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Convergence Dynamics of Over-Parameterized Score Matching for a Single Gaussian

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 6.7Download Report PDF
over-parameterizationglobal convergencenon-convex optimizationmixtures of Gaussiansscore-based generative models
Abstract:

Score matching has become a central training objective in modern generative modeling, particularly in diffusion models, where it is used to learn high-dimensional data distributions through the estimation of score functions. Despite its empirical success, the theoretical understanding of the optimization behavior of score matching, particularly in over-parameterized regimes, remains limited. In this work, we study gradient descent for training over-parameterized models to learn a single Gaussian distribution. Specifically, we use a student model with nn learnable parameters, motivated by the structure of a Gaussian mixture model, and train it on data generated from a single ground-truth Gaussian using the population score matching objective. We analyze the optimization dynamics under multiple regimes. When the noise scale is sufficiently large, we prove a global convergence result for gradient descent, which resembles the known behavior of gradient EM in over-parameterized settings. In the low-noise regime, we identify the existence of a stationary point, highlighting the difficulty of proving global convergence in this case. Nevertheless, we show convergence under certain initialization conditions: when the parameters are initialized to be exponentially small, gradient descent ensures convergence of all parameters to the ground truth. We further give an example where, without the exponentially small initialization, the parameters may not converge to the ground truth. Finally, we consider the case of random initialization, where parameters are sampled from a Gaussian distribution far from the ground truth. We prove that, with high probability, only one parameter converges while the others diverge to infinity, yet the loss still converges to zero with a 1/τ1/\tau rate, where τ\tau is the number of iterations. We also establish a nearly matching lower bound on the convergence rate in this regime. This is the first work to establish global convergence guarantees for Gaussian mixtures with at least three components under the score matching framework.

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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.
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Overview

Overall Novelty Assessment

The paper analyzes gradient descent dynamics for over-parameterized score matching when learning a single Gaussian distribution, using a student model with n learnable parameters motivated by Gaussian mixture structure. Within the taxonomy, it occupies the sole position in the 'Over-Parameterized Regime Convergence Analysis' leaf under 'Theoretical Analysis of Score Matching Optimization'. This leaf contains only the original paper itself, indicating a relatively sparse research direction focused specifically on convergence guarantees in over-parameterized settings for Gaussian targets, distinct from the broader statistical theory and algorithmic extension branches.

The taxonomy reveals three main branches: theoretical convergence analysis, statistical generalization bounds, and algorithmic extensions. The paper's leaf sits within the first branch, which emphasizes optimization dynamics rather than sample complexity or practical variants. Neighboring leaves include 'Denoising Score Matching Under Manifold Assumptions' (statistical bounds under geometric constraints) and 'Particle-Based Variational Inference' plus 'Energy-Based Latent Variable Model Training' (algorithmic methods for structured models). The scope notes clarify that convergence analysis excludes generalization bounds and algorithmic design, positioning this work as foundational theory for understanding training behavior before addressing broader distributional or architectural questions.

Among 16 candidates examined across three contributions, no refutable prior work was identified. The global convergence guarantee under large noise examined 4 candidates with 0 refutations; the low-noise exponentially small initialization analysis examined 2 candidates with 0 refutations; and the convergence rate analysis with random initialization examined 10 candidates with 0 refutations. This limited search scope suggests that within the top-16 semantically similar papers, none provide overlapping theoretical results for over-parameterized score matching on Gaussian distributions. The absence of sibling papers in the same taxonomy leaf further indicates that this specific convergence analysis angle has received minimal prior attention.

Based on the limited literature search of 16 candidates, the work appears to address a relatively unexplored theoretical question within score matching optimization. The taxonomy structure shows active research in related areas (manifold constraints, latent variable training, distillation), but the specific focus on over-parameterized convergence for Gaussian targets occupies a sparse niche. The analysis does not claim exhaustive coverage of all optimization theory or score matching literature, only that among semantically proximate papers, this particular convergence perspective has not been directly addressed.

