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The Potential of Second-Order Optimization for LLMs: A Study with Full Gauss-Newton

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.5Download Report PDF
LLMsoptimization
Abstract:

Recent efforts to accelerate LLM pretraining have focused on computationally-efficient approximations that exploit second-order structure. This raises a key question for large-scale training: how much performance is forfeited by these approximations? To probe this question, we establish a practical upper bound on iteration complexity by applying full Gauss-Newton (GN) preconditioning to transformer models of up to 150M parameters. Our experiments show that full GN updates yield substantial gains over existing optimizers, achieving a 5.4x reduction in training iterations compared to strong baselines like SOAP and Muon. Furthermore, we find that a precise layerwise GN preconditioner, which ignores cross-layer information, nearly matches the performance of the full GN method. Collectively, our results suggest: (1) the GN approximation is highly effective for preconditioning, implying higher-order loss terms may not be critical for convergence speed; (2) the layerwise Hessian structure contains sufficient information to achieve most of these potential gains; and (3) a significant performance gap exists between current approximate methods and an idealized layerwise oracle.

Disclaimer
This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
NOTE that some papers exist in multiple, slightly different versions (e.g., with different titles or URLs). The system may retrieve several versions of the same underlying work. The current automated pipeline does not reliably align or distinguish these cases, so human reviewers will need to disambiguate them manually.
If you have any questions, please contact: mingzhang23@m.fudan.edu.cn

Overview

Overall Novelty Assessment

The paper establishes a practical upper bound on iteration complexity by applying full Gauss-Newton preconditioning to transformer models up to 150M parameters, achieving a 5.4x reduction in training iterations over strong baselines. It resides in the 'Full and Layerwise Gauss-Newton Preconditioning' leaf, which currently contains only this paper as its sole member. This places the work in a relatively sparse research direction within the broader second-order optimization landscape, where most efforts have focused on diagonal or block-diagonal approximations rather than full or layerwise Gauss-Newton methods.

The taxonomy reveals a crowded ecosystem of related approaches in sibling categories. 'Diagonal Hessian-Based Optimizers' (e.g., Sophia) and 'Full-Matrix and Structured Preconditioners' (e.g., Shampoo variants) represent neighboring directions that trade off curvature fidelity for scalability. The paper's focus on layerwise structure positions it between these extremes, exploring whether cross-layer curvature information is necessary for convergence gains. Nearby branches in parameter-efficient fine-tuning and post-training compression demonstrate alternative applications of curvature information, but the core optimization methods branch remains the most directly relevant context for assessing novelty.

Among 15 candidates examined, the contribution on establishing iteration complexity upper bounds shows no clear refutation (0 of 3 candidates). However, the memory-feasible implementation using Jacobian-vector products faces substantial prior work (3 of 10 candidates can refute), and the layerwise variant isolating cross-layer importance is clearly anticipated by existing methods (2 of 2 candidates refute). The limited search scope means these statistics reflect top semantic matches rather than exhaustive coverage. The iteration complexity analysis appears most novel, while the implementation techniques and layerwise decomposition have more established precedents in the examined literature.

Based on the top-15 semantic matches, the work's primary novelty lies in empirically quantifying the performance gap between idealized layerwise oracles and current approximate methods for transformer pretraining. The implementation and layerwise decomposition contributions build on recognizable prior techniques, though the specific application context and scale may differ. The sparse population of its taxonomy leaf suggests this precise combination of full Gauss-Newton analysis at 150M parameter scale represents a relatively underexplored direction, even if individual components have precedents.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
15
Contribution Candidate Papers Compared
5
Refutable Paper

