S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Our hope is that these methods may also be able to give bounds that are more realistic than previous PAC estimates. Finally we remark that recently a very nice result concerning the stability of the VC dimension on a random sample has been obtained using concentration of measure techniques [6]. We conjecture that one may be able to obtain REFERENCES 25 refinements of the results of the present paper concerning empirical covering numbers using those techniques. ....

S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Preprint, CNRS-Universit'e Paris-Sud, April 1999.

....c 2001 Academie des sciences Editions scientifiques et medicales Elsevier SAS. Tous droits reserves. 1 O. Bousquet 2. R esultats Principaux Le premier resultat donne une inegalite de type Bennett qui peut etre consideree comme une generalisation d un theoreme de Boucheron, Lugosi et Massart [1]. THEOREM 2.1. Soient (Z; Z 0 1 ; Z 0 n ) une s equence de v.a. A mesurables et (Z k ) k=1; n une s equence de v.a. respectivement A k n mesurables. Supposons qu il existe un r eel positif u tel que, pour tout k = 1; n les in egalit es suivantes sont vraies Z 0 k 6 ....

....k 1 ; X k 1 ; Xn ) We denote by E k n [ the expectation taken conditionally to A k n . Let h(x) 1 x) log(1 x) x, x) e x 1 x and (x) 1 (1 x)e x . 2. Main Results The first result can be considered as a generalization of a result of Boucheron, Lugosi and Massart [1] since it gives a Bennett type concentration inequality for Z under less restrictive conditions. THEOREM 2.1. Let (Z; Z 0 1 ; Z 0 n ) be a sequence of A measurable r.v. and let (Z k ) k=1; n be a sequence of r.v. respectively A k n measurable. Assume that there exists u 0 ....

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S. Boucheron, G. Lugosi, and P. Massart, A sharp concentration inequality with applications, Random Structures and Algorithms 16 (2000), no. 3, 277--292.

....explanations have also been proposed [16, 3] but more work on this aspect is still needed. Two important theoretical tools for studying the generalization performance of learning machines are the leave one out (or cross validation) error of the machines, and the stability of the machines [2, 1]. The second, although an older tool [7, 6] has become only important recently with the work of [11, 2] In this paper we study the generalization performance of ensembles of kernel machines using both leave one out and stability arguments. We consider the general case where each of the machines ....

S. Boucheron, G. Lugosi, and P.M assart. A sharp concentration inequality with applications. Random Structures and Alg--KC)fi , 16:277--292, 2000.

....which could be explored in the same spirit. ffl Formulate and prove similar bounds in a functional setting. ffl Compute tight bounds on the expected value of the maximal deviation, which is a question of growing interest since the impressive recent results on concentration inequalities (see [3], 20] and their references) Moreover, we point out that theoretical analysis on VC bounds and VC dimension could benefit of an empirical study. Indeed, we have proposed in [29] to use computer simulations to estimate the probability of the event fsup C2 Gamma j n (C) Gamma (C)j fflg for ....

S. Boucheron, G. Lugosi, and P. Massart, A Sharp Concentration Inequality with Applications, Random Structures and Algorithms 16 (2000), 277--292.

....any r # R and t # 0, P(X # r t) P(X # r) # e t 2 4#(r) 5) In particular, if m is a median of X, then for every t # 0, P(X # m t) # 2e t 2 4#(m) 6) and P(X # m t) # 2e t 2 4#(m t) 7) 4 Remark 1. A recent inequality of Boucheron, Lugosi and Massart [2] is sometimes an interesting alternative to Talagrand s inequality; in several applications it yields essentially the same result (with better constants) We do not, however, see any way to use their inequality in the set up treated here. 2.2 Kim Vu concentration via average smoothness ....

S. Boucheron, G. Lugosi & P. Massart, A sharp concentration inequality with applications. Random Struct. Alg. 16 (2000), 277--292.

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S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.

....F k , then the upper bound has an improved rate of convergence. The disadvantage of the simple penalty de ned above is that instead of the expected shatter coecients, a distribution free (and therefore suboptimal) upper bound appears for each class F k . Recently, Boucheron, Lugosi and Massart [4] proved that log S k (X ) concentrates sharply around its mean. For example, we have the following inequalities: Proposition 3.3. For all 0, n 1, P [E log S k (X 1 ) 2 log S k (X P [log S k (X 1 ) 2E log S k (X Moreover, for each n 1, 1 ) log ES k (X 1 ) ln ....

