S. Boucheron and D. Gardy, An urn model from learning theory, Random Struct. Alg. 10, 43-67, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in establishing a Sanov like theorem in a dependent context. Moreover, the asymptotic behavior of the random allocation model is interesting because random allocations models arise naturally in many applications running from algorithm analysis [29] to statistics and statistical learning theory [6]. In this paper, we illustrate our main result by deriving a large deviation principle (ldp) for the degree distribution of sparse random graphs. This complements recent results by McKay and Wormald on moderate deviations [18] 1.1. The limitations of Poisson approximation. The random allocations ....

S. Boucheron and D. Gardy. An urn model from learning theory. Random Structures and Algorithms, 1, 4367, (1997).

....variants of the classical random allocation process. Keywords: Stochastic differential equations, Infinite dimensional diffusions, Markov processes, Diffusion approximation, Urn models. 1. MOTIVATIONS Random allocation problems are naturally encountered in areas like statistical inference [21, 7, 4] (where they are implicitly used to to prove lower rates of convergence for regression or estimation problems) data structure analysis [9, 10, 11] or average case analysis of hard combinatorial problems [15, 20] Analyzing random allocation problems consist in investigating the random occupancy ....

....measure is called the empirical occupancy measure. As this random measure will be analyzed in an asymptotic setting, for large k and N , it is an element of 1 , the set real sequences (x i ) indexed by N, such that P i jx i j 1. As some the quantities investigated in applications [21, 7, 4, 11] are continuous (amd often linear) functionals of the empirical occupancy measure, it may be worth to get a characterization of the asymptotic behavior of this empirical occupancy measure. Then using the continuous mapping theorem, this characterization can be used to provide with ....

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S. Boucheron and D. Gardy. An Urn Model from Learning Theory. Random Structures and Algorithms, 10(1):43-25 1997.

....ad hoc treatment. Urns of balance q : There are again two types of balls; the balance of an urn is the relative di erence between the numbers of balls of each type, and the valuation is the number of urns with the speci ed balance : f(k 1 ; k 2 ) 1 [k1 =k2 q] Such models were introduced in [3] to study the behaviour of a learning process; they also appear in [6] The model we consider in the present paper di ers somewhat, in that here the number of balls of each type is known, whereas the former study assumed that only the total number of balls was known. It should be mentioned that ....

S. Boucheron and D. Gardy, An urn model from learning theory, Random Struct. Alg. 10, 43-67, 1997.

....in establishing a Sanov like theorem in a dependent context. Moreover, the asymptotic behavior of the random allocation model is interesting because random allocations models arise naturally in many applications running from algorithm analysis [29] to statistics and statistical learning theory [6]. In this paper, we illustrate our main result by deriving a large deviation principle (ldp) for the degree distribution of sparse random graphs. This complements recent results by McKay and Wormald on moderate deviations [18] 1.1. The limitations of Poisson approximation. The random allocations ....

S. Boucheron and D. Gardy. An urn model from learning theory. Random Structures and Algorithms, 1, 4367, (1997).

....of blue balls and the number of red balls, and the set E of required balances is a subset of the set of relative integers. For example, balanced urns are obtained for E = f0g, urns with balance r for E = frg, and urns with positive balance for E = N. For previous work dealing with this case see [3]. We shall prove in this paper that, when the balls are thrown at each unit time, the process associated to the number of urns with a specified number of balls or a specified balance converges weakly towards a Gaussian process, whose covariance matrix can be explicitely computed. 2. A General ....

.... of the urn, K l (z 1 ; z l ) X n1 ; n l I n1 (2z 1 ) I n l (2z l ) where the sum is for n 1 ; n l such that n 1 62 E , n 1 Delta Delta Delta n l Gamma1 62 E and n 1 Delta Delta Delta n l 2 E , and where the I n (z) are Bessel functions (see [3] for details) where we recall that for q 0 we have I q (2z) X n0 z 2n q n (n q) and K d 1 (z 1 ; z d ) X n1 ; n d I n1 (2z 1 ) I nd (2z d ) where the sum is for n 1 ; n d such that n 1 62 E , n 1 Delta Delta Delta n d 62 E . We will ....

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S. Boucheron and D. Gardy, An urn model from learning theory, Random Struct. Alg. 10, 43--67, 1997.

....of blue balls and the number of red balls, and the set E of required balances is a subset of the set of relative integers. For example, balanced urns are obtained for E = f0g, urns with balance r for E = frg, and urns with positive balance for E = N. For previous work dealing with this case see [3]. We shall prove in this paper that, when the balls are thrown at each unit time the process associated to the number of urns with a specified number of balls or a specified balance converges weakly towards a Gaussian process, whose covariance matrix can be explicitely computed. 2. A General ....

.... the urn, K l (z 1 ; z l ) X n1 ; n l I n1 (2z 1 ) I n l (2z l ) where the sum is for n 1 ; n l such that n 1 62 E , n 1 Delta Delta Delta n l Gamma1 62 E and n 1 Delta Delta Delta n l 2 E , and where the I n (z) are Bessel functions (details see [3]) and K d 1 (z 1 ; z d ) X n1 ; n d I n1 (2z 1 ) I n l (2z l ) where the sum is for n 1 ; n d such that n 1 62 E , n 1 Delta Delta Delta n d 62 E . We will show the following theorem: Theorem 2.1. Let Xm (bmtc) t 0, be the process associated to ....

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S. Boucheron and D. Gardy, An urn model from learning theory, Random Struct. Alg. 10, 43--67, 1997.

....balls of two colors, and where the state of an urn is related to its balance, i.e. to the difference Good ball Bad ball bad good good bad neutral neutral good Figure 4: Allocation of balls of two types and corresponding urn types between the numbers of balls of each color. We refer the reader to [3] for a detailed presentation and give below a brief summary of the problem and of its modelization. Let us consider the following (very) simple problem : Assume that we want to learn some boolean function on p boolean variables, whose domain has cardinality m = 2 p . Associate an urn to each ....

S. BOUCHERON and D GARDY. An urn model from learning theory. Random Structures and Algorithms, January 1997. Special issue on the analysis of algorithms.

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