Randomized Algorithms (Spring 2010)/Random sampling

Conductance

Recap

• A Markov chain with finite space [math]\displaystyle{ \mathcal{S} }[/math], where [math]\displaystyle{ N=|\mathcal{S}| }[/math], transition matrix [math]\displaystyle{ P }[/math], whose eigenvalues are [math]\displaystyle{ \lambda_1\ge\cdots\ge\lambda_N }[/math], stationary distribution [math]\displaystyle{ \pi }[/math].
• The mixing time [math]\displaystyle{ \tau(\epsilon)\, }[/math]: time to be close to [math]\displaystyle{ \pi }[/math] within total variation distance [math]\displaystyle{ \epsilon }[/math], starting from a worst-case state.

Conditions:

• Lazy random walk: [math]\displaystyle{ P_{i,i}\ge\frac{1}{2} }[/math] for any [math]\displaystyle{ i\in\mathcal{S} }[/math], so [math]\displaystyle{ 1=\lambda_1\ge\cdots\ge\lambda_N\ge 0 }[/math] and [math]\displaystyle{ \lambda_{\max}=\max(|\lambda_2|,|\lambda_N|)=\lambda_2\, }[/math].
• The Markov chain is time-reversible: [math]\displaystyle{ \pi_{i}P_{i,j}=\pi_{j}P_{j,i} }[/math] for all [math]\displaystyle{ i,j\in\mathcal{S} }[/math].
• The stationary [math]\displaystyle{ \pi }[/math] is the uniform distribution, that is, [math]\displaystyle{ \pi_i=\frac{1}{N} }[/math] for all [math]\displaystyle{ i\in\mathcal{S} }[/math].

Then:

Theorem The mixing time [math]\displaystyle{ \tau(\epsilon)=O\left(\frac{\ln N+\ln(1/\epsilon)}{1-\lambda_2}\right) }[/math]

Conductance and the mixing time

For many problems, such as card shuffling, the state space is exponentially large, so the estimation of [math]\displaystyle{ \lambda_2 }[/math] becomes very difficult. The following technique based on conductance is due to Jerrum and Sinclair, which is fundamental for the theory of rapid mixing random walks.

Definition (conductance) The conductance of a irreducible Markov chain with finite state space [math]\displaystyle{ \Omega }[/math], transition matrix [math]\displaystyle{ P }[/math], and stationary distribution [math]\displaystyle{ \pi }[/math], is defined as [math]\displaystyle{ \Phi=\min_{\overset{S\subset\Omega}{0\lt \pi(S)\le\frac{1}{2}}} \frac{\sum_{i\in S, j\not\in S}\pi_ip_{ij}}{\pi(S)} }[/math] where [math]\displaystyle{ \pi(S)=\sum_{i\in S}\pi_i }[/math] is the probability density of [math]\displaystyle{ S\subset \Omega }[/math] under the stationary distribution [math]\displaystyle{ \pi }[/math].

The definition of conductance looks quite similar to the expansion ratio of graphs. In fact, for the random walk on a undirected [math]\displaystyle{ d }[/math]-regular graph [math]\displaystyle{ G }[/math], there is a straightforward relation between the conductance [math]\displaystyle{ \Phi }[/math] of the walk and the expansion ratio [math]\displaystyle{ \phi(G) }[/math] of the graph

[math]\displaystyle{ \Phi=\frac{\phi(G)}{d}. }[/math]

Very informally, the conductance can be seen as the weighted normalized version of expansion ratio.

The following theorem states a Cheeger's inequality for the conductance.

Theorem (Jerrum-Sinclair 1988) In a reversible Markov chain, [math]\displaystyle{ 1-2\Phi\le\lambda_2\le 1-\frac{\Phi^2}{2} }[/math].

The inequality can be equivalent written for the spectral gap:

[math]\displaystyle{ \frac{\Phi^2}{2}\le1-\lambda_2\le 2\Phi }[/math]

thus a large conductance implies a large spectral gap, which in turn implies the rapid mixing of the random walk.