OpenNovelty.org

Hyperbolic Aware Minimization: Implicit Bias for Sparsity

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
SparsityImplicit biasSign flipExponential updateTraining dynamicsBregman function
Abstract:

Understanding the implicit bias of optimization algorithms is key to explaining and improving the generalization of deep models. The hyperbolic implicit bias induced by pointwise overparameterization promotes sparsity, but also yields a small inverse Riemannian metric near zero, slowing down parameter movement and impeding meaningful parameter sign flips. To overcome this obstacle, we propose Hyperbolic Aware Minimization (HAM), which alternates a standard optimizer step with a lightweight hyperbolic mirror step. The mirror step incurs less compute and memory than pointwise overparameterization, reproduces its beneficial hyperbolic geometry for feature learning, and mitigates the small–inverse-metric bottleneck. Our characterization of the implicit bias in the context of underdetermined linear regression provides insights into the mechanism how HAM consistently increases performance --even in the case of dense training, as we demonstrate in experiments with standard vision benchmarks. HAM is especially effective in combination with different sparsification methods, advancing the state of the art.

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 proposes Hyperbolic Aware Minimization (HAM), an optimization method that alternates standard gradient steps with hyperbolic mirror steps to induce beneficial geometric structure for feature learning and sparsity. It resides in the 'Specialized Optimization Methods for Sparsity' leaf, which contains five papers total. This leaf sits within the broader 'Optimization Algorithm Design and Training Dynamics' branch, indicating a moderately populated research direction focused on novel training procedures that exploit or enhance implicit sparsity beyond standard gradient descent analysis.

The taxonomy reveals that HAM's leaf neighbors other specialized methods such as sharpness-aware and scale-invariant approaches, cyclic and alternating sparse training strategies, and implicit sparsification techniques. These sibling papers primarily focus on pruning schedules, flatness-based regularization, or training dynamics that naturally drive weights toward zero. HAM diverges by introducing hyperbolic geometry and Riemannian metric considerations, connecting conceptually to the 'Implicit Regularization Mechanisms' branch (which includes diagonal linear network dynamics and overparameterization theory) while remaining distinct in its algorithmic design and geometric motivation.

Among the three contributions analyzed, the HAM optimization method itself was examined against one candidate with no refutation found. The theoretical characterization of HAM's implicit bias via Riemannian gradient flow was examined against ten candidates, with one appearing to provide overlapping prior work on implicit bias analysis in related geometric settings. The mitigation of the vanishing inverse metric problem was examined against ten candidates with no refutations identified. These statistics reflect a limited search scope of twenty-one total candidates, suggesting that while the core algorithmic proposal appears relatively novel, the theoretical analysis intersects with existing work on implicit regularization in overparameterized or geometrically structured models.

Based on the top-twenty-one semantic matches examined, HAM's combination of hyperbolic geometry with practical sparsification appears less explored than standard implicit bias theory or conventional pruning methods. The analysis does not cover exhaustive literature on Riemannian optimization or mirror descent variants outside the implicit bias context, nor does it capture potential overlaps in broader optimization geometry literature. The contribution-level findings suggest that the algorithmic innovation is more distinctive than the theoretical framing, which aligns with established implicit regularization research.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
21
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: implicit bias and optimization for neural network sparsity. The field explores how gradient-based training naturally induces sparse solutions and how optimization algorithms can be designed or analyzed to control this behavior. The taxonomy reveals several major branches: Implicit Regularization Mechanisms examines how standard optimizers like SGD inherently favor certain solution structures (e.g., Step Size Bias[14], Norm Scaling Bias[38]); Optimization Algorithm Design focuses on specialized methods that explicitly target sparsity or leverage implicit biases during training (e.g., Sharpness-Aware Scale-Invariant[8], Implicit Sparsification[16]); Application-Specific work adapts these ideas to domains such as inverse problems, multi-task learning, and graph neural networks (e.g., GNN Explanation Regularization[10], Multi-Task Implicit Regularization[12]); Robustness and Generalization investigates how sparsity interacts with overfitting and adversarial settings (e.g., Benign Overfitting Sparse[5], Adversarial Robust Bias[19]); and Theoretical Foundations provides rigorous analysis of convergence, sample complexity, and the geometry of sparse recovery (e.g., Diagonal Linear Networks[30], High-Dimensional Linear Regression[29]). A particularly active line of work centers on specialized optimization methods that go beyond vanilla gradient descent to achieve or exploit sparsity. Hyperbolic Aware Minimization[0] sits within this branch, proposing a geometry-aware approach that contrasts with more conventional sparsity-inducing techniques like Alternating Sparse Training[41] or Cyclic Sparse Training[48], which alternate between pruning and retraining phases. Nearby efforts such as Sharpness-Aware Scale-Invariant[8] emphasize flatness and scale invariance to improve generalization, while Implicit Sparsification[16] investigates how certain training dynamics naturally drive weights toward zero without explicit penalties. The main trade-offs revolve around computational overhead, theoretical guarantees, and the degree to which sparsity emerges implicitly versus being enforced explicitly. Hyperbolic Aware Minimization[0] distinguishes itself by leveraging hyperbolic geometry to guide the optimization trajectory, offering a fresh perspective on how curvature and metric structure can influence the implicit regularization path toward sparse solutions.

Claimed Contributions

Hyperbolic Aware Minimization (HAM) optimization method

The authors introduce HAM, a plug-and-play optimization algorithm that alternates between any standard optimizer step and a hyperbolic mirror step. This method captures the beneficial hyperbolic geometry of pointwise overparameterization (m ⊙ w) while avoiding its vanishing inverse metric problem and without requiring explicit parameter doubling.

1 retrieved paper
Theoretical characterization of HAM's implicit bias via Riemannian gradient flow

The authors provide a theoretical analysis of HAM's training dynamics using Riemannian gradient flow for linear regression. They characterize HAM's implicit bias as interpolating between L2 and L1 regularization, and show how it facilitates parameter sign flips while maintaining faster convergence than pointwise overparameterization.

10 retrieved papers
Can Refute
Mitigation of vanishing inverse metric problem

The authors demonstrate that HAM resolves the small inverse metric bottleneck near zero that affects pointwise overparameterization methods. HAM maintains an inverse metric g^(-1)(θ) = 1 + α|θ| that stays bounded away from zero, enabling effective parameter movement and sign flips without the computational overhead of explicit overparameterization.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Hyperbolic Aware Minimization (HAM) optimization method

The authors introduce HAM, a plug-and-play optimization algorithm that alternates between any standard optimizer step and a hyperbolic mirror step. This method captures the beneficial hyperbolic geometry of pointwise overparameterization (m ⊙ w) while avoiding its vanishing inverse metric problem and without requiring explicit parameter doubling.

Contribution

Theoretical characterization of HAM's implicit bias via Riemannian gradient flow

The authors provide a theoretical analysis of HAM's training dynamics using Riemannian gradient flow for linear regression. They characterize HAM's implicit bias as interpolating between L2 and L1 regularization, and show how it facilitates parameter sign flips while maintaining faster convergence than pointwise overparameterization.

Contribution

Mitigation of vanishing inverse metric problem

The authors demonstrate that HAM resolves the small inverse metric bottleneck near zero that affects pointwise overparameterization methods. HAM maintains an inverse metric g^(-1)(θ) = 1 + α|θ| that stays bounded away from zero, enabling effective parameter movement and sign flips without the computational overhead of explicit overparameterization.

Table of Contents

Hyperbolic Aware Minimization: Implicit Bias for Sparsity | Novelty Validation