Title: Convergence Dynamics of Over-Parameterized Score Matching for a Single Gaussian
Primary area: optimization
Keywords: over-parameterization, global convergence, non-convex optimization, mixtures of Gaussians, score-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 $n$ 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 $1/\tau$ rate, where $\tau$ 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.