D. Aldous, "On the markov chain simulation method for uniform combinatorial distributions and simulated annealing," Probab. Engrg. Inform. Sci., 1, pp 33-46, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= 1) or the constant zero function (h(x) 0) if the number of positive examples seen differs significantly from the number of negatives seen. Otherwise, their algorithm outputs the single variable function (f(x) x i ) that has highest observed correlation with the data. By results of Aldous [Ald86] there must exist some variable with correlation Omega Gammar =n) and thus their error is 1=2 Gamma Omega Gamma7 =n) Bshouty and Tamon [BT96] improve on this guarantee using results of Kahn, Kalai, and Linial [KKL88] They demonstrate an algorithm which outputs linear threshold functions ....

D Aldous. On the markov chain simulation method for uniform combinatorial distributions and simulated annealing. Technical Report 60, Univ California at Berkeley, 1986.

....the size of an arbitrary connected graph in O(n ) time and a regular graph in O(n ) time. COMPONENT was used as a subroutine for testing graph connectivity. For expander graphs, we can apply Gillman [21] s Chernoff bound for random walks. Adapting Gillman s modified Aldous s procedure [22], we can show that, in O(# n log n) time and with O(# n) messages, a node can estimate the size of the graph within error #n with high probability. b) Covering Walk: We are currently working on a new distributed estimation algorithm based on long lived random walkers on the graph. A walker ....

D. Aldous, "On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing," Probability in the Engineering and Informational Sciences, vol. 1, pp. 33--46, 1987.

....group G and symmetric generator set S of G let (G;S) be the undirected connected Cayley graph of G generated by S. Let d denote the diameter of (G;S) with respect to the shortest path metric. The simple random walk on (G;S) produces nearly uniform distribution after O(jSjd log jGj) steps [1, 2]. The diameter d can be as large as O(jGj) an example is a cyclic group generated by f1; 1g. In this case the necessary number of steps for a simple random walk on a Cayley graph to approximate the uniform distribution is a polynomial of jGj. Our goal is to nd faster algorithms which can ....

D. Aldous, On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Probability in Engineering and Informational Sciences 1 (1987), pp. 33-46.

....= 1) or the constant zero function (h(x) 0) if the number of positive examples seen differs significantly from the number of negatives seen. Otherwise, their algorithm outputs the single variable function (f(x) x i ) that has highest observed correlation with the data. By results of Aldous [1] there must exist some variable with correlation Omega Gammao =n) and thus their error is 1=2 Gamma Omega Gamma7 =n) Bshouty and Tamon [4] improve on this guarantee using results of Kahn, Kalai, and Linial [5] They demonstrate an algorithm which outputs linear threshold functions and ....

D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Technical Report 60, Univ California at Berkeley, 1986.

....trees with O(log n) labels and O(log n) depth. This is always the case, in the context of a base change, and was the basis for a fast randomized base change algorithm [18, 22] The following key observation was discovered independently, but variations of it have appeared from as early as 1965 [1, 7, 25]. Let G act transitively on For xed P and random g 2 G, E(jP g P j) jP j j nP j=n: We identify P with the current nodes of a partially built Schreier tree. As long as fewer than half of the elements of are in the tree, each random element has a xed positive probability of expanding ....

D. Aldous, On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Prob. Eng. and Informational Sciences 1 (1987), 33-46.

....of the present section is to explain the connection between the edge expansion of a graph and the second largest eigenvalue of a certain matrix, which will be relevant in Section 3. This connection originates in Alon s and Milman s work [4, 5] and was speci cally adapted for our context by Aldous [3]. Our treatment closely follows Behrend s book [7] Let G = V; E) be a graph (without loops or multiple edges) on n : jV j nodes. We de ne a random walk (i.e. transition probabilities for all edges in both directions) on G in a canonical way. Let max be the maximum degree of a vertex in G. ....

D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Eng. Inf. Sci., 1:33-46, 1987.

