C. McDiarmid. On the method of bounded differences, Surveys in Combinatorics, London Math. Soc. Lecture Notes Series 141:148--188, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....properties of the sequence fM x ( GammaK) K 2 Ng, we cannot expect to control this speed using large deviations or central limit theorems. In fact, we apply a martingale method based on Azuma s inequality. For a survey on the method, called there method of bounded differences , see McDiarmid [33]. For an utilization in a closely related model (percolation) and the proof of a stronger inequality than the one used here, see Kesten [24] First of all, we need to truncate the service times. We define, for a 0, s(i; j) min s(i; j) jij jjj) 34) We denote by S ; M x ; ....

....random variable and let fY n ; n = 1; Ng be the associated martingale difference sequence. Then for all u 0, P fjX Gamma E[X ]j ug = P fj n=1 Y n j ug 2 exp n=1 kY n k where kY k = inffc : jY j c; P Gamma a:s:g. For a proof of Lemma 5. 12, see for instance [33], Lemma 4.1. Let F k ; k 2 N ; be the sub oe algebra of F generated by fs(i; j) Gammak i j Gamma1g and let F 0 = g. Any directed path intersects the set fs(i; j) i j = Gammakg at one point at most. For a given K 2 N , the r.v. M x ( GammaK) K is F [ 1 x)K] measurable. We ....

C. McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, number 141 in London Math. Soc. Lect. Note Series, pages 148--188. Camb. Univ. Press, 1989.

....variables taking values in f0; 1g. Let f : CN Gamma R be a function such that jf(x) Gamma f(y)j 1 whenever dist(x; y) 1 and let j be the random variable f( 1 ; N ) Then for any ffi 0 j : jj Gamma E (j)j ffi Gamma2ffi Proof. This is a special case of Lemma 1. 2 of [McDiarmid 89] The next lemma provides a useful scaling trick. CN = C n Theta : Theta C n = C n ) Thus a point x 2 CN is identified with a k tuple x = x 1 ; x k ) where x i 2 C n for i = 1; k. For a subset A ae C n , let B = A Theta : Theta A = A dist(x i ; A) ....

....x = 1 ; N ) and l = 1 ; N ) Let i i = i i . Then i i , i = 1; N are independent random variables such that P fi i = 1g = p=2 and P fi i = 0g = 1 Gamma p=2. Hence i 1 : i N r(ffl) by a corollary of Hoeffding s inequality (see Corollary 5. 6 of [McDiarmid 89] Now we are ready to prove the first part of Theorem 4.5. Proof of inequality (4.5.1) Let us choose a positive integer m, let N = mn, let , and let N = n ) ae CN as in Lemma 6.3. Let us choose an ff 0. Applying Lemma 6.4, we obtain N : d l (x; B) pN(1 Gamma jBje ....

C. McDiarmid, On the method of bounded differences, in: Surveys in Combinatorics, 1989 (Norwich, 1989), 148--188, London Math. Soc. Lecture Note Ser., 141, Cambridge Univ. Press, Cambridge, 1989.

.... x(j) In order to enhance the understanding consider the collection C (Xl, hl) x2, hl) x3, 1) x4, 1) and a ranking function f such that f(x3) f(x2) f(xl) f(x4) Then [x(1)x(2)x(3)x(4) x3,x2,xl,x4] x(1) x(2) x2,xl] fl ,f2 ] If (x2) f (Xl) and [i(1) i(2) [2,3]. In order to evaluate the quality of the ranking produced by f in C, we are going to use the previously defined measures of precision and recall. For this we introduce a threshold b R and construct the new classification function: hb(X) sign (f(x) b) 1.3) Precision and recall now ....

....m L. zooo Figure 2.1: Contour plot of (2.2) for Ay (C) The probability that the expected average precision lies within of the found average precision is represented for a range of , ra and six different cases of ra . In our notation, the results of McDiarmid reads as follows (see [1] or [3] for more details) For any i, let Ti be defined such that v (x,y) vc: k: IA (C) Af (Ci (x,y) Ti, 2.1) where Ci(x,y) is the document collection with the ith example replaced by (x, y) In other words, i bounds the maximal change in average precision for a ranking function f if the ....

C. McDiarmid. On the method of bounded differences. In Survey in Combinatorics, pages 148-188. Cambridge University Press, 1989.

....absence and presence of noise, is retained as shown below. The results given below are proven almost identically to their counterparts by simply ap plying Hoeffding and Chernoff style bounds for bounded real random variables. The following is a simple extension of results contained in McDiarmid [23]: 1The range [0, M] is used so that we can derive efficient simulations of relative error SQ algorithms. For additive error SQ algorithms, one may consider any interval [a, b] where M = b a. 67 Extensions Theorem 14 Let X1, X2, X m be independent and identically distributed random ....

Colin McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, pages 149-188. Cambridge University Press, Cambridge, 1989. London Mathematical Society LNS 141.

