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A Derandomization Framework for Structure Discovery: Applications in Neural Networks and Beyond

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.0Download Report PDF
Neural NetworksOptimizationStructure DiscoveryCompressibilityDerandomizationMultiple Index ModelJohnson LindenstraussMAXCUT
Abstract:

Understanding the dynamics of feature learning in neural networks (NNs) remains a significant challenge. The work of (Mousavi-Hosseini et al., 2023) analyzes a multiple index teacher-student setting and shows that a two-layer student attains a low-rank structure in its first-layer weights when trained with stochastic gradient descent (SGD) and a strong regularizer. This structural property is known to reduce sample complexity of generalization. Indeed, in a second step, the same authors establish algorithm-specific learning guarantees under additional assumptions. In this paper, we focus exclusively on the structure discovery aspect and study it under weaker assumptions, more specifically: we allow (a) NNs of arbitrary size and depth, (b) with all parameters trainable, (c) under any smooth loss function, (d) tiny regularization, and (e) trained by any method that attains a second-order stationary point (SOSP), e.g. perturbed gradient descent (PGD). At the core of our approach is a key derandomization\textit{derandomization} lemma, which states that optimizing the function Ex[gθ(Wx+b)]E_{x} \left[g_{\theta}(Wx + b)\right] converges to a point where W=0W = 0, under mild conditions. The fundamental nature of this lemma directly explains structure discovery and has immediate applications in other domains including an end-to-end approximation for MAXCUT, and computing Johnson-Lindenstrauss embeddings.

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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 contributes a derandomization lemma showing that optimizing averaged functions over random inputs drives certain weight matrices to zero under mild conditions, thereby establishing structure discovery at second-order stationary points. It sits in the Overparameterized Network Training and Generalization leaf, which contains only one sibling paper examining convergence guarantees in deeper architectures. This is a relatively sparse research direction within the broader taxonomy of 32 papers across multiple branches, suggesting the specific focus on structure discovery via second-order conditions in overparameterized settings remains underexplored compared to landscape analysis or Hessian-based methods.

The taxonomy tree reveals neighboring work in Structure Optimization and Architecture Search, which integrates structural learning with parameter updates, and in Hessian Spectral Analysis and Structure, which investigates low-rank phenomena and eigenvalue distributions. The paper diverges from these by emphasizing derandomization arguments rather than explicit Hessian decomposition or architecture search heuristics. Its connection to Optimization Landscape Analysis is evident through shared interest in critical point characterization, yet it operates under weaker assumptions than typical landscape studies, which often restrict to linear networks or specific loss geometries.

Among 17 candidates examined, the derandomization lemma and structure discovery under weaker assumptions show no clear refutation across 7 and 5 candidates respectively, suggesting these contributions occupy relatively novel ground within the limited search scope. The applications to MAXCUT and Johnson-Lindenstrauss embeddings, however, encountered 2 refutable candidates out of 5 examined, indicating more substantial prior work in these specific application domains. The search scale is modest, so these statistics reflect top-K semantic matches rather than exhaustive coverage of the field.

Based on the top-17 semantic matches and taxonomy structure, the core theoretical contributions appear more novel than the application examples. The analysis does not cover broader optimization literature outside the neural network context or recent preprints that may address similar derandomization ideas. The sparse leaf placement and limited refutation counts suggest the work occupies a distinct niche, though the small search scope precludes definitive claims about field-wide novelty.

Taxonomy

Core-task Taxonomy Papers
32
3
Claimed Contributions
17
Contribution Candidate Papers Compared
2
Refutable Paper

Research Landscape Overview

Core task: structure discovery in neural networks through second-order stationary points. The field organizes around four main branches that collectively address how neural networks navigate complex loss landscapes. Optimization Landscape Analysis and Critical Point Characterization examines the geometric properties of loss surfaces, identifying benign versus problematic critical points in settings ranging from deep linear networks to sensor localization and phase retrieval problems. Hessian-Based Analysis and Second-Order Methods focuses on curvature information, exploring Hessian eigenstructure, rank properties, and outlier phenomena that reveal implicit biases and guide pruning strategies. Training Methods and Algorithmic Approaches encompasses practical techniques for leveraging second-order information, including Newton-type methods, stochastic stabilization schemes, and strategies for overparameterized regimes. Domain-Specific Applications translates these insights into specialized contexts such as graph neural networks, physics-informed models, and optical computing architectures. A particularly active line of work investigates how overparameterization shapes the loss landscape and enables efficient training. Overparameterized Beyond Two Layers[3] explores convergence guarantees in deeper architectures, while studies like Deep Linear Loss Landscape[1] and Deep Linear Critical Points[11] characterize the structure of critical points in simplified settings. Derandomization Structure Discovery[0] sits within this cluster, emphasizing how second-order stationary points can be systematically identified in overparameterized networks. Compared to Overparameterized Beyond Two Layers[3], which focuses on gradient descent convergence, Derandomization Structure Discovery[0] appears more concerned with the explicit characterization of critical point structure through derandomization techniques. Meanwhile, works like Hessian Rank Insights[18] and Implicit Regularization Low Rank[19] reveal that curvature properties often exhibit low-rank structure, suggesting that second-order analysis can uncover hidden regularities even in high-dimensional parameter spaces.

Claimed Contributions

Derandomization lemma for structure discovery

The authors introduce a derandomization lemma showing that for functions of the form Ex[gθ(Wx + b)] + λ||W||²F, any second-order stationary point (SOSP) satisfies W ≈ 0 under mild conditions. This lemma forms the theoretical foundation for discovering low-rank structure in neural networks and has applications beyond neural networks.

7 retrieved papers
Structure discovery in neural networks under weaker assumptions

The authors extend prior work on structure discovery in neural networks by showing that low-rank first-layer weights emerge at SOSPs under significantly more general conditions, including arbitrary network depth, trainable biases, any smooth loss, and arbitrarily small regularization, using only the requirement of reaching a SOSP.

5 retrieved papers
Applications to MAXCUT and Johnson-Lindenstrauss embeddings

The authors demonstrate the generality of their derandomization lemma by applying it to obtain a deterministic MAXCUT approximation matching the Goemans-Williamson guarantee and a deterministic construction for Johnson-Lindenstrauss embeddings, showing the lemma's applicability beyond neural networks.

5 retrieved papers
Can Refute

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Derandomization lemma for structure discovery

The authors introduce a derandomization lemma showing that for functions of the form Ex[gθ(Wx + b)] + λ||W||²F, any second-order stationary point (SOSP) satisfies W ≈ 0 under mild conditions. This lemma forms the theoretical foundation for discovering low-rank structure in neural networks and has applications beyond neural networks.

Contribution

Structure discovery in neural networks under weaker assumptions

The authors extend prior work on structure discovery in neural networks by showing that low-rank first-layer weights emerge at SOSPs under significantly more general conditions, including arbitrary network depth, trainable biases, any smooth loss, and arbitrarily small regularization, using only the requirement of reaching a SOSP.

Contribution

Applications to MAXCUT and Johnson-Lindenstrauss embeddings

The authors demonstrate the generality of their derandomization lemma by applying it to obtain a deterministic MAXCUT approximation matching the Goemans-Williamson guarantee and a deterministic construction for Johnson-Lindenstrauss embeddings, showing the lemma's applicability beyond neural networks.

Table of Contents

A Derandomization Framework for Structure Discovery: Applications in Neural Networks and Beyond | Novelty Validation