....matrix with (i; j) entry ffi i(j) this is one if item j is ranked in position i and zero otherwise. Given a function f : X N, then the 4 Theta 4 matrix t = X 2Sn f( T ( is the first order summary reported above. We identify f with the monomial Q x f( in the variables x = [ 1 2 3 4 ], 2 X . The permutation group was ordered using lex order (1234 1243 Delta Delta Delta 4321) Then grevlex order was used on monomials in K[X ] The computer program MACAULAY found a Grobner basis containing 199 binomials. There were 18 quadratic relations (example [3421] 4312] ....

Aldous, D. (1987). On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Prob. in Eng. and Info. Sci. 1, 33-46.

....pseudorandomness is expansion. Graphs with good expansion have the nice property, among other things, that a random walk on the nodes of the graph is rapidly mixing, i.e. it gets very close to the stationary distribution after a number of steps that is only polylogarithmic on the size of the graph [Alo86, Bro86, Ald87, JS88]. Definition 1 The expansion v of a graph G(V; E) with n vertices is: v = min jSj jV j 2 ae jN (S)j jSj oe (6) where N (S) is the set of vertices in S which are adjacent to some vertex in S. Let P be the transition matrix of an ergodic Markov Chain X with states V = f1; 2; ng. ....

....the chain converges independently of the initial distribution. Denote by Delta(t) the distance between t and at time t: Delta(t) max i j i;t Gamma i j i . The lemma that relates the expansion of a graph to the speed of a convergence to the stationary distribution is: Lemma 6 ([Alo86, Bro86, Ald87, JS88]) Let G be a graph on n vertices and P the transition matrix of a random walk on G. If G has expansion v then Delta(t) 1 Gamma v 2 6 ) t min i i Corollary 2 For t = 6 v 2 (ln ffl Gamma1 ln Gamma1 min ) Delta(t) ffl. At this point we will replace the assumption of ....

D. Aldous. On the markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences, 1:33--46, 1987.

....the subset S of the state space in one step, given that it is initially in S . Thus Phi measures the ability of the chain to escape from any small region of the state space, and hence to make rapid progress to equilibrium. The following result formalising this intuition is from [23, 24] see also [2, 3, 4, 16, 19, 21] for related results. Theorem 2 The second eigenvalue 1 of a reversible Markov chain satisfies 1 Gamma 2 Phi 1 1 Gamma Phi 2 2 : Note that Phi characterises the rapid mixing property: a Markov chain is rapidly mixing, in the sense of the previous section, if and only if Phi ....

Aldous, D. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences 1 (1987), pp. 33--46. 19

....trees with O(log n) labels and O(logn) depth. This is always the case, in the context of a base change, and was the basis for a fast randomized base change algorithm [18, 22] The following key observation was discovered independently, but variations of it have appeared from as early as 1965 [1, 7, 25]. Let G act transitively on Omega Gamma For fixed P Omega and random g 2 G, E(jP g Gamma P j) jP j j Omega nP j=n: We identify P with the current nodes of a partially built Schreier tree. As long as fewer than half of the elements of Omega are in the tree, each random element has ....

D. Aldous, On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Prob. Eng. and Informational Sciences 1 (1987), 33--46.

....there is an edge in E connecting the two vertices in question) See [Bollobas 1979] for details and other standard terminology. Alon and Milman also proved a discrete version of Cheeger s inequality [1970] related to isoperimetric inequalities on manifolds. Connecting expansion with mixing time, Aldous [1987] showed that random walks on graphs with good expansion properties of low degree mix rapidly. Building on these ideas, Jerrum and Sinclair [1989; Sinclair and Jerrum 1989] defined the conductance of a random walk, and showed that large conductance implies fast mixing rate. Our aim here is to ....

Aldous, D. (1987). On the markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Engrg. Inform. Sci. 1, 33--46.

....we will have a second sample which is nearly independent of the first and is from a distribution with M distance at most again to Q. This improves steps needed for subsequent samples by a factor of O (n) Something akin to this happens for general Markov Chains, as first pointed out in [1] for discrete chains. Here, we will follow the simple proof in [13] First, we need a definition of a particular weak form of independence that turns out to be sufficient for us. Definition : Suppose X;Y are two random variables (with values in two possibly different oe Gamma algebras) and any ....