....proof uses ideas similar to Theorem 6 of [4] Remark 1. Theorem 3 is, modulo a small constant factor, always at least as good as Theorem 1. This may be seen by observing that by concentration of sup c: k, n 2 J 2 (f) which can be easily quantified using the bounded difference inequality [14]) fc with very large probability. This is essentially the same quantity as E supyc: Ln(f) n(f) appearing in the proof of Theorem 2; only the sam ple size n has now been replaced by n 2. 4 Convex hulls In this section we consider an important class of special cases. Let 7 be a class of ....

....that all f :7: satisfy 1 sup n 2(h) V is at least 1 5, where 2 (1) h 2 n12 n12V : Zi=1I sgn(h(Xi) 1 2) The proof is based on arguments of [12] and concentration inequalities. Remark 2. To interpret this new bound note that, for all 5 0, by the bounded difference inequality [14], with probability at least 1 5, The expectation on the right hand side may be further bounded by the Vapnik Chervonenkis inequality (see [9] for this version) j(2) j(1) hh 8ElogS (XI ) Esup k 2 12v] where 7 (X) is the random shatter coefficient, that is, the number of ....

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C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics lgSg, pages 148-188. Cambridge University Press, Cambridge, 1989.

....range of p. Their paper was the first and still one of the most exciting applications of the Janson Inequality. 1.2 Martingale Inequalities Martingales have a long history in probability theory but their usefulness in our context is quite new. We refer to Colin McDiarmid s excellent survey [16] at this meeting for a more detailed examination. For our purposes we consider a martingale to be a sequence X 0 ; Xm of random variables (on a common space) so that for any 0 i m and value a E[X i 1 jX i = a] a. We further assume X 0 = a constant. Then = E[X i ] for all i. ....

C. McDiarmid, On the method of bounded differences, in J. Siemons, ed., Surveys in Combinatorics 1988.

....absence and presence of noise, is retained as shown below. The results given below are proven almost identically to their counterparts by simply applying Hoeffding and Chernoff style bounds for bounded real random variables. The following is a simple extension of results contained in McDiarmid [23]: The range [0; M ] is used so that we can derive efficient simulations of relative error SQ algorithms. For additive error SQ algorithms, one may consider any interval [a; b] where M = b Gamma a. 67 68 Extensions Theorem 14 Let X 1 ; X 2 ; Xm be independent and identically ....

Colin McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, pages 149--188. Cambridge University Press, Cambridge, 1989. London Mathematical Society LNS 141.

.... nicely packaged generalization of a martingale inequality known as Azuma s inequality, allows one to consider any function f(X 1 ; X n ) under the additional bounded difference requirement that changing the choice of one of the variables does not affect the final value of Y by too much [18]. More precisely, the result states that if, for all vectors A and B differing only in the i th coordinate, jf(A) Gamma f(B)j c i then 4 jY Gamma Ex[Y ]j =2 5 2e : 1) This result was significantly strengthened by Kim for the case of 0 1 random variables and further ....

C. McDiarmid, On the method of bounded differences, In Surveys in Combinatorics, J.Siemons ed., London Math. Society Lecture Note Series 141 (1989), 148--188.

.... i: max j Exp(f(Y ) j Y 1 ; Y 2 ; Y i 1 ) Gamma Exp(f(Y ) j Y 1 ; Y 2 ; Y i ) j c i then the probability that j f Gamma Exp(f) j t is at most: 2 exp Gammat 2 2 P c 2 i For more details on this corollary and an excellent discussion of Martingale arguments see either [16] or [2] Remark 1.4 We often apply this corollary to show concentration of variables which are functions of random permutations. Typically, these applications have the following flavour. Suppose Z is a function of a random permutation OE : W U , where W = fw 1 ; w n g. We choose our ....

C. McDiarmid. On the Method of Bounded Differences. Surveys in Combinatorics, Proceedings of the Twelfth British Combinatorial Conference (1989), 148 - 188.

.... j Sigma 1 ; Sigma 2 ; Sigma i ) j c i where E(f) denotes the expected value of f , then the probability that j f Gamma E(f) j t is at most: 2 exp Gammat 2 2 P c 2 i For more details on this corollary and an excellent discussion of martingale arguments see either [16] or [5] In order to prove Lemma 3, we will analyze the Markov process described in Section 1. Recall that X i is the number of open vertex copies after i pairs of our configuration have been exposed. Similarly, we let Y i be the number of backedges formed, and C i be the number of components ....

C. McDiarmid. On the Method of Bounded Differences. Surveys in Combinatorics, Proceedings of the Twelfth British Combinatorial Conference (1989), 148 - 188.

....The phenomenon of measure concentration has recently received distinguished attention due to its much better understanding and its spectacular power and simplicity in applications. The basic methods for proving concentration inequalities have been (1) martingale methods see McDiarmid [22], 23] for excellent surveys; 2) information theoretic methods, see Alhswede, G acs, and Korner [1] Marton [16] 17] 18] Dembo [4] and Massart [20] 3) Talagrand s induction method [27] 25] 26] which led to a large variety of powerful new inequalities. Recently, a new proof technique ....

