperparameters follow Table 2.The reported configurations vary across CounterFact, WikiBigEdit, ZsRE, ZsRE10k, and the Qwen-2.5-1.5B-Instruct experiments.
  • Implementation details: Masking prompt tokens during K-FAC calculation produced suboptimal CrispEdit-Seq performance, possibly because of a relaxed token-independence assumption.The implementation instead calculates next-token predictions for edit samples.
  • Implementation details: Baselines use EasyEdit code and hyperparameters supplied by their original authors and tuned for LLaMA-3-8B-Instruct.This fixes the baseline implementation source and tuning context.

Generation Output

The generation case studies illustrate that editing methods can produce repetitive outputs, while CrispEdit returns concise target-aligned generations in the Cebu flowerpecker example.

  • Model Editing Case Study 1: In the Marina Rebeka case, Adam-NSCL and FT repeat “mezzo-srano” throughout their generated outputs.Both methods produce visibly degenerate repetition after the edit prompt.
  • Model Editing Case Study 2: For the Cebu flowerpecker prompt, the edit target is “endangered species.”The case study compares several methods’ generated answers against this target.
  • Model Editing Case Study 2: CrispEdit generates only “endangered species” for the Cebu flowerpecker edit, without the repetition shown by several alternatives.Adam-NSCL, FT, and other methods produce longer or different generations in the same case study.
  • Tabulated comparison: Table 3 compares editing reliability, generality, and specificity alongside base performance under a teacher-forcing pipeline.The comparison covers 3,000 samples from three representative datasets.

G Additional tables

Additional tables examine scaling, model transfer, prompt masking, and hyperparameter sensitivity, reporting robustness across capability-dataset sizes and energy thresholds.

  • Additional evaluations: Table 4 evaluates scaling to 10,000 ZsRE editing samples using WILD editing metrics and five representative base-capability benchmarks.The table reports reliability and generality alongside capability preservation.
  • Additional evaluations: Table 5 compares CrispEdit with existing methods on Qwen-2.5-1.5B-Instruct using 3,000 ZsRE samples and five base-capability benchmarks.Reliability and generality are included among the reported measures.
  • Ablations: Table 6 shows that prompt masking during K-FAC calculation leads to suboptimal CrispEdit-Seq performance even with more tokens.This isolates the effect of prompt masking in the K-FAC computation.
  • Hyperparameter sensitivity: CrispEdit remains robust across a wide range of capability-dataset sizes n when γ = 0.9.Table 7 uses 3,000 ZsRE edits and evaluates five representative base benchmarks.
  • Hyperparameter sensitivity: CrispEdit remains robust across a wide range of energy thresholds γ when n = 10,000.Table 8 evaluates three representative datasets with WILD editing metrics and five base-capability benchmarks.
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