....us estimate the integrals of such functions over the orthant. The connection between sampling and integration is in a sense a natural and old one, but the point here is to describe a scheme which given a polynomial time sampling algorithm, can estimate the integral in polynomial time. For p 2 [0 1], define f(p) Z R n e Gammapx T Ax Gamma(1 Gammap)jxj 2 dx = Z R n e Gammax T Mx dx; where M = pA (1 Gamma p)I is a positive definite matrix. f (0) is known and we are interested in f(1) Let X be a (n Gammavector) random variable with values in R n and with ....

D. Aldous, "On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing," Probability in the Engineering and Informational Sciences 1, pp. 33-46, 1987.

....and denote by Z the random variable Z = X n X n 1 . X 1 which is the product of the various sample means. Then, since the random variables X i are independent, the expectation 2 For a more detailed discussion of the problem of inferring information from observations of a Markov chain, see [Ald87, Gill93, Kah94]. 488 CHAPTER 12 THE MARKOV CHAIN MONTE CARLO METHOD of Z is E Z = # n # n 1 . # 1 = # (b) 1 , and Var Z (E Z) 2 = n Y i =1 1 Var X i # 2 i # 1# 1 # 2 17n n 1# # 2 16 , assuming # # 1. By Chebyshev s inequality, this implies that Pr (1 # 2) # ....

....state space in one step, given that it is initially in S; thus, # measures the readiness of the chain to escape from any small enough region of the state space, and hence to make rapid progress towards equilibrium. This intuitive connection can be given a precise quantitative form as follows. See [Ald87, Alon86, AM85, Che70, LS88] for related results. PROPOSITION 12.2 Let M be a finite, reversible, ergodic Markov chain with loop probabilities P(x,x) # 1 2 for all states x . Let # be the conductance of M as defined in (12.3) Then the mixing time of M satisfies # x (#) # 2# 2 (ln#(x) 1 ln# 1 ) for any choice ....

D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Probability in the Engineering and Informational Sciences, 1:33--46, 1987.

....M(S;B) the bases exchange graph G(M) possesses cutsetexpansion inverse polynomial in jSj, then for any matroid M(S;B) there exists an efficient algorithm to approximate jBj, given an independence oracle for M. The above assertion follows by standard techniques on random walks on expanders [1] [16] 21] and the well known equivalence of uniform generation and approximate counting for self reducible combinatorial structures [3] 12] The reader is referred to [18] for further explanations. In particular, several unsolved counting problems (including network reliability which is known ....

D. Aldous, On the Markov chain Simulation Method for Uniform Combinatorial Distributions and Simulated Annealing, Probability in Eng. and Inf. Sci. 1, 1987, pp 33-46.

....probability of the set A, and by S l = A (X 1 ) A (X 2 ) Delta Delta Delta A (X l ) the number of steps the Markov chain is inside A. It is known [2] that, for any initial distribution, the fraction S l =l converges almost surely to (A) as l goes to infinity. This lead Aldous [3] to propose the following sampling technique: A) can be estimated by simulating the Markov chain for l steps and computing the fraction S l =l of steps it spends in A. Typically, the size of A is exponential in the input size (e.g. A is the set of matchings of a given size of a graph) and thus ....

....work was done while the author was at the Massachusetts Institute of Technology. time. It is therefore important to establish a bound on the probability that S l =l exceeds (A) by a given amount. A bound on the variance of S l in terms of the mixing properties of the chain was established in [3, 14]. An exponential bound on the tail of S l =l was established in [5, 11] in a special case, and in a more general setting in [7] In this paper, we establish a bound on the tail of the distribution of S l =l that beats the previously known bounds. As the previous bounds, ours depends on the second ....

D. Aldous. On the Markov Chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Information Sciences, 1:33--46, 1987.

....stationary distribution is the uniform over S. 3. Convergence to stationarity is rapid, in that after poly(log jSj) steps, the distance of the chain from the uniform stationary distribution is negligible for all practical purposes. Such a Markov chain is called rapidly mixing (see, e.g. [1] for technical details. Now the simulation of a rapidly mixing Markov chain can be used as an efficient sampling scheme for S, simply by simulating the chain for poly(log jSj) steps and using the final point of the chain as a random point of S. This method has been used extensively and with ....

