• Corpus ID: 203610262

# An Efficient Sampling Algorithm for Non-smooth Composite Potentials

@article{Mou2019AnES,
title={An Efficient Sampling Algorithm for Non-smooth Composite Potentials},
author={Wenlong Mou and Nicolas Flammarion and Martin J. Wainwright and Peter L. Bartlett},
journal={ArXiv},
year={2019},
volume={abs/1910.00551}
}
• Published 1 October 2019
• Mathematics, Computer Science
• ArXiv
We consider the problem of sampling from a density of the form $p(x) \propto \exp(-f(x)- g(x))$, where $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth and strongly convex function and $g: \mathbb{R}^d \rightarrow \mathbb{R}$ is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis-Hastings framework, and prove that it mixes to within TV distance $\varepsilon$ of the target density in at most $O(d \log (d/\varepsilon))$ iterations. This guarantee extends…

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## References

SHOWING 1-10 OF 88 REFERENCES

### Structured Logconcave Sampling with a Restricted Gaussian Oracle

• Computer Science, Mathematics
COLT
• 2021
A reduction framework is developed, inspired by proximal point methods in convex optimization, which bootstraps samplers for regularized densities to improve dependences on problem conditioning and gives algorithms for sampling several structured logconcave families to high accuracy.

### Convergence of Langevin MCMC in KL-divergence

• Computer Science, Mathematics
ALT
• 2018
By considering the Langevin diffusion as a gradient flow in the space of probability distributions, an elegant analysis is obtained that applies to the stronger property of convergence in KL-divergence and gives a conceptually simpler proof of the best-known convergence results in weaker metrics.

### Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlo

• Mathematics, Computer Science
NeurIPS
• 2018
This work shows that the conjecture that Hamiltonian Monte Carlo can be run in gradient evaluations when sampling from strongly log-concave target distributions that satisfy a weak third-order regularity property associated with the input data, and suggests that leapfrog HMC performs better than its competitors when this condition is satisfied.

### The geometry of logconcave functions and sampling algorithms

• Computer Science, Mathematics
Random Struct. Algorithms
• 2007
These results are applied to analyze two efficient algorithms for sampling from a logconcave distribution in n dimensions, with no assumptions on the local smoothness of the density function.

### Algorithmic Theory of ODEs and Sampling from Well-conditioned Logconcave Densities

• Mathematics, Computer Science
ArXiv
• 2018
This paper gives an improved contraction bound for the exact HMC process and logarithmic bounds on the degree of polynomials that approximate solutions of the differential equations arising in implementing HMC to obtain a nearly linear implementation of HMC for a broad class of smooth, strongly logconcave densities.

### Proximal Markov chain Monte Carlo algorithms

This paper presents a new Metropolis-adjusted Langevin algorithm (MALA) that uses convex analysis to simulate efficiently from high-dimensional densities that are log-concave, a class of probability

### Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

• Mathematics, Computer Science
• 2015
For both constant and decreasing step sizes in the Euler discretization, non-asymptotic bounds for the convergence to the target distribution $\pi$ in total variation distance are obtained.

### A Hamiltonian Monte Carlo Method for Non-Smooth Energy Sampling

• Computer Science, Mathematics
IEEE Transactions on Signal Processing
• 2016
This paper addresses the problem of using Hamiltonian dynamics to sample from probability distributions having non-differentiable energy functions such as those based on the l1 norm and shows its ability to accurately sample according to various multivariate target distributions.

### Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting

• Computer Science
48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)
• 2007
It is proved any non-adaptive cooling schedule has length at least O*(ln A), and an algorithm to find an adaptive schedule of length O* (radicln A) and a nearly matching lower bound is presented.

### Rates of convergence of the Hastings and Metropolis algorithms

• Mathematics, Computer Science
• 1996
Recent results in Markov chain theory are applied to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and it is shown geometric convergence essentially occurs if and only if $pi$ has geometric tails.