R Tyrrell Rockafellar. Convex analysis. Princeton, N.J., Princeton University Press, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....convex. From this we see that # n is either everywhere on ( #, 0) or it is finite on ( #, 0] In the rest of this section we will only be considering the latter case. Note that if # # n (D) for any D, then # n must be finite on ( #, 0] # n is a proper ( #) closed (l.sc. convex function [13] and continuous from the left. It is finite and C on ( #, 0) # # n , the derivative with w.r.t. #, is nondecreasing and # # n (#) sup = D ave , # # n (#) inf = D min . 14 Furthermore, if D min D ave , then # n ( is strictly convex on ( #, 0) From the definition ....

....that the assumption # # n (D) means that # n has all of the nice properties detailed in Section A.2. In particular, the strict convexity of # n implies that there is a unique #D 0 with # # n (# D ) D. We have just seen that # # n (D) is the conjugate of # n at D, so Rockafellar (1970) [13][Theorem 23.5, Corollary 23.5.1, Theorem 25.1] gives # n (# D ) and # # n (D) #D 0. This shows that # # n is finite, strictly convex and C on (D min , D ave ) Since #D 0 as D ave , # # n is di#erentiable at 0 and so it is C #) The last thing we have to prove is the claim about ....

R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1970.

....g, the Bregman divergence is de ned as: d g ( 1 ; 2 ) d g ( 1 ; 2 ) Note that in the above de nition, f) in general denotes a subgradient of a convex function (f) at f . A subgradient p of a convex function (f) at p is a value such that (q) p) p (q p) for all q (see [11] Section 23) Clearly, by de nition, the Bregman divergence is always non negative. However, in general a subgradient of a convex function at a point may not always exist, and even when it exists it may not be unique. To avoid such diculties, in this paper we only use Bregman divergence for ....

.... f n ( f n ( k f n (X k )Y k )j h(X denotes a subgradient of . Proof. The minimizer f n of (15) lies in the nite dimensional space VX spanned by g i (x) i x 1) 2 H h (i = 1; n) Use the linear representation f(X i ) hf; g i i, and Theorem 23.8 in [11], we know that there exist subgradients f n (X i )Y i ) i = 1; n) such that the following rst order condition holds: h f n ; g i iY i )g i Y i n f n = 0: 16) We also have f n (X i )Y i ) X i ) f n (X i ) Y i n h ....

R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970.

....that f(a; b; c) is a convex function of a. For simplicity, we assume that a solution of (2) exists, but may not be unique. To be able to treat classi cation and regression under the same general framework, we shall consider convex functions from the general convex analysis point of view, as in [9]. Especially we allow a convex function to take the 1 value which is equivalent to a constraint. Consider a convex function p(u) R d R , where R is the real line, and R denotes the extended real line R [ f 1g. However, we assume that convex functions do not achieve 1. We also assume ....

....more focused on the main topic of leave one out analysis) we would like to point out that for most speci c learning formulations we considered later in the paper, the existence of solution is guaranteed. Since the solution n of (2) achieves the minimum of Ln ( we have (see page 264 in [9]) 0 2 i Ln ( for all i (the sub di erential is with respect to each i ) By Theorem 23.8 in [9] for each i, we can nd a subgradient of f( i ; x i ; y i ) at i with respect to i such that r i f( i ; x i ; y i ) j K(x i ; x j ) 0: i = 1; n) 3) We may now ....

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R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970.

.... convex programming problem involving linear transformations of the primal variable w, a dual form can be obtained by introducing auxiliary variables i for each data point x i : w; arg inf where k( Delta) is the dual transform of f( Delta) assume f is lower semi continuous, see [11]) k(v) sup (uv Gamma f(u) It is well known that k is convex. By switching the order of inf w and sup , which is valid for the above minimax convex concave programming problem (a proof of this interchangeability, i.e. strong duality, is given in Appendix A) we obtain = arg sup ....

