Blocking Conductance and Mixing in Random Walks

@article{Kannan2006BlockingCA,
  title={Blocking Conductance and Mixing in Random Walks},
  author={Ravi Kannan and L{\'a}szl{\'o} Mikl{\'o}s Lov{\'a}sz and Ravi Montenegro},
  journal={Combinatorics, Probability and Computing},
  year={2006},
  volume={15},
  pages={541 - 570}
}
The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum–Sinclair bound (where the average is taken over subsets of states with different sizes). Furthermore, instead of just the conductance… 
Mathematical Aspects of Mixing Times in Markov Chains
TLDR
The strength of the main techniques are illustrated by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.
Two conductance theorems, two canonical path theorems, and two walks on directed Cayley graphs
We show two Conductance-like theorems for mixing time of non-reversible non-lazy walks. These bounds involve a measure of expansion which expresses how well ergodic flow is distributed among
Random Walks and Evolving Sets: Faster Convergences and Limitations
TLDR
This work defines a combinatorial analog of the spectral gap, and uses it to prove the convergence of non-lazy random walks, and proves that random walks converge faster when the robust vertex expansion of the graph is larger.
Stochastic localization + Stieltjes barrier = tight bound for log-Sobolev
TLDR
It is proved that the log-Sobolev constant of any isotropic logconcave density in Rn with support of diameter D is Ω(1/D), resolving a question posed by Frieze and Kannan in 1997 and is asymptotically the best possible estimate.
Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing
We show that the Cheeger constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the previous best bound of $O(n^{1/3}\sqrt{\log n}).$ As corollaries we obtain the
The simple random walk and max-degree walk on a directed graph
We bound total variation and L∞ mixing times, spectral gap and magnitudes of the complex valued eigenvalues of general (nonreversible nonlazy) Markov chains with a minor expansion property. The
Conductance and canonical paths for directed non-lazy walks
We show two Conductance-like theorems for mixing time of finite non-reversible non-lazy Markov Chains, i.e. when the Markov kernel is neither self-adjoint nor positive definite. The first holds for
Random walks on polytopes and an affine interior point method for linear programming
TLDR
An affine interior point algorithm is designed that does a <i>single</i> random walk to solve linear programs approximately and has a run-time that is provably polynomial.
The simple random walk and max‐degree walk on a directed graph
TLDR
It is found that within a factor of two or four, the worst case of each of these mixing times, spectral gap and magnitudes of the complex valued eigenvalues of a general Markov chain is a walk on a cycle with clockwise drift.
Fast MCMC Sampling Algorithms on Polytopes
TLDR
It is shown that the Vaidya walk mixes in significantly fewer steps than the logarithmic-barrier based Dikin walk studied in past work, and an improved variant of it could achieve a mixing time of $\mathcal{O}(d^2\cdot\text{polylog}(n/d))$.
...
...

References

SHOWING 1-10 OF 48 REFERENCES
Evolving sets and mixing
TLDR
It is proved that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation of the distribution at time n from the stationary distribution π.
The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume
  • L. LovászM. Simonovits
  • Mathematics
    Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
  • 1990
TLDR
The authors generalize a bound on the mixing rate of time-reversible Markov chains in terms of their conductance by not assuming time reversibility and using a weaker notion of conductance and prove an isoperimetric inequality for subsets of a convex body.
Spectral gap and log-Sobolev constant for balanced matroids
  • M. JerrumJung-Bae Son
  • Mathematics
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
TLDR
Tight lower bounds are computed on the log-Sobolev constant of a class of inductively defined Markov chains, which contains the bases-exchange walks for balanced matroids studied by Feder and Mihail and improved upper bounds for the mixing time of a variety of Markov Chains are obtained.
On the mixing time of a simple random walk on the super critical percolation cluster
Abstract. We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Zd. We show that for d≥2 and p>pc(Zd), the
Random Walks in a Convex Body and an Improved Volume Algorithm
TLDR
A randomized algorithm using O(n7 log’ n) separation calls to approximate the volume of a convex body with a fixed relative error is given and the mixing rate of Markov chains from finite to arbitrary Markov Chains is analyzed.
Elementary bounds on Poincaré and log-Sobolev constants for decomposable Markov chains
We consider finite-state Markov chains that can be naturally decomposed into smaller ``projection'' and ``restriction'' chains. Possibly this decomposition will be inductive, in that the restriction
LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS
This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic
Vertex and edge expansion properties for rapid mixing
We show a strict hierarchy among various edge and vertex expansion properties of Markov chains. This gives easy proofs of a range of bounds, both classical and new, on chi‐square distance, spectral
Random walks and an O * ( n 5 ) volume algorithm for convex bodies
TLDR
This algorithm introduces three new ideas: the use of the isotropic position (or at least an approximation of it) for rounding; the separation of global obstructions and local obstructions for fast mixing; and a stepwise interlacing of rounding and sampling.
Efficient stopping rules for Markov chains
TLDR
A stopping rule is given which runs in time polynomial in the maximum hitting time of Al and achieves the stationary distribution exactly, even though the transition probabilities of the chain are unknown.
...
...