Accelerating diffusions

Abstract

Let U be a given function defined on R^d and π(x) be a density function proportional to \exp -U(x). The following diffusion X(t) is often used to sample from π(x), dX(t)=-\nabla U(X(t)) dt+\sqrt2 dW(t),\qquad X(0)=x_0. To accelerate the convergence, a family of diffusions with π(x) as their common equilibrium is considered, dX(t)=\bigl(-\nabla U(X(t))+C(X(t))\bigr) dt+\sqrt2 dW(t),\qquad X(0)=x_0. Let L_C be the corresponding infinitesimal generator. The spectral gap of L_C in L^2(π) (λ(C)), and the convergence exponent of X(t) to πin variational norm (ρ(C)), are used to describe the convergence rate, where λ(C)= Sup{real part of μ\dvtxμis in the spectrum of L_C, μis not zero}, {-2.8cm}ρ(C) = Inf\biggl{ρ\dvtx\int | p(t,x,y) -π(y)| dy \le g(x) e^{ρt}\biggr}.Roughly speaking, L_C is a perturbation of the self-adjoint L_0 by an antisymmetric operator C\cdot\nabla, where C is weighted divergence free. We prove that λ(C)\le λ(0) and equality holds only in some rare situations. Furthermore, ρ(C)\le λ(C) and equality holds for C=0. In other words, adding an extra drift, C(x), accelerates convergence. Related problems are also discussed.

Published at http://dx.doi.org/10.1214/105051605000000025 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• General state space Markov chains and MCMC algorithms
• Existence of the spectral gap for elliptic operators
• Ergodic convergence rates of Markov processes--eigenvalues, inequalities and ergodic theory

Cited by in corpus (41)

• The free energy principle made simpler but not too simple
• Generalisation of the Eyring-Kramers transition rate formula to irreversible diffusion processes
• Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion
• Variance Reduction using Nonreversible Langevin Samplers
• Irreversible Langevin samplers and variance reduction: a large deviation approach
• Violation of detailed balance accelerates relaxation
• Improving the convergence of reversible samplers
• Characterising the nonequilibrium stationary states of Ornstein-Uhlenbeck processes
• Acceleration of convergence to equilibrium in Markov chains by breaking detailed balance
• Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions
• Role of current fluctuations in nonreversible samplers
• Methods and Conversations in (Post)Modern Thermodynamics
• Analytic theory of Itô-stochastic differential equations with non-smooth coefficients
• Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization
• Geometric Allocation Approaches in Markov Chain Monte Carlo
• Irreversible Samplers from Jump and Continuous Markov Processes
• Asymmetric Thermal Relaxation in Driven Systems: Rotations go Opposite Ways
• Accelerating, to some extent, the $p$-spin dynamics
• Irreversible Boltzmann samplers in dense liquids: weak-coupling approximation and mode-coupling theory
• Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents
• Filtering Dynamical Systems Using Observations of Statistics
• Transverse forces and glassy liquids in infinite dimensions
• Control of probability flow in Markov chain Monte Carlo -- Nonreversibility and lifting
• Non-reversible lifts of reversible diffusion processes and relaxation times
• Non-Convex Optimization via Non-Reversible Stochastic Gradient Langevin Dynamics
• Irreversible swap algorithms for soft sphere glasses
• Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach
• Lifted directed-worm algorithm
• Hessian-Free High-Resolution Nesterov Acceleration for Sampling
• A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit
• A Unifying and Canonical Description of Measure-Preserving Diffusions
• An improved variant of simulated annealing that converges under fast cooling
• On hitting time, mixing time and geometric interpretations of Metropolis-Hastings reversiblizations
• On the asymptotic variance of reversible Markov chain without cycles
• On the convergence of an improved and adaptive kinetic simulated annealing
• Large Deviations of Irreversible Processes
• Particular flows and attracting sets: A comment on "How particular is the physics of the Free Energy Principle?" by Aguilera, Millidge, Tschantz and Buckley
• Uniqueness in law for singular degenerate SDEs with respect to a (sub-)invariant measure
• Markov chain entropy games and the geometry of their Nash equilibria
• Improved annealing for sampling from multimodal distributions via landscape modification
• Adaptive Non-reversible Stochastic Gradient Langevin Dynamics