Fast Convergence for Langevin Diffusion with Matrix Manifold Structure
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In this paper, we study the problem of sampling from distributions of the form p(x) ∝ e^-β f(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 study one 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. We give a recipe for proving mixing time bounds of Langevin dynamics in order to sample from manifolds of local optima of the function f in settings where the distribution is well-concentrated around them. We specialize our arguments to classic matrix factorization-like Bayesian inference problems where we get noisy measurements A(XX^T), X ∈ R^d × k of a low-rank matrix, i.e. f(X) = A(XX^T) - b^2_2, X ∈ R^d × k, and β the 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.
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