Mixing Times 5 – Cesaro Mixing

We have just finished discussing chapters 11 and 12 of Markov Chains and Mixing Times, the end of the ‘core material’. I thought that, rather than addressing some of the more interesting but technical spectral methods that have just arisen, it would be a good subject for a quick post to collate some of the information about Cesaro mixing, which is spread throughout this first section.

Idea

A main result in the introductory theory of Markov chains is that for an irreducible aperiodic chain X, the distribution of X_t\rightarrow \pi, the (unique) equilibrium distribution. The mixing time gives the rate at which this first mode of convergence takes place. We have freedom over the initial state, so we typically consider the ‘worst case scenario’, ie the slowest convergence. The most appropriate metric is given by the total variation distance, which is defined in previous posts. The most important point to note is that the mixing time should be thought of as the correct timescale for convergence, rather than some threshold. In particular, the time at which the chain is within 1/4 of the equilibrium distribution in the TV metric has the same order magnitude (in n, some parameter controlling the number of states) as the time at which it is within 1/20 of the equilibrium distribution.

But this isn’t the only result about convergence in distribution of functionals of Markov chains. Perhaps more intuitive is the ergodic theorem which asserts that the proportion of time spent in a particular state also converges to the equilibrium probability as time advances. We might write:

\frac{1}{t}\sum_t \mathbf{1}(X_t=x)\rightarrow \pi(x),\quad \forall x\in \Omega.

Note that if the state space \Omega is finite, then we can also assume that this occurs uniformly in x. We can also think of the LHS of this convergence as a measure on \Omega varying in time

\frac{1}{t}\sum_t \mathbf{1}(X_t=\cdot),

and the mixing time for this sequence of measures is defined as for the conventional mixing time, and is called the Cesaro mixing time, at least in the Levin / Peres / Wilmer text.

Advantages

There are some obvious advantages to considering Cesaro mixing. Principally, a main drawback of conventional mixing is that we are unable to consider periodic chains. This property was the main content of the previous post, but to summarise, if a chain switches between various classes in a partition of the state space in a deterministic periodic way, then the distribution does not necessarily converge to equilibrium. The previous post discusses several ways of resolving this problem in specific cases. Note that this problem does not affect Cesaro mixing as the ergodic theorem continues to hold in the periodic case. Indeed the form of the distribution (which we might call an occupation measure in some contexts) confirms the intuition about viewing global mixing as a sort of sum over mixing modulo k in time.

Other advantages include the fact that the dependence on the initial state is weaker. For instance, consider the occupation measures for a chain started at x which moves first to y, versus a chain which starts at y then proceeds as the original. It requires very little thought to see that for O(1) values of t, this difference in occupation measures between these chains becomes small.

Another bonus is that we can use so-called stationary times to control Cesaro mixing. A stationary time is a stopping time such that X_\tau\stackrel{d}{=}\pi. It is clear that if we wait until \tau, then run the chain for a further \alpha\tau, the chain will have spent \frac{\alpha}{1+\alpha} of its duration in the equilibrium distribution, and so using Markov’s inequality and bounding the total variation distance between occupation measure and \pi by 1 up until the stationary time, we can get good bounds for the Cesaro mixing time in terms of \mathbb{E}\tau.

Why does this fail to work for normal mixing? The key to the above argument was that by taking an average over time up to some T>>\tau, the dependence on the actual value of X_\tau was suppressed. Consider the deterministic walk on the cycle \mathbb{Z}_n, which advances by 1 modulo n on each go. Now sample independently a random variable Z distributed uniformly on \mathbb{Z}_n. By definition, the random hitting time \tau_Z is a stationary time, but in fact the chain’s distribution does not converge. The condition we actually require for normal mixing is that \tau be a strong stationary time, meaning that X_\tau\stackrel{d}{=}\pi, and the value of X_\tau is independent of \tau. With this definition we can proceed with a similar result for normal mixing. An example of a strong stationary time would be for shuffling a pack of cards by repeatedly inserting the top card into a random place in the rest of the pile. Then consider the moment at which the original bottom card first reaches the top of the pile. It does not take too much to reassure oneself that after the next move, we have a strong stationary time, since every card has been randomised at least once, and the position of the other cards is independent of how long it took the original bottom card to rise to the top.

