We investigate the Langevin dynamics for Wigner matrices with a spherical spike, in the regime where the signal-to-noise ratio $θ$ is large, but order one. For large, order-$1$, signal-to-noise, the (worst-case) mixing time undergoes a sharp transition around the critical inverse temperature $β_c(θ) = \frac{1}θ$. Namely, if $β= α/θ$, and $α<1$ then at large $θ$ the mixing time is $O(\log N)$, and if $α>1$ it is exponential in $N$. We show that initialized from the uniform-at-random spherical prior, however, the mixing time in the low-temperature $α>1$ regime circumvents the exponential bottleneck and the mixing time is $O(\log N)$. In fact, this fast mixing holds for any initialization that is symmetric with respect to the top eigenvector of the spiked matrix. Using this, we are able to show a low-temperature metastability picture, pinning down the exact exponential rate of the (worst-case initialization) mixing time for low temperatures, showing it is given by the difference of the free energies of the spiked and null models.
Langevin dynamics for high-dimensional optimization: the case of multi-spiked tensor PCA
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Unité de Mathématiques Pures et Appliquées (UMPA), ENS Lyon（里昂高等师范学院纯数学与应用数学研究所）
Langevin algorithms are gradient descent methods augmented with additive noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling, optimization, and machine learning. In recent years, the non-asymptotic analysis of Langevin algorithms for non-convex learning has been extensively explored. For constrained problems with non-convex losses over a compact convex domain with IID data variables, the projected Langevin algorithm achieves a deviation of $O(T^{-1/4} (\log T)^{1/2})$ from its target distribution [27] in $1$-Wasserstein distance. In this paper, we obtain a deviation of $O(T^…
Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in t…
Bayesian methods of sampling from a posterior distribution are becoming increasingly popular due to their ability to precisely display the uncertainty of a model fit. Classical methods based on iterative random sampling and posterior evaluation such as Metropolis-Hastings are known to have desirable long run mixing properties, however are slow to converge. Gradient based methods, such as Langevin Dynamics (and its stochastic gradient counterpart) exhibit favorable dimension-dependence and fast mixing times for log-concave, and "close" to log-concave distributions, however also have long escape…
Microsoft Research（微软研究院）；Weizmann Institute（魏茨曼科学研究所）；Université Paris-Dauphine（巴黎第九大学）
We extend the Langevin Monte Carlo (LMC) algorithm to compactly supported measures via a projection step, akin to projected Stochastic Gradient Descent (SGD). We show that (projected) LMC allows to sample in polynomial time from a log-concave distribution with smooth potential. This gives a new Markov chain to sample from a log-concave distribution. Our main result shows in particular that when the target distribution is uniform, LMC mixes in $\tilde{O}(n^7)$ steps (where $n$ is the dimension). We also provide preliminary experimental evidence that LMC performs at least as well as hit-and-run,…
Duke University（杜克大学）；Princeton University（普林斯顿大学）；Massachusetts Institute of Technology（麻省理工学院）
A key task in Bayesian statistics is sampling from distributions that are only specified up to a partition function (i.e., constant of proportionality). However, without any assumptions, sampling (even approximately) can be #P-hard, and few works have provided "beyond worst-case" guarantees for such settings. For log-concave distributions, classical results going back to Bakry and Émery (1985) show that natural continuous-time Markov chains called Langevin diffusions mix in polynomial time. The most salient feature of log-concavity violated in practice is uni-modality: commonly, the distributi…
National University of Singapore（新加坡国立大学）；Nanjing University（南京大学）
Given a target function $H$ to minimize or a target Gibbs distribution $π_β^0 \propto e^{-βH}$ to sample from in the low temperature, in this paper we propose and analyze Langevin Monte Carlo (LMC) algorithms that run on an alternative landscape as specified by $H^f_{β,c,1}$ and target a modified Gibbs distribution $π^f_{β,c,1} \propto e^{-βH^f_{β,c,1}}$, where the landscape of $H^f_{β,c,1}$ is a transformed version of that of $H$ which depends on the parameters $f,β$ and $c$. While the original Log-Sobolev constant affiliated with $π^0_β$ exhibits exponential dependence on both $β$ and the en…
Massachusetts Institute of Technology（麻省理工学院）；Carnegie Mellon University（卡内基梅隆大学）
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 …
Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization
Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T. Hong Kong（系统工程与工程管理系，香港中文大学（深圳））；Department of Management Science and Information Systems and the DIMACS Institute, Rutgers University, Piscataway, NJ-08854, United States of America（管理科学与信息系统系及DIMACS研究所，罗杰斯大学）；Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL-32306, United States of America（数学系，佛罗里达州立大学）
We show that mixing for local, reversible dynamics of mean field spin glasses is exponentially slow in the low temperature regime. We introduce a notion of free energy barriers for the overlap, and prove that their existence imply that the spectral gap is exponentially small, and thus that mixing is exponentially slow. We then exhibit sufficient conditions on the equilibrium Gibbs measure which guarantee the existence of these barriers, using the notion of replicon eigenvalue and 2D Guerra Talagrand bounds. We show how these sufficient conditions cover large classes of Ising spin models for re…