....which would imply that they can be used as efficient sampling schemes for C, D, and C l , thus yielding polynomial sampling and total complexities for g 3 (m) g 4 (m) and g 5 (m) in Theorems 3.1, 3.2, 3.3, 3.4, and 3. 5 Various measures of convergence have been used elsewhere (e.g. see [1, 7]) here we use the natural measure that fits our algorithmic purposes: ffl For MC2, where G is the graph in question (recall G is in general G e1 111e k ) let d e (t) jf t : e 2 H( gj a t 0 t 1 jC e j a jCj : 16) ffl Where R is an integer sampling rate, let R denote ....

D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences, 1(1):33--46, 1987.

....[1996a, 1996b] for more information. The original Cheeger inequality, found independently by Cheeger [1970] and Poly a and Szego [1951] bounds the eigenvalues of the Laplacian on a Riemannian manifold. An early discrete version is due to Alon and Millman [1985] and Alon [1986] Aldous [1987] noted the relevance to mixing times. See Diaconis and Stroock [1991] for the exact version cited below and Chung [1997] for discussion. Define the Cheeger constant h of a Markov chain by h = min (S)1=2 Q(S Theta S C ) S) where the minimum is taken over subsets S of X, the vertex set of ....

....total number of path edges in Gamma (counted with multiplicity and directed) leaving any vertex is equal to P x;y2G d(x; y) jGj jGjD: The lemma follows because every directed edge leaves some vertex. 2 Bounds similar to the Poincar e eigenvalue bound in Lemma 4 have been obtained by Aldous [1987]. For random walks on groups generated by conjugacy classes, Diaconis and Saloff Coste [1996b] can get bounds of the form fi 1 1 Gamma 1 D 2 , thus eliminating the factor of d. Intuitively this is reasonable, since their proof technique uses flows (averaging over paths) and one could average ....

Aldous, D. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Engrg. Inform. Sci. 1 33-46.

....spent in A, and it quantifies the rate of convergence to #(A) in each L p norm, 1 # p # [Kah] It also sharpens a theorem of Ajtai, Komlos, and Szemeredi [AKS] see also [CW] and [IZ] which showed that the probability of a deviation of constant size decays exponentially in n. Aldous [Ald87] bounded the variance of the fraction of time spent in A in terms of the eigenvalue gap. Lovasz and Simonovits [LS] gave a similar result for arbitrary measure spaces. Those results give quadratic bounds on the deviation probability via Chebyshev s inequality. Goldreich et al. G # ] have given a ....

....the rapid mixing property of the random walk on G to generate a single nearly random sample point from #. The random walk is repeated to generate the number of independent sample points Cherno# s bound requires [DFK, JS89, LS] We compare this with the alternative procedure analyzed by Aldous in [Ald87] which was the first to generate a nearly random point in G and then to continue the random walk from that point, sampling every subsequent vertex. This procedure is commonly used in statistical biology and physics, usually without rigorous analysis of its reliability; for convenience we will ....

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<F3.746e+05> D.<F3.827e+05> Aldous,<F3.405e+05> On the Markov chain simulation method for uniform combinatorial distributions and simulated<F3.827e+05> annealing, Probab. Engrg. Inform. Sci., 1 (1987), pp. 33--46.

....time Z. Theorem 7. 2 For any starting distribution oe, the rule Gamma produces a distribution satisfying (1 Gamma ) and has mean length O(Tmix log(1= We note that similar convergence speed can be achieved using other averaging rules, e.g. the Poisson stopping time of Aldous [2]. 8 Exact Mixing in an Unknown Chain In the previous section we saw that there is at least one practical stopping rule for approximating the stationary distribution when all that is known about the chain is its mixing time (or a good upper bound for it) Even this requirement can be ....

D.J. Aldous, On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Probability in the Engineering and Informational Sciences, 1 (1987), 33--46.

....the pattern of faults or the topology of the network. Continuous methods. The techniques that we use to prove our results are related to those used by Mihail in [Mih89] to bound the convergence rate of random walks on graphs in terms of the expansion of the graph. Other previous work [Ald83, Ald87, Alo86, AD87, Mih89, JS88] related the convergence rate to the second eigenvalue of the underlying graph and then related the second eigenvalue to the conductance. Informally, Mihailshowed that a step in a random walk corresponds to a type of averaging of probabilities across edges in the ....

