V. G. Vovk and C. Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory (COLT-98), pages 12--23, New York, 1998. ACM Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....prediction [Sol64, Sol78] and function inversion and minimization [Sol86] Together with Chaitin [Cha66, Cha75] this was the invention of what is now called Algorithmic Information theory. For further literature and many applications see [LV97] Other interesting applications can be found in [Cha91, Sch99, VW98]. Related topics are the Weighted Majority Algorithm invented by Littlestone and Warmuth [LW94] universal forecasting by Vovk [Vov92] Levin search [Lev73] pac learning introduced by Valiant [Val84] and Minimum Description Length [LV92a, Ris89] Resource bounded complexity is discussed in ....

V. G. Vovk and C. Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory (COLT-98), pages 12--23, New York, 1998. ACM Press.

....p i ) 2 and the logloss function (x i ; p i ) Gamma log p i if x i = 1 and (x i ; p i ) Gamma log(1 Gamma p i ) if x i = 0. Here log means logarithm to the base 2. Other loss functions are considered in Vovk [6] Haussler, Kivinen, Warmuth [1] log loss, Hellinger, etc. Vovk and Watkins [8] (financial theory) Yamanishi [13] logistic, etc) It is natural to suppose that all predictions are given according to a prediction strategy (or prediction algorithm) p i = S(x 1 x 2 : x i Gamma1 ) The total loss incurred by Predictor who follows the strategy S over the first n trials x 1 ....

....; x n is defined Loss S (x 1 x 2 : x n ) n X i=1 (x i S(x 1 x 2 : x i Gamma1 ) The main task is to minimize the total loss suffered on a sequence x = x 1 x 2 : x n of outcomes. The corresponding game theoretic interpretation is given in Vovk [7] or in Vovk and Watkins [8]. Predictive complexity 2 Let us fix j 0 (the learning rate) and put fi = e Gammaj 2 (0; 1) A generalized prediction is defined to be any function on finite binary sequences taking real values. Three important classes of generalized predictions g are given by the following definitions from ....

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Vovk, V., Watkins, C.J.H.C. (1998) Universal portfolio selection, Proceedings of the 11th Annual Conference on Computational Learning Theory, 12--23.

.... log loss function (x i ; p i ) Gamma log p i if x i = 1 and (x i ; p i ) Gamma log(1 Gamma p i ) if x i = 0 (in the following log means the logarithm to the base 2) Other loss functions are considered in Vovk [6] Haussler, Kivinen, Warmuth [2] log loss, Hellinger, etc) Vovk and Watkins [8] (financial theory) Yamanishi [11] logistic) etc] An interesting theoretical possibility (and sometimes an unsightly phenomenon of the real life) is snooping , or choosing the learning algorithm depending on the sequence x. Of course, we can always built x into the learning algorithm and ....

....: x n is defined Loss S (x 1 x 2 : x n ) n X i=1 (x i ; S(x 1 x 2 : x i Gamma1 ) The main task is to minimize the total loss suffered on a sequence x = x 1 x 2 : x n of outcomes. The corresponding game theoretic interpretation is given in Vovk [7] or in Vovk and Watkins [8]. Honest learning algorithms S are those with small Kolmogorov complexity. Knowing x allows us to choose more complicated S (tuned for this particular x) but we will impose the restriction K(S) ff, where ff is a positive constant reflecting the degree of snooping allowed. We impose the ....

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Vovk, V, Watkins, C.J.H.C. (1998) Universal portfolio selection, Proceedings of the 11th Annual Conference on Computational Learning Theory, 12--23.

....Abstract A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from day to day. There has been much work on Cover s Universal algorithm, which is competitive with the best constant rebalanced portfolio determined in hindsight [3, 9, 2, 8, 14, 4, 5, 6]. While this algorithm has good performance guarantees, all known implementations are exponential in the number of stocks, restricting the number of stocks used in experiments [9, 4, 2, 5, 6] We present an efficient implementation of the Universal algorithm that is based on non uniform random ....