....b R F to E sup f2F jL(f) b L(f)j. By a classical symmetrization device (cf. Gin e and Zinn [10] or Van der Vaart and Wellner [25] 2E b RF : 4.7) Also, b R F is known to concentrate sharply around its mean. For example, we have, by results of Boucheron, Lugosi, and Massart [4], 5] the following bounds. Proposition 4.4. For all 0, n 1, b RF k 2E b RF k 6n =5 b RF k E b RF k Proof. De ne Z = n b RF k , then it follows from Boucheron, Lugosi, and Massart [4] that log E exp( Z EZ) EZ(e 1 ) which implies further that for 0 3 ....

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S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277-292, 2000.

.... Moment inequalities, Concentration inequalities; Empirical processes; Random graphs Supported by EU Working Group RAND APX, binational PROCOPE Grant 05923XL The work of the third author was supported by the Spanish Ministry of Science and Technology and FEDER, grant BMF2003 03324 Lugosi [6, 7], Bousquet [8] Devroye [14] Massart [30, 31] Rio [36] for various applications. To derive moment inequalities for general functions of independent random variables, we elaborate on the pioneering work of Latala and Oleszkiewicz [25] and describe so called # Sobolev inequalities which ....

....theorems which provide explicit estimates under various typical conditions on the behavior of V , V , or V . The first corollary, obtained from Theorem 3, is concerned with functionals Z satisfying V Z. Such functionals were at the center of attention in Boucheron, Lugosi, and Massart [6] and Rio [36] where they were called self bounded functionals. They encompass sums of bounded non negative random variables, suprema of nonnegative empirical processes, configuration functions in the sense of Talagrand [39] and conditional Rademacher averages [7] see also Devroye [14] for other ....

S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.

....Working Group RAND APX, binational PICASSO Grant 025 43RM The work of the second author was supported by DGI grant BMF2000 0807 Ledoux [3] Massart [24] Rio [28] and Bousquet [5] for proving sharp concentration bounds for maxima of empirical processes. Recently Boucheron, Lugosi, and Massart [4] pointed out that the methodology may be used e ectively outside of the context of empirical process theory as well. In this paper we follow the line of [4] to develop new easy to apply inequalities for general functions of independent random variables. These inequalities may be considered as ....

....Rio [28] and Bousquet [5] for proving sharp concentration bounds for maxima of empirical processes. Recently Boucheron, Lugosi, and Massart [4] pointed out that the methodology may be used e ectively outside of the context of empirical process theory as well. In this paper we follow the line of [4] to develop new easy to apply inequalities for general functions of independent random variables. These inequalities may be considered as exponential versions of the well known Efron Stein inequality. One of the goals of this paper is to point out that the methodology based on logarithmic Sobolev ....

[Article contains additional citation context not shown here]

S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications in random combinatorics and learning. Random Structures and Algorithms, 16:277{ 292, 2000.

....results of Shawe Taylor and Williamson [16] who obtained generalization bounds in terms of the margin error (f) and empirical covering numbers, but the new bounds are sharper and more general. The improvement was made possible by some recent concentration inequalities for combinatorial entropies [7]. The rest of the paper is organized as follows: In Sections 2 and 3 we present the main data dependent upper bounds for the probability of misclassification. In Section 4 we also develop an alternative data based bound which provides a more easily computable data dependent bound on the ....

....hence each g Q, 2(7 l) maps X into C. Thus by the special choice a = if 2 of the quantization step size, T(x) is just the cardinality of the set Q, 2( c R. More explicitly, T(xD: I( g(Xl) g(x) e C: g e (10) implying (since ICI = 5) that log, T(x) is a combinatorial entropy in the sense of [7]. Thus by [7, Theorem 2] the random variable log, T(X) satisfies the concentration inequality t 2 P(log, T(X)E[log, T(X) t) exp(2E[log(X) 11) Thus, for any u 0 and v 0, using (9) we obtain P(sup(L(f) f) SlnT(X) u fe n ( SlnT(X) p 2ElnT(X ) The first probability ....

S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277-292, 2000.

....Working Group RAND APX, binational PICASSO Grant 025 43RM 2 The work of the second author was supported by DGI grant BMF2000 0807 1 2 S. Boucheron, G. Lugosi and P. Massart [21] for proving sharp concentration bounds for maxima of empirical processes. Recently Boucheron, Lugosi, and Massart [3] pointed out that the methodology may be used e#ectively outside of the context of empirical process theory as well. In this paper we follow the line of [3] to develop new easy to apply inequalities for general functions of independent random variables. These inequalities may be considered as ....