Disadvantages

So why do we not consider Cesaro mixing rather than the conventional variety? Well, mainly because of how we actually use mixing times. The Metropolis algorithm gives a way to generate chains with a particular equilibrium distribution, including ones for which it is hard to sample directly. Mixing time theory then gives a quantitative answer to the question of how long it is necessary to run such a chain for before it gives a good estimate to the equilibrium distribution. In many cases, such a random walk on a large unknown network, the main aim when applying such Monte Carlo procedures is to minimise the difficulty of calculation. For Cesaro mixing, you have to store all the information about path states, while for conventional mixing you only care about your current location.

The other phenomenon that is lacking in Cesaro mixing is cutoff. This is where the total variation distance

d_n(t)=||P^t(x,\cdot)-\pi||_{TV}

converges to 0 suddenly. More formally, there is some timescale f(n) such that

\lim_{n\rightarrow\infty}d_n(cf(n))=\begin{cases}1& c<1\& c>1,\end{cases}

so in the n-limit, the graph of d looks like a step-function. Several of the shuffling chains exhibit this property, leading to the statements like “7 shuffles are required to mix a standard pack of cards”. Cesaro mixing smooths out this effect on an f(n) timescale.

A Further Example

Perhaps the best example where Cesaro mixing happens faster than normal mixing is in the case of a lazy biased random walk on \mathbb{Z}_n. (Ex. 4.10 in MTMC) Here, we stay put with probability 1/2, otherwise move clockwise with probability p>1/4 and counter-clockwise with probability 1/2 – p < 1/4. This chain is not reversible, as we can determine the direction (or arrow) of time by examining a path. Roughly speaking, the chain will drift clockwise at rate 2p – 1/2 > 0. In particular, at some time Kn, where K is large, we would expect to have completed ~ \frac{K}{2p-\frac12} circuits of the vertices, and so the occupation measure will be close to the uniform equilibrium distribution if we choose K large enough.

On the other hand, the distribution of X at time Kn is still fairly concentrated. If we assume we are instead performing the random walk on \mathbb{Z}, the distribution after Kn i.i.d. increments is

X_t\sim N(Kn(2p-\frac12), Kn\sigma^2).

That is, the standard deviation is O(\sqrt{n})<<n. So, even once we return to considering the random walk on the cyclic group, if we view it as a circle, we expect most of the probability mass to be concentrated near Kn(2p-\frac12) \mod n. By an identical heuristic argument, we see that the mixing time is achieved when the variance has order n, that is when time has order n^2.

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Mixing Times 4 – Avoiding Periodicity

A Markov chain is periodic if you can partition the state space such it is possible to be in a particular class only at certain, periodic times. Concretely, suppose we can find a decomposition into classes \Omega=V_1\cup\ldots\cup V_k such that conditional on X_t\in V_i, we have \mathbb{P}(X_{t+1}\in V_{i+1})=1, where the indices of the Vs are taken modulo k. Such a chain is called periodic with period k. In most cases, we would want to define the period to be the maximal such k.

Why is periodicity a problem? It prevents convergence to equilibrium. The distribution at time t has some fairly strong dependence on the initial distribution. For example, if the initial distribution is entirely supported on V_1 as defined above, then the distribution at time t will be entirely supported on V_i, where i\equiv t \mod k. In particular, this cannot converge to some equilibrium.

Aperiodicity thus becomes a necessary condition in any theorem on convergence to equilibrium. Note that by construction this is only relevant for chains in discrete time. In an first account of Markov chains, most of the examples will either have a small state space, for which the transition matrix will have to contain lots of zeros before it stands a chance of being periodic, or obviously aperiodic birth-death or queue type processes. But some of the combinatorially motivated chains we consider for interesting mixing properties are more likely to be periodic. In particular, for a random walk on a group say, the generator measure may well be supported only on a small subset of the whole group, which is completely natural (eg transpositions as a subset of the symmetric group). Then it becomes more plausible that periodicity might arise because of some underlying regularity or symmetry in the group structure.

My first claim is that periodicity is not a disaster for convergence properties of Markov chains. Firstly, by the definition above, P^k(x,y) for x,y\in V_1 is an irreducible (aperiodic if k is maximal) transition matrix on V_1, and so we have convergence to some equilibrium distribution on V_1 of (X_{kt+a}) or similar. An initial distribution mixed between classes gives a mix of such equilibria. Alternatively, we could think about large-time ergodic properties. By taking an average over all distributions up to some large t, the periodic problems get smoothed out. So, for mixing on a periodic chain, it might be possible to make headway with Cesaro mixing, which looks at the speed of convergence of the ergodic average distribution.