D. Aldous. On the markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in Eng. and Inf. Sci., 1:33--46, 1987.

....is very little rigorous work on the random walk method besides West s observation concerning the decoupled protocol. In Markov chain theory, it is now well understood that many common random walks do not converge in any satisfactory way and a theory of those that do is emerging [10] 1] [2] [14] 16] 6] 11] 12] In this context, West s example of a decoupled system is easily identified as the hypercube, the most simple case of a rapidly mixing random walk. The aim therefore is to use Markov chain theory to separate those protocols which can be effectively tested by random walk ....

Aldous, D., "On the Markov Chain Simulation Method for uniform combinatorial distributions and simulated annealing", Probability in Eng. and Inf. Sci. 1, 1987, pp 33-46.

....= 1) or the constant zero function (h(x) 0) if the number of positive examples seen differs significantly from the number of negatives seen. Otherwise, their algorithm outputs the single variable function (f(x) x i ) that has highest observed correlation with the data. By results of Aldous [1] there must exist some variable with correlation Omega Gammao =n) and thus their error is 1=2 Gamma Omega Gamma7 =n) Bshouty and Tamon [4] improve on this guarantee using results of Kahn, Kalai, and Linial [5] They demonstrate an algorithm which outputs linear threshold functions and ....

D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Technical Report 60, Univ California at Berkeley, 1986.

....useful indications of how K depends on the chain, and so don t help with Open Problem 30. The variance results in Proposition 29 are presumably classical, being straightforward consequences of the spectral representation. Their use in algorithmic settings such as Corollary 31 goes back at least to [2]. Section 4.3. Systematic study of the optimal choice of weights in the Cauchy Schwarz argument for Theorem 32 may lead to improved bounds in examples. Alan Sokal has unpublished notes on this subject. Section 5.1. The quantity 1= c , or rather this quantity with the alternate definition of c ....

D.J. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Engineering Inform. Sci., 1:33--46, 1987.

....of the graph. ii) c(A) 2dae(A) for all A with ae(A) 1, where ae(A) j min v2V max w2A d(v; w) is the radius of A. 15 Note that sup A ae(A) is bounded by Delta but not in general by Delta=2 (consider the cycle) so that (ii) implies (i) with an extra factor of 2. Part (i) is from Aldous [2] and (ii) is from Babai [5] Proof. i) Fix A. Because 1 n X v2V jA Avj = jAj 2 =n there exists some v 2 V such that jA Avj jAj 2 =n, implying jAv n Aj jAjjA c j=n: 23) We can write v = g 1 g 2 : g ffi for some sequence of generators (g i ) and some ffi Delta, and jAv n ....

D.J. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Engineering Inform. Sci., 1:33--46, 1987.

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D. Aldous, "On the markov chain simulation method for uniform combinatorial distributions and simulated annealing," Probab. Engrg. Inform. Sci., 1, pp 33-46, 1987.

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D. Aldous, "On the markov chain simulation method for uniform combinatorial distributions and simulated annealing," Probab. Engrg. Inform. Sci., 1, pp 33-46, 1987.

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D. Aldous. On the markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences, 1:33--46, 1987.

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D. Aldous, "On the markov chain simulation method for uniform combinatorial distributions and simulated annealing," Probab. Engrg. Inform. Sci., 1, pp 33-46, 1987.

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D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. University of California at Berkeley Statistics Department, technical report number 60, 1986.

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D. Aldous. On the markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probab. Engrg. Inform. Sci., 1:33-46, 1987.

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Aldous, D. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences 1 (1987), 33-- 46.

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Aldous D., \On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing", Probab. Eng. Inform. Sci. 1 (1987) 33-46.

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Aldous, D. (1987), "On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing", Probability in Eng. and Inf. Sci. 1, 33-46.

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D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Probability in the Engineering and Informational Sciences, 1:33--46, 1987.

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Aldous, D. (1987) On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. Pr. Eng. Inf. 1, 33--46.

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D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. University of California at Berkeley Statistics Department, technical report number 60, 1986.

No context found.

D. Aldous, On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing, Probability in the Engineering and Informational Sciences 1 (1987), 33-46.

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