....Lagrangian multipliers as in Section 3. The main reason of this treatment is for numerical considerations. However, for the mathematical proof of strong duality in Appendix A which does not have any direct numerical consequence, we use the generalized definition of convex functions and duality in [11] without assuming differentiability. For example, in the general case, a constraint on a convex function c(z) can be regarded as a modification of c(z) so that c(z) 1 when z does not satisfies the constraint. Substituting (6) into (5) we obtain (k( Gamma i ) i y i ) w rg(w) ....

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R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970.

....size n 1. Furthermore, we give a large deviation bound on the rate of convergence. From (4) we obtain the following first order condition: ED fi( x; y)xy w ( 0; 8) where fi( x; y) f (w ( xy Gamma 1) and f (z) 2 [ Gamma1; 0] denotes a member of the subgradient of f at z [9]. In the finite sample case, we can also interpret fi( x; y) in (8) as a scaled dual variable ff: fi = Gammaff=C, where ff appears in the dual (or Kernel) formulation of an SVM (for example, see chapter 10 of [11] The convexity of f implies that f(z 1 ) z 2 Gamma z 1 )f (z 1 ) f(z ....

R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970.

.... linear representation of p(x) we differentiate (2) at the optimal solution p( which leads to the following first order condition: E x;y L 1 ( p q x ; y)q x p = 0; 4) where L 1 (a; b) is the derivative of L(a; b) with respect to a if L is smooth; it denotes a subgradient (see [4]) otherwise. Since we have assumed that L(a; b) is a convex function of a, we know that L(a 1 ; b) a 2 a 1 )L 1 (a 1 ; b) L(a 2 ; b) This implies the following inequality: 1 ( p(x i ) y i ) p n (x i ) p(x i ) L( p n (x i ) y i ) which is equivalent ....

R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970.

....let k k 2 . Thus L(x j ) L(x j a) a 1 a 2 k x k log(a 1 a 2 k ) log(x k ) 6) Theorem 1: L(x j a) is strictly concave. Proof of Theorem 1: In order to determine whether L(x j a) is strictly concave, we may determine whether L(x j a) is strictly convex. From [7], we know that L(x j a) is strictly convex if and only if its Hessian matrix is positive de nite. The Hessian matrix of L(x j a) is Q 4 5 : Let c k x k (a 1 a 2 k Then P N k=1 c k 0 P N : 7) For notational simplicity, let w c k y z ....

R. Tyrrell Rockafellar, Convex Analysis, Princeton University Press, Princeton, New Jersey, 1111.

....but satisfies our axioms, the functional I that represents is not necessarily Gateaux di#erentiable. However, it does have a generalized set valued derivative. It is the notion of Clarke di#erential , developed by Clarke [5] as an extension of the concept of superdi#erential (e.g. Rockafellar [25]) to functionals that do not satisfy concavity. It is natural to maintain that the generalization of the previous remark should hold in this case: the Clarke di#erential describes the DM s understanding of the collection of all possible probabilistic scenarios (more than one, in this case) Our ....

....by Gilboa and Schmeidler [16] it can be represented by maxmin expected utility over a set of priors D. In this case I is not necessarily Gateaux differentiable. However, it does everywhere have directional derivatives and a nonempty superdi#erential, as defined below (see, e.g. Rockafellar [25]) Definition 13 Given a concave functional I : B 0 (#) R, its directional derivative in # in the direction # is defined by dI(#; #) t . The superdi#erential of I at # is the set of linear functionals that dominate the directional derivative dI(#; dI(#; #) B 0 (#) For ....

R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton, New Jersey, 1970.

....under which # is di#erentiable at a, and this #a is simply ##(a) 25 Whenever t 1, e.g. when a = 0, we have t = n) determining a descent direction (or that none exists) is much harder. One general method uses the notion of the subgradient ##(a) of a convex function # at a , defined as [Roc72, Cla83] ##(a) # ##(a) a) a . 60) ##(a) can be shown to be nonempty, compact, and convex, and moreover #a is a descent direction at a if and only #a g 0 ##(a) 61) so that descent directions correspond precisely to hyperplanes through the origin with the ....