In most cases, though, we prefer to alter the chain directly to remove periodicity, or even any chance of periodicity. The preferred method in many contexts is to replace the transition matrix P with \frac12 (P+I). This says that at every time t, we toss an independent fair coin, and with probability 1/2 make the transition suggested by P, and with probability 1/2 we stay where we are. Note that if a chain is irreducible, and some P(x,x)>0, then it is definitely aperiodic, as x cannot be in more than one class as per the definition of periodicity.

If you want to know about the mixing time of the original chain, note that this so-called lazy chain moves at half the speed of the original, so to get exact asymptotics (eg in the case of cutoff, that is mixing speed faster than the scale of the mixing time) you must multiply by 2. Also note that all of the eigenvalues of \frac12 (P+I) are non-negative, and in fact, the eigenvalues are subject to a linear transform in the construction of the lazy transition matrix \lambda\mapsto \frac12(1+\lambda).

Note that choosing 1/2 as the parameter is unnecessary. Firstly, it would suffice to take some P(x,x)=\epsilon and rescale the rest of the row appropriately. Also, in some cases, a different constant gives a more natural interpretation of the underlying mechanism. For example, one model worth considering is the Random Transposition Random Walk on the symmetric group, where at time t we multiply (ie compose) with a transposition chosen uniformly at random. This model is interesting partly because the orbits of an element resemble, at least initially, the component size process of a Erdos-Renyi random graph, on the grounds that when the number of transpositions is small, they don’t interact too much, so can be viewed as independent edges. Anyway, some form of laziness is needed in RTRW, otherwise the chain will alternative between odd and even permutations. In this case, 1/2 is not the most natural choice. The most sensible way to sample random transpositions is to select the two elements of [n] to be transposed uniformly and independently at random. Thus each transposition is selected with probability \frac{2}{n^2}, while the identity, which corresponds to ‘lazily’ staying at the current state in the random walk, is selected with probability 1/n.

The lazy chain is also useful when the original chain has a lot of symmetry involved. In particular, if the original chain involves ‘switching’ say one coordinate. The best example is the random walk on the vertices of the n-hypercube, but there are others. Here, the most helpful way to visualise the configuration is to choose a coordinate uniformly at random and then flip its value (from 0 to 1 or 1 to 0). Now the lazy chain can be viewed similarly, but note that the dependence on the current value of the coordinate is suppressed. That is, having chosen the coordinate to be affected, we set it to be 0 with probability 1/2 and 1 with probability 1/2, irrespective of the prior value at that coordinate. Thus instead of viewing the action on coordinates to be ‘stay or switch’, we can view the action on the randomly chosen coordinate to be ‘randomly resample’, to use statistical terminology. This is ideal for coupling, because from the time coordinate j is first selected, the value at that coordinate is independent of the past, and in particular, the initial value or distribution. So we can couple arbitrary initial configurations or distributions, and we know that as soon as all coordinates have been selected (a time that can be described as a coupon collector problem), the chains are well coupled, that is, the values are the same.

Note that one way we definitely get periodicity is if the increment distribution for random walk on a group is supported entirely in a single coset of a normal subgroup. Why? Well if we take H\lhd G to be the normal subgroup, and gH to be the relevant coset, then P^t(g',\cdot) is supported entirely on g^tg'H, so is periodic with period equal to the order of gH in the quotient group G/H. Note that if the coset is the normal subgroup itself, then it might well include support on the identity, which immediately makes the chain aperiodic. However, there will be then be no transitions between cosets, so the chain is not irreducible on G.

The previous paragraph is the content of Remark 8.3 in the book we are reading. My final comment is that normality is precisely what is needed for this to hold. The key idea is that the set of subsets {gH, gHgH, gHgHgH, … } forms a partition of the group. This is certainly true if H is normal and gH generates G/H. If the latter statement is not true, then the set of subsets still forms a partition, but of some subset of G. The random walk is then neither irreducible nor aperiodic on the reduced state space. If H is not normal, then there are no such restrictions. For example, gHgH might be equal to the whole group G. Then the random walk is aperiodic, as this would imply we can move between any pair of states in two steps, and so by extension between any pair of states in three steps. (2,3)=1, hence the chain is aperiodic. As a concrete example, consider

\tau=\langle (1 2)\rangle \leq S_3,

the simplest example of a non-normal subgroup. Part of the problem is that cosets are different in the left-case and the right-case. Consider the left coset of \tau given by \sigma\tau=\{(1 2 3),(2 3)\}. These elements have order three and two respectively, and so by a similar argument to the general one above, this random walk is aperiodic.