Global Convergence of Gradient Descent for Score Matching in Gaussian Mixtures via Reverse Fisher Divergence

Abstract: The score matching problem is a central training objective in modern generative modeling, diffusion models, fitting unnormalized statistical models, and inverse problems. A standard approach is to minimize the forward Fisher divergence, where the expectation is taken with respect to the teacher distribution. However, recent results show that even in simple Gaussian mixture model settings, this objective can lead to undesirable and initialization-dependent convergence behavior. In this paper, we study an alternative objective: the reverse Fisher divergence, where the expectation is taken with respect to the student distribution. We analyze gradient descent (GD) for fitting Gaussian mixture models and show that this change in the objective leads to significantly better optimization properties. First, when the teacher distribution is a single Gaussian and the student is a Gaussian mixture model with fixed weights and identity covariances, we prove the global convergence of GD from arbitrary initializations. Second, we extend the analysis to the case where the teacher is also a Gaussian mixture model and prove global convergence guarantees under a global random initialization scheme and a $\widetildeΩ(1)$-separation assumption on the target means. In particular, with high probability, each student component converges near its closest teacher component, and we provide conditions under which the student distribution converges in total variation distance. Our proofs rely on a new Lyapunov-based analysis of the gradient descent dynamics, showing that the reverse Fisher divergence has a much more favorable optimization landscape than the forward Fisher divergence.

• The paper demonstrates that using reverse Fisher divergence allows gradient descent to achieve global convergence in score matching for GMMs from arbitrary initializations.
• It provides explicit convergence rates with logarithmic dependence on the initial distance and independence from high-dimensional ambient spaces.
• The work validates reverse Fisher divergence as a robust loss for generative modeling by eliminating spurious stationary points and ensuring stable optimization.

Global Convergence Analysis of Gradient Descent for Score Matching in GMMs via Reverse Fisher Divergence

Overview

This paper presents a detailed theoretical investigation of the optimization properties of gradient descent (GD) when applied to the score matching objective under the reverse Fisher divergence for Gaussian mixture models (GMMs). The central thesis is that, contrary to the commonly used forward Fisher divergence, the reverse Fisher divergence yields a significantly more favorable optimization landscape, admitting global convergence guarantees from arbitrary initializations and providing dimension-independent rates in high-dimensional settings.

Score Matching and Divergence Objectives

Score matching is a pervasive training objective in modern generative modeling and diffusion models, with the goal of learning a model whose score function (gradient of log-density) closely approximates that of a target distribution, particularly valuable for unnormalized statistical models.

Two primary Fisher divergence-based objectives are considered:

• Forward Fisher Divergence: The expectation is taken with respect to the teacher distribution. Extant literature demonstrates that, for GMMs, GD applied to this objective often results in convergence behavior highly sensitive to initialization and can even exhibit pathological stationary points. This instability is notable even in the simple case of overparameterized mixtures targeting a single Gaussian component.
• Reverse Fisher Divergence: The expectation is instead computed over the student distribution. This paper rigorously proves that switching to this objective eliminates spurious stationary points and guarantees global convergence of GD, both for single-component and multi-component teacher GMMs.

Main Theoretical Results

Single Gaussian Target

When the teacher is a single Gaussian and the student is a GMM with fixed weights and isotropic covariances, the reverse Fisher divergence enables GD to achieve global convergence from any initialization, with each student mean converging to the target mean. Explicit convergence rates show logarithmic dependence on initial distance and independence from the ambient dimension (when $d$ is sufficiently large):

• GD iterates satisfy $$|\mu_{k,i} - \mu_1|^2 \leq \max \left\{ (1 - 2\gamma)^{k/2} n(R_0)^2, \frac{8n^2}{\min\{n,d\}} \right\}$$, where $R_0$ is the worst-case initial distance.
• The Lyapunov-based proof establishes monotonicity and sublinear decay of the error.

Multiple Gaussian Components

When both teacher and student are GMMs with arbitrary numbers of components, the paper establishes high-probability global convergence under random initialization and a separation condition on the target means:

• If initial student means are sampled uniformly from a large-radius sphere, with high probability, each student mean converges to a neighborhood of its closest teacher mean.
• Provided that the target means are sufficiently separated ($\delta_{\text{min}}\geq 2\Omega(1)$) and $d \geq 2\Omega(n)$, the convergence rate is $O(n/\varepsilon)$ with only logarithmic dependence on initialization radius and independence from dimension for $d \gg n$.
• The proof relies on a two-stage error control, with linear contraction in the global stage and refined convergence in the local stage as components cluster.

Total Variation and Component Allocation

For sufficiently many student components ($n \gg m$ and $n > m^2 \log 2m$), the paper quantifies the convergence in total variation distance: the student distribution approaches the teacher's up to error controlled by initialization and the allocation probability of student means. In symmetric cases (e.g., equally spaced, orthogonal means), the student GMM matches the teacher GMM in TV up to arbitrarily small $\varepsilon$.

Algorithmic Feasibility

Despite the reverse Fisher divergence requiring expectation over the student, the paper shows that, for GMMs, gradients are tractable and comparable to the forward Fisher case via analytic expressions; practical computation can exploit sampling or quadrature methods. The authors propose zeroth-order approximations for more general models.

Implications and Discussion

The theoretical guarantees provided in this paper stand in stark contrast to the known limitations of the forward Fisher divergence, which often necessitates careful initialization, local convergence analysis, and suffers from non-convexity-induced artifacts. The results here validate the reverse Fisher divergence as an exceptionally robust loss for score matching in mixture models, unlocking the following implications:

• Generative Modeling: Robust convergence from arbitrary initialization is salient for training diffusion models and score-based generative models, particularly when component count and separation are unknown or overparameterized.
• Algorithm Design: The favorable optimization landscape motivates the development and adaptation of algorithms for reverse divergence-based training, potentially with stochastic gradient variants and extensions to broader distribution families.
• Theoretical Developments: The Lyapunov and matrix-based analysis technique may generalize to more intricate non-isotropic cases and non-Gaussian mixtures.
• Practical Deployment: The independence from high dimensionality and stability under random initialization will facilitate scalable and automated training procedures in large-scale applications.

Contradictions and Bold Claims

The claim of global convergence for GD under the reverse Fisher divergence from arbitrary initialization directly contradicts established impossibility results for the forward divergence even in basic settings. This stark improvement is grounded in the paper's rigorous analytic proofs and numerical experiments.

Future Directions

The extension to general distribution classes, optimization under stochastic gradient noise, and adaptation to practical large-scale generative modeling scenarios are identified as important directions for further research. Analysis of the impact of low target separation and more general covariance structures remains an open challenge.

Conclusion

This paper substantiates the reverse Fisher divergence as an optimally-behaved objective for score matching in GMMs. The elimination of local maxima and spurious stationary points, combined with strong global convergence properties and algorithmic practicality, position this objective as a superior alternative to the classic forward divergence in generative modeling. The theoretical machinery presented offers a path forward for robust training and further advances in both theoretical and applied machine learning.