Taxonomy

Core-task Taxonomy Papers
4
3
Claimed Contributions
16
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: optimization dynamics of over-parameterized score matching for Gaussian distributions. The field of score matching has evolved into a rich landscape with three main branches. The first branch, Theoretical Analysis of Score Matching Optimization, investigates convergence guarantees and the behavior of gradient-based methods in over-parameterized regimes, often examining how network width and depth influence training trajectories. The second branch, Statistical Theory and Generalization Bounds, focuses on sample complexity, estimation error, and the interplay between model capacity and data requirements. The third branch, Algorithmic Extensions and Structured Model Learning, explores practical variants such as manifold-constrained methods and hierarchical or bilevel formulations that adapt score matching to more complex settings. Together, these branches reflect a maturing understanding of both the theoretical underpinnings and the algorithmic flexibility of score-based models. Recent work has highlighted several active themes and trade-offs. One line examines how semi-implicit or continuous-time perspectives, as in Semi-Implicit Gradient Flow[1], can yield sharper convergence analyses by smoothing discrete optimization steps. Another explores geometric constraints, with Manifold Score Matching[2] addressing data that lie on lower-dimensional structures. Meanwhile, Bilevel Score Matching[3] and related hierarchical approaches tackle scenarios where the score function itself must be learned in a nested optimization framework. Within this landscape, Overparameterized Score Matching[0] sits squarely in the convergence analysis branch, emphasizing how excess parameters enable faster or more stable training for Gaussian targets. Its focus on explicit Gaussian settings contrasts with the more general geometric or bilevel perspectives of neighbors like Manifold Score Matching[2] and Bilevel Score Matching[3], offering a controlled testbed for understanding over-parameterization effects before extending to richer data distributions.

Claimed Contributions

Global convergence guarantee for over-parameterized score matching under large noise

The authors establish a global convergence result for gradient descent when training an over-parameterized model (n learnable parameters) to learn a single Gaussian distribution under the score matching objective, specifically when the noise scale is sufficiently large. This extends the connection between DDPM and gradient EM to the over-parameterized setting.

4 retrieved papers
Convergence analysis under exponentially small initialization in low-noise regime

The authors prove that in the low-noise regime, if all student parameters are initialized exponentially close to zero, then all parameters converge to the ground truth. They introduce a technique for tracking the evolution of the geometric center of parameters and provide a counterexample showing this exponentially small initialization is necessary.

2 retrieved papers
Convergence rate analysis and lower bound for random initialization

For random Gaussian initialization far from the ground truth, the authors prove that with high probability one parameter converges to the ground truth while others diverge, yet the loss converges at rate O(1/τ). They establish a nearly matching lower bound of Ω(1/τ^(1+ε)), contrasting sharply with linear convergence in the exactly parameterized case.

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

Global convergence guarantee for over-parameterized score matching under large noise

The authors establish a global convergence result for gradient descent when training an over-parameterized model (n learnable parameters) to learn a single Gaussian distribution under the score matching objective, specifically when the noise scale is sufficiently large. This extends the connection between DDPM and gradient EM to the over-parameterized setting.

Contribution

Convergence analysis under exponentially small initialization in low-noise regime

The authors prove that in the low-noise regime, if all student parameters are initialized exponentially close to zero, then all parameters converge to the ground truth. They introduce a technique for tracking the evolution of the geometric center of parameters and provide a counterexample showing this exponentially small initialization is necessary.

Contribution

Convergence rate analysis and lower bound for random initialization

For random Gaussian initialization far from the ground truth, the authors prove that with high probability one parameter converges to the ground truth while others diverge, yet the loss converges at rate O(1/τ). They establish a nearly matching lower bound of Ω(1/τ^(1+ε)), contrasting sharply with linear convergence in the exactly parameterized case.

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