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Better LMO-based Momentum Methods with Second-Order Information

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Second-order MomentumLinear Minimization OracleStochastic Optimization
Abstract:

The use of momentum in stochastic optimization algorithms has shown empirical success across a range of machine learning tasks. Recently, a new class of momentum-based stochastic algorithms has emerged within the Linear Minimization Oracle (LMO) framework--leading to methods such as Muon, Scion, and Gluon--for effectively solving deep neural network training problems. However, traditional stochastic momentum methods offer convergence guarantees no better than O(1/K1/4)\mathcal{O}(1/K^{1/4}). While several approaches--such as Hessian-Corrected Momentum (HCM)--have aimed to improve this rate, their theoretical results are generally restricted to the Euclidean norm setting. This limitation hinders their applicability in the problems where arbitrary norms are often required. In this paper, we extend the LMO-based framework by integrating HCM, and provide the convergence guarantees under relaxed smoothness and arbitrary norm settings. Specifically, we establish improved convergence rates of O(1/K1/3)\mathcal{O}(1/K^{1/3}) for HCM, thereby surpassing the classical momentum rate and allowing the algorithms to better adapt to the geometry of the problem. Experimental results on training Multi-Layer Perceptrons (MLPs) and Long Short-Term Memory (LSTM) networks support our theoretical findings, demonstrating that the proposed LMO-based algorithms with HCM significantly outperform their vanilla algorithms with traditional momentum.

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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 proposes integrating Hessian-Corrected Momentum into the Linear Minimization Oracle framework, targeting improved convergence rates under relaxed smoothness and arbitrary norms. It resides in the 'LMO-Based and Second-Order Momentum Methods' leaf, which currently contains only this paper as a sibling. This leaf sits within the broader 'Specialized Momentum Techniques and Extensions' branch, indicating a relatively sparse research direction compared to more populated areas like 'Adam and AMSGrad Variants' or 'Momentum Methods for Non-Smooth Non-Convex Problems', which contain multiple sibling papers.

The taxonomy reveals that neighboring leaves address related but distinct momentum formulations: 'Heavy-Ball and Nesterov Momentum Variants' explores classical momentum schemes, 'Model-Based and Proximal Momentum Methods' focuses on proximal operators for weakly convex problems, and 'Shuffling and Finite-Sum Optimization' targets finite-sum settings. The paper's LMO-based approach diverges from these gradient-centric methods by substituting gradient steps with structured subproblem solves, positioning it at the intersection of second-order information and oracle-based optimization—a boundary explicitly noted in the taxonomy's exclude criteria for first-order methods.

Among thirty candidates examined, the contribution on improved convergence rates shows one refutable candidate, while the LMO-based framework and empirical validation contributions each examined ten candidates with no clear refutations. The convergence rate claim appears to have more substantial prior work overlap within the limited search scope, whereas the LMO-arbitrary-norm integration and empirical validation on MLPs and LSTMs show fewer direct precedents among the examined papers. This suggests the algorithmic framework may occupy a less explored niche, though the convergence rate improvement builds on more established theoretical territory.

