[Paper Review] Fast Convergence for Langevin Diffusion with Manifold Structure

Ankur Moitra, Andrej Risteski|arXiv (Cornell University)|Feb 13, 2020

Markov Chains and Monte Carlo Methods33 references4 citations

TL;DR

This paper establishes fast mixing time bounds for Langevin diffusion in nonlog-concave distributions with manifold structure due to symmetries, such as orthogonal invariance in matrix factorization problems. By analyzing the geometry of the level sets and leveraging stochastic differential equations, the authors prove polynomial-time sampling guarantees under mild conditions, extending rapid mixing beyond convexity to symmetric, nonconvex settings relevant in Bayesian inference and energy-based models.

ABSTRACT

In this paper, we study the problem of sampling from distributions of the form p(x) \propto e^{-βf(x)} for some function f whose values and gradients we can query. This mode of access to f is natural in the scenarios in which such problems arise, for instance sampling from posteriors in parametric Bayesian models. Classical results show that a natural random walk, Langevin diffusion, mixes rapidly when f is convex. Unfortunately, even in simple examples, the applications listed above will entail working with functions f that are nonconvex -- for which sampling from p may in general require an exponential number of queries. In this paper, we focus on an aspect of nonconvexity relevant for modern machine learning applications: existence of invariances (symmetries) in the function f, as a result of which the distribution p will have manifolds of points with equal probability. First, we give a recipe for proving mixing time bounds for Langevin diffusion as a function of the geometry of these manifolds. Second, we specialize our arguments to classic matrix factorization-like Bayesian inference problems where we get noisy measurements A(XX^T), X \in R^{d imes k} of a low-rank matrix, i.e. f(X) = \|A(XX^T) - b\|^2_2, X \in R^{d imes k}, and βthe inverse of the variance of the noise. Such functions f are invariant under orthogonal transformations, and include problems like matrix factorization, sensing, completion. Beyond sampling, Langevin dynamics is a popular toy model for studying stochastic gradient descent. Along these lines, we believe that our work is an important first step towards understanding how SGD behaves when there is a high degree of symmetry in the space of parameters the produce the same output.

Motivation & Objective

• To address the challenge of efficient sampling from nonlog-concave distributions arising in Bayesian inference and energy-based models.
• To identify structural conditions—specifically, symmetry-induced manifolds of equal probability—that enable fast mixing despite nonconvexity.
• To develop a general framework linking the geometry of the distribution’s level sets to the mixing time of Langevin diffusion.
• To establish provable polynomial-time sampling guarantees for matrix factorization-like problems with orthogonal invariance.

Proposed method

• Derives a general recipe for bounding mixing time of Langevin diffusion based on the Riemannian geometry of the distribution’s level sets and symmetry manifolds.
• Analyzes the continuous-time Langevin SDE $ dX_t = -\beta \nabla f(X_t) dt + \sqrt{2} dB_t $, showing that curvature and symmetry reduce mixing time.
• Uses Itô’s lemma to bound moments of the process, proving $ \mathbb{E}[\|X_t\|_F^p] \leq \text{poly}(d, \|M\|_F) $, which controls the discretization error.
• Establishes a discretization error bound via a coupling argument, showing $ \text{KL}(\hat{p}_T, p_T) \leq \text{poly}(d, \|M\|_F) \cdot h^2 $, where $ h $ is the step size.
• Applies the framework to matrix factorization problems with $ f(X) = \|\mathcal{A}(XX^T) - b\|_2^2 $, showing invariance under orthogonal transformations.
• Uses Pinsker’s inequality and moment bounds to prove that the discretized Langevin dynamics mixes in polynomial time when $ f $ has manifold symmetry.

Experimental results

Research questions

• RQ1 Can Langevin dynamics achieve fast mixing in nonlog-concave distributions that exhibit symmetry-induced manifolds of equal probability?
• RQ2 How does the geometry of the level sets and symmetry manifolds affect the mixing time of Langevin diffusion?
• RQ3 What conditions on the function $ f $, such as matrix manifold structure, allow for provably fast sampling despite nonconvexity?
• RQ4 To what extent can the mixing time of discrete Langevin dynamics be bounded in terms of the step size and the underlying geometry?
• RQ5 How does the presence of orthogonal invariance in matrix factorization problems influence the convergence of sampling algorithms?

Key findings

• The paper establishes a general framework linking the Riemannian geometry of symmetry manifolds to the mixing time of Langevin diffusion.
• For matrix factorization problems with orthogonal invariance, the mixing time of Langevin dynamics is polynomial in dimension and problem parameters.
• The authors prove $ \mathbb{E}[\|X_t\|_F^p] \leq \|X_0\|_F^p e^{-\frac{4}{5}t \frac{p}{d^2}} + 25d^2\left(\frac{\|M\|_F}{p} + \frac{1}{\beta^2}dp\right) $, ensuring moment control.
• Discretization error is bounded as $ \text{KL}(\hat{p}_T, p_T) \leq \text{poly}(d, \|M\|_F) \cdot h^2 $, enabling polynomial-time sampling.
• The framework applies to concrete problems like matrix completion, matrix sensing, and noisy matrix factorization, all with provable fast mixing.
• The results suggest that symmetry and curvature in the parameter space can enable efficient sampling even when $ f $ is nonconvex, challenging the conventional view that nonlog-concavity implies intractability.