....set is the independence number of the graph and it is denoted by ff(G) To show that the independence number can be regarded as a configuration function, we merely have to check that the G(n; p) model may be regarded as an inhomogeneous product probability space. This is well known (see, e.g. [22]) the i th component of the probability space just defines the set of edges between vertex i and vertices with index j i. Thus, Z = ff(G) satisfies the inequalities of Theorem 2, regardless of the values of n and p. Results concerning the average value of ff(G) and concentration of ff(G) ....

C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics

....= x 0 ) 1 k (6) and so E X (Z) p(p Gamma 1) k : It follows from (5) that E (Z 1 ) k Gamma 1) p Gamma1 X i=i0 1 i exp ae Gamma 4(k Gamma 2) log k k oe p 2 k 3 (7) for large k; p. We continue by using the Azuma Hoeffding Martingale tail inequality see for example [1, 2, 7, 8, 9]. Let x 0 be fixed and for a given X let X be obtained from X by replacing x j by randomly chosen x j . For j 1 let d j = max X fjE x j (Z(X) Gamma Z( X) jg: Then for any t 0 we have P(jZ 0 Gamma E (Z 0 )j t) 2 exp ae Gamma 2t 2 d 2 1 Delta Delta Delta d 2 ....

C.J.H. McDiarmid, On the method of bounded differences, Surveys in Combinatorics,

....follows trivially from the fact that, by the definition of domponents, the arc e being added (removed) cannot be traversed (have been traversed) during the determination of the domponents more than log 3 n times. Given the above claim, we can apply the following inequality of McDiarmid [29] to get that the probability of each D i deviating by n 3=4 is bounded by exp( n 1=5 ) The union bound then implies that w.h.p. no D i deviates by that much. 17 Theorem 6.6 ( 29] Let X = X 1 ; X 2 ; X n ) be a family of independent random variables with each X k taking values in ....

....domponents more than log 3 n times. Given the above claim, we can apply the following inequality of McDiarmid [29] to get that the probability of each D i deviating by n 3=4 is bounded by exp( n 1=5 ) The union bound then implies that w.h.p. no D i deviates by that much. 17 Theorem 6. 6 ([29]) Let X = X 1 ; X 2 ; X n ) be a family of independent random variables with each X k taking values in a set A k . Suppose that the real valued function f defined on Q A k satisfies jf(x) f(x 0 )j c k whenever the vectors x and x 0 differ only in the k th coordinate. Let be ....

C. J.H. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, Proceedings of the 12th British Combinatorial Conference, pages 148--188. 1989.

....space for which these facts apply is the product of the random planted bisection graph model with the random choices made by the algorithm. Sections 4.1 through 4.3 analyze successive phases of the algorithm. The following version of Azuma s Inequality, which is in a form due to McDiarmid [18] (see also [12] is used throughout. Theorem 1 (Azuma s Inequality) Let Z 1 ; Z n be independent random variables, with Z k taking values in a set A k for each k. Suppose that the (measurable) function f : Q A k R satisfies jf(x) Gamma f(x 0 )j c k whenever the vectors x and x 0 ....

C. McDiarmid. "On the method of bounded differences," in London Society Lecture Note Series, Volume 141, Cambridge University Press, 1989, 148--188.

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C. McDiarmid. On the method of bounded differences, Surveys in Combinatorics, London Math. Soc. Lecture Notes Series 141:148--188, 1989.

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McDiarmid, C. J. H. (1989). On the method of bounded differences. In Surveys in Combinatorics: Invited Papers at the 12th British Combinatorial Conference (pp. 148--188).

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C. McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, volume 141 of LMS Lecture Notes Series, pages 148--188. 1989.

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C. J.H. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics, Proceedings of the 12th British Combinatorial Conference, pages 148--188. Cambridge University Press, 1989.

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C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989.

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C. McDiarmid, "On the method of bounded differences," in Surveys in Combinatorics 1989.

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C. McDiarmid. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, volume 141 of London Math. Soc. Lecture Note Series, pages 148--188. Cambridge University Press, Cambridge, 1989.

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C. McDiarmid.On the method of bounded differences.In Surveys in Combinatorics 1989.

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C. McDiarmid, On the method of bounded differences, in Surveys in Combinatorics 1989, 141 London Mathematical Society Lecture Notes Series, Cambridge University Press, Cambridge, (1989) 148-188.

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McDiarmid C: On the method of bounded differences. In Siemons J (ed) Surveys in Combinatorics, London Math. Society Lecture Note Series 141, 1989, pp 148--188

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C. McDiarmid, "On the Method of Bounded Differences", in J. Siemons (ed) Surveys in Combinatorics, London mathematical Society lecture Note Series 141, 1989.

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