....Lugosi and P. Massart [21] for proving sharp concentration bounds for maxima of empirical processes. Recently Boucheron, Lugosi, and Massart [3] pointed out that the methodology may be used e#ectively outside of the context of empirical process theory as well. In this paper we follow the line of [3] to develop new easy to apply inequalities for general functions of independent random variables. These inequalities may be considered as exponential versions of the well known Efron Stein inequality. One of the goals of this paper is to point out that the methodology based on logarithmic Sobolev ....

[Article contains additional citation context not shown here]

S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications in random combinatorics and learning. Random Structures and Algorithms, 16:277-- 292, 2000.

....of a subsample of that satis es the hereditary property p. As cubes are drawn independently, generic concentration of measure arguments for con guration function tell us that the typical deviations of X d n should be at most of order 0( p E[X d n ] Namely, applying results from[10] or [2]: Proposition 1.2. The independence number of a random sample of n squares in T d satis es for all positive t: P X d n E[X d n ] t exp t 2 2(E[X d n ] t=3) P X d n E[X d n ] t exp t 2 2E[X d n ] Con guration functions show up frequently ....

Boucheron, S., Lugosi, G. and Massart, P. (2000) A sharp concentration inequality with applications. Random Structures and Algorithms. 16, 277-292.

....Ef(X 1 ; Xn ) tg e 2t 2 P n i=1 c 2 i : McDiarmid s inequality is convenient when f( has variance ( P n i=1 c 2 i ) In other situations when the variance of f is much smaller, the following inequality might be more appropriate. Theorem 4 (BOUCHERON, LUGOSI, AND MASSART [10]) Suppose that X 1 ; Xn are independent random variables taking values in a set A, and that f : A n R is such that there exists a function g : A n 1 R such that for all x 1 ; xn 2 A (1) f(x 1 ; xn ) 0; 2) 0 f(x 1 ; xn ) g(x 1 ; x i 1 ; x i 1 ....

....2 is used with R n;k = b Ln ( b f k ) r 12 log S k (X n 1 ) log 4 n and m = n=80. Then EL(fn ) L min k r 12E log S k (X n 1 ) log 4 n inf f2Fk L(f) L 8:95 r log k n # 8:23 p n : The key ingredient of the proof is a concentration inequality from [10] for the random VC entropy, log 2 S k (X n 1 ) Proof. We need to check the validity of Assumption 1. It is shown in [10] that f(x 1 ; xn ) log 2 S k (x n 1 ) satisfies the conditions of Theorem 4. First note that ES k (X 2n 1 ) E 2 S k (X n 1 ) and therefore log ES k (X ....

[Article contains additional citation context not shown here]

S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications in random combinatorics and learning. Random Structures and Algorithms, to appear, 2000.

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S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.

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S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.

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Boucheron, S., Lugosi, G., Massart, P.: A sharp concentration inequality with applications. Random Structures and Algorithms 16 (2000) 277--292

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S. Boucheron, G. Lugosi, and P. Massart, A sharp concentration inequality with applications, Random Struct. Algorithms (2000), 277--292.

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S. Boucheron, G. Lugosi, and P. Massart, A sharp concentration inequality with applications, Random Struct. Algorithms (2000), 277--292.

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S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.

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S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with applications. Technical Report NC2-TR-1999-057, NeuroCOLT2, 1999.

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S. Boucheron, G. Lugosi, and S. Massart. A sharp concentration inequality with applications. Random Structures and Algorithms, 16:277--292, 2000.

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S. Boucheron, G. Lugosi & P. Massart, A sharp concentration inequality with applications. Random Struct. Alg. 16 (2000), 277-292.

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S. Boucheron, G. Lugosi and P. Massart. A sharp concentration inequality with applications in random combinatorics and learning. Random Structures and Algorithms, 16:277-292, 2000.

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Boucheron, S., Lugosi, G., Massart, P., A sharp concentration inequality with applications, Random Structures Algorithms, 16 (2000), 277 - 292.

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Boucheron, S., Lugosi, G., Massart, P., A sharp concentration inequality with applications, Random Structures Algorithms, 16 (2000), 277 - 292.

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