Based on the top-thirty semantic matches and taxonomy structure, the work appears to address a relatively underexplored intersection of LMO methods, second-order momentum, and arbitrary norms. The limited sibling count in its taxonomy leaf and the sparse refutation rate for two of three contributions suggest potential novelty, though the analysis does not cover exhaustive literature beyond the examined candidates. The convergence rate contribution warrants closer scrutiny given the identified overlap, while the framework integration and empirical scope appear less directly anticipated by prior work within the search scope.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
30
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: stochastic optimization with momentum under relaxed smoothness conditions. The field has evolved to address scenarios where classical Lipschitz-smoothness assumptions fail, organizing itself into several major branches. Momentum Methods under Generalized Smoothness explores how momentum-based algorithms behave when gradient Lipschitz constants are unbounded or problem-dependent, with works like Parameter-agnostic Smoothness[1] and Unbounded Momentum[19] establishing convergence under weaker regularity. Adaptive Optimization Methods investigates how adaptive step-size schemes such as Adam and AdaGrad can be extended beyond standard smoothness, exemplified by Adam Relaxed Assumptions[12] and AdaGrad Relaxed[13]. Distributed and Communication-Efficient Optimization tackles large-scale settings where communication costs and decentralized architectures demand specialized momentum techniques, as seen in Communication-efficient Smoothness[2] and Distributed Stochastic Consensus[3]. Bilevel and Compositional Optimization addresses nested or multi-level problems, Theoretical Analysis and Convergence Guarantees provides foundational results on rates and complexity, Specialized Momentum Techniques and Extensions covers novel momentum variants and second-order methods, and Empirical and Application-Oriented Studies bridges theory with practice. A particularly active line of work contrasts standard first-order momentum schemes with more sophisticated variants that exploit problem structure or alternative oracles. For instance, Heavy-tailed Momentum[5] and Unbounded SignSGD[6] explore robustness to heavy-tailed noise and gradient clipping, while Momentum Coupled Compositional[9] and Bilevel Unbounded Smoothness[11] extend momentum to hierarchical objectives. Within this landscape, LMO Momentum[0] occupies a distinctive niche in the Specialized Momentum Techniques branch by leveraging linear minimization oracles rather than gradient steps, offering a second-order perspective that complements gradient-based methods like Nonlinear Preconditioned Gradient[10]. This approach contrasts with the broader first-order momentum literature, such as Momentum Methods Survey[7] and Momentum Nonsmooth Nonconvex[8], by trading gradient evaluations for structured subproblem solves, thus addressing relaxed smoothness through a fundamentally different algorithmic lens.

Claimed Contributions

LMO-based methods with second-order momentum under arbitrary norms

The authors extend LMO-based optimization algorithms by integrating two variants of second-order momentum (Hessian-corrected momentum). This generalizes prior second-order momentum methods from the Euclidean norm setting to arbitrary norm settings, allowing the algorithms to adapt to the geometry of the problem.

10 retrieved papers
Improved O(1/K^(1/3)) convergence rate under relaxed smoothness

The authors prove that their LMO-based methods with second-order momentum achieve an O(1/K^(1/3)) convergence rate in the expected gradient norm under relaxed smoothness conditions and arbitrary norms. This improves upon the O(1/K^(1/4)) rate of existing LMO-based methods with Polyak momentum and matches known rates for second-order momentum under standard smoothness in Euclidean settings.

10 retrieved papers
Can Refute
Empirical validation on MLP and LSTM training tasks

The authors conduct experiments on nonconvex problems including MLP and LSTM training to validate their theoretical findings. The empirical results demonstrate that LMO-based methods with second-order momentum significantly outperform methods using Polyak momentum and extrapolated momentum.

10 retrieved papers

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

LMO-based methods with second-order momentum under arbitrary norms

The authors extend LMO-based optimization algorithms by integrating two variants of second-order momentum (Hessian-corrected momentum). This generalizes prior second-order momentum methods from the Euclidean norm setting to arbitrary norm settings, allowing the algorithms to adapt to the geometry of the problem.

Contribution

Improved O(1/K^(1/3)) convergence rate under relaxed smoothness

The authors prove that their LMO-based methods with second-order momentum achieve an O(1/K^(1/3)) convergence rate in the expected gradient norm under relaxed smoothness conditions and arbitrary norms. This improves upon the O(1/K^(1/4)) rate of existing LMO-based methods with Polyak momentum and matches known rates for second-order momentum under standard smoothness in Euclidean settings.

Contribution

Empirical validation on MLP and LSTM training tasks

The authors conduct experiments on nonconvex problems including MLP and LSTM training to validate their theoretical findings. The empirical results demonstrate that LMO-based methods with second-order momentum significantly outperform methods using Polyak momentum and extrapolated momentum.

Table of Contents