Research Landscape Overview

Core task: second-order optimization for large language model pretraining. The field has evolved into a rich ecosystem of methods that exploit curvature information to improve training efficiency, model compression, and adaptation. At the highest level, the taxonomy distinguishes core optimization algorithms—such as full and layerwise Gauss-Newton or Kronecker-factored approaches—from downstream applications including parameter-efficient fine-tuning, post-training compression, data selection via influence analysis, unlearning, and curvature-based interpretability. Works like Sophia[1] and Scalable Second Order[4] exemplify efforts to make second-order preconditioning computationally feasible at scale, while branches focused on compression (e.g., VPTQ[5], QuIP[16]) and fine-tuning (e.g., Sensitivity-LoRA[7]) demonstrate how curvature estimates can guide resource-constrained model adaptation. Theoretical foundations and scaling analyses provide the mathematical underpinnings, and emerging topics explore novel intersections with hardware constraints and information geometry. Several active lines of work reveal key trade-offs between computational overhead and convergence speed. Full Gauss-Newton methods promise strong curvature approximations but require careful memory management and efficient Hessian computation strategies, as seen in Gauss-Newton for LLMs[0] and related efforts like Second Order Transformers[9] and Practical Second Order[35]. In contrast, diagonal or block-diagonal approximations sacrifice some curvature fidelity for scalability, a theme echoed in adaptive gradient methods such as Adaptive Gradient Scaling[6] and AdaFish[42]. Gauss-Newton for LLMs[0] sits squarely within the core optimization branch, emphasizing layerwise preconditioning to balance accuracy and efficiency during pretraining. Compared to Sophia[1], which uses a lightweight Hessian diagonal estimate, Gauss-Newton for LLMs[0] pursues a more structured approximation, aiming to capture richer curvature information without the full cost of exact second-order updates. This positioning highlights an ongoing exploration of how much curvature structure is necessary to meaningfully accelerate large-scale pretraining.

Claimed Contributions

Establishing practical upper bound on iteration complexity via full Gauss-Newton preconditioning

The authors apply full Gauss-Newton preconditioning to transformer models to determine the best achievable iteration complexity for second-order optimization methods. This serves as a performance benchmark for evaluating approximate second-order methods in LLM training.

3 retrieved papers
Memory-feasible Gauss-Newton implementation using Jacobian-vector products

The authors develop an implementation that avoids materializing the full Hessian by using Jacobian-vector products and optimizing a second-order Taylor approximation of the loss on a first-order Taylor approximation of the model. This makes full Gauss-Newton optimization computationally tractable for studying performance limits.

10 retrieved papers
Can Refute
Layerwise Gauss-Newton variant for isolating cross-layer curvature importance

The authors introduce a layerwise variant of Gauss-Newton that ignores cross-layer curvature information to determine whether layer-local Hessian structure is sufficient for achieving performance gains. This helps identify which structural properties of the Hessian are essential for optimization improvements.

2 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Within the taxonomy built over the current TopK core-task papers, the original paper is assigned to a leaf with no direct siblings and no cousin branches under the same grandparent topic. In this retrieved landscape, it appears structurally isolated, which is one partial signal of novelty, but still constrained by search coverage and taxonomy granularity.

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Establishing practical upper bound on iteration complexity via full Gauss-Newton preconditioning

The authors apply full Gauss-Newton preconditioning to transformer models to determine the best achievable iteration complexity for second-order optimization methods. This serves as a performance benchmark for evaluating approximate second-order methods in LLM training.

Contribution

Memory-feasible Gauss-Newton implementation using Jacobian-vector products

The authors develop an implementation that avoids materializing the full Hessian by using Jacobian-vector products and optimizing a second-order Taylor approximation of the loss on a first-order Taylor approximation of the model. This makes full Gauss-Newton optimization computationally tractable for studying performance limits.

Contribution

Layerwise Gauss-Newton variant for isolating cross-layer curvature importance

The authors introduce a layerwise variant of Gauss-Newton that ignores cross-layer curvature information to determine whether layer-local Hessian structure is sufficient for achieving performance gains. This helps identify which structural properties of the Hessian are essential for optimization improvements.

Table of Contents