....an algorithm for (57) which involves even less computation per iteration than the subgradient method, and thus may be useful for very large systems. Kamenetskii and Pyatnitskii consider the function F (a, x) x ( a i Q i )x. 67) Recall that a, x is said to be a saddle point of F [AHU58, Roc72] if F (a, x) a, x. It is easy to see that a, x is a saddle point of F if and only if #(a) 0 and Q i x = 0, i = 0, r, which we assume without loss of generality occurs only if x = 0 (otherwise A is clearly not SSLS) This is theorem 2 of [KP87] The Kamenetskii Pyatnitskii ....

R. T. Rockefellar. Convex Analysis. Princeton University Press, Princeton, second edition, 1972.

....g, the Bregman divergence is de ned as: d g (q 1 ; q 2 ) d g (q 1 ; q 2 ) Note that in the above de nition, f 0 (p) in general denotes a subgradient of a convex function f(p) at p. A subgradient p of a convex function f(p) is a value such that f(p 0 ) f(p) p (p 0 p) see [6] Section 23) Clearly, by de nition, the Bregman divergence is always non negative. However, in general a subgradient of a convex function at a point may not always exist, and even when it exists it may not be unique. To avoid such diculties, in this paper we only use Bregman divergence for ....

R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970.

....g, the Bregman divergence is de ned as: d g (q 1 ; q 2 ) d g (q 1 ; q 2 ) Note that in the above de nition, f 0 (p) in general denotes a subgradient of a convex function f(p) at p. A subgradient p of a convex function f(p) is a value such that f(p 0 ) f(p) p (p 0 p) see [5] Section 23) Clearly, by de nition, the Bregman divergence is always non negative. However, in general a subgradient of a convex function at a point may not always exist, and even when it exists it may not be unique. To avoid such diculties, in this paper we only use Bregman divergence for ....

R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ, 1970. 25

....a whose components are: a i j trA i ae 1 : 2.45) Then, letting also b i j trA i ae 2 ; 2.46) the entanglement fidelity of the convex combination of ae 1 and ae 2 may be written F e (ae 1 (1 Gamma )ae 2 ; E) jja (1 Gamma )bjj 2 : 2. 47) The norm is easily shown to be convex (see e.g. [18] for the real case) hence so is its square and the theorem follows. Note that with this representation of the entanglement fidelity, the freedom to choose an environment basis (equivalently, the freedom to move to a different operator decomposition of a given operation) corresponds to performing ....

R. Tyrrell Rockefellar, Convex Analysis, Princeton University Press, Princeton, 1970.

.... of a complex vector a whose components are: a i tr A i 1 : 25) Then, letting also b i tr A i 2 ; 26) the entanglement delity of the convex combination of 1 and 2 may be written F e ( 1 (1 ) 2 ; E) jj a (1 )bjj 2 : 27) Any norm is easily shown to be convex (see e.g. [32] for real vector spaces) and since a norm is positive its square is also convex and the lemma follows. Note that with this representation of the entanglement delity, the freedom to choose an environment basis (equivalently, the freedom to move to a di erent operator decomposition of a given ....

R. Tyrrell Rockefellar, Convex Analysis, Princeton University Press, Princeton, 1970.

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R Tyrrell Rockafellar. Convex analysis. Princeton, N.J., Princeton University Press, 1970.

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R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970.

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R. T. Rockefellar. Convex Analysis, Princeton University Press, 1970.

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R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton NJ, 1970.

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R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, N.J., 1970.

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R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, N.J., 1970.

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R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton, N.J., 1970.

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R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton, NJ, USA, 1970. xviii+451 pp. ISBN 0-691-08069-0.

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R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, Princeton, New Jersey, 1970.

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