....the same distribution of wealth among a set of stocks from day to day. That is, the proportion of total wealth in a given stock is the same at the beginning of each day. Recently there has been work on on line investment strategies which are competitive with the best CRP determined in hindsight [3, 9, 2, 8, 14, 4, 5, 6]. Specifically, the daily performance of these algorithms on a market approaches that of the best CRP for that market, chosen in hindsight, as the lengths of these markets increase without bound. As an example of a useful CRP, consider the following market with just two stocks [9, 5] The price ....

V. Vovk and C.J.H.C. Watkins. Universal Portfolio Selection. In Proceedingsof the 11th Annual Conferenceon Computational Learning Theory, pages 12-23, 1998.

.... of London) related to this new principle, which we call the complexity approximation principle (CAP) Both MDL and MML principles can be interpreted as Kolmogorov complexity approximation principles (as explained in Rissanen [1, 2] and Wallace and Freeman [3] see also [4] It is shown in [5] and [6] that it is possible to generalize Kolmogorov complexity to describe the optimal performance in different games of prediction . Using this general notion, called predictive complexity,itis straightforward to extend the MDL and MML principles to our more general CAP. In Section 2 we define ....

....of predictions is a superprediction. We say that our game is perfectly mixable if it is # mixable for some # 0. It is known that many popular games, such as the log loss game, square loss game, Cover s game, long short game, Kullback Leibler game, # 2 game, Hellinger game etc. see, e.g. [6, 11, 12, 13]; some of these games will be described below) are perfectly mixable. LEMMA 1. There exists a universal measure of predictive complexity for any perfectly mixable game. Remember that we always assume that our games satisfy assumptions of computability which we do not specify explicitly. The ....

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Vovk, V. and Watkins, C. J. H. C. (1998) Universal portfolio selection. In Proc. 11th Ann. Conf. Computational Learning Theory, pp. 12--23. ACM Press, New York.

.... the simplest game of this kind is Cover s game: Omega = 0; 1) K ; Gamma = f(p 1 ; p K ) 2 [0; 1) K j p 1 Delta Delta Delta p K = 1g; y 1 ; y K ) p 1 ; p K ) Gamma ln K X k=1 y k p k ; the financial meaning of this game is described in, eg, [18, 58]. Like the Kullback Leibler game, Cover s game is also a generalization of the log loss game (with a finite sample space) it suffices to consider degenerate 2 Omega of the form (0; 0; 1; 0; 0) Such degenerate can be regarded as crisp events ; they can be generalized ....

....generalized to fuzzy events = y 1 ; y K ) 2 [0; 1] K , where y k is interpreted as the weight of evidence in favour of k occurring. Cover s universal portfolio algorithm [17, 18] is the AA applied to this game and a particular decision pool, the constant rebalanced portfolios (cf [58]) however, the AA and Cover s algorithm were found independently of each other (the AA was obtained by bridging the Weighted Majority Algorithm and the Bayesian mixing rule) Cover s game ignores transaction costs and the possibility of short selling in financial markets. The long short game, ....

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Volodya Vovk and Chris J H C Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory, pages 12--23, New York, 1998. Assoc Comput Mach.

....Properties. 9 4.2 Differential Criteria. 10 4.3 The Legendre Transformation and Expectations of Complexity. 11 Introduction 3 1 Introduction In this paper, we investigate the problem of existence of predictive complexity. In Section 2, we give the definition of predictive complexity. Unlike [VW98], where the original definition was given, we define predictive complexity for generalised games, which correspond to classes of traditional games. In Section 3, we discuss necessary and sufficient conditions for the existence of predictive complexity. The most general of known sufficient ....

....1 4, then its set of superpredictions is a generalised game. Conversely, every generalised game is the set of superpredictions for a game which has the possible predictions space [0; 1] and satisfies the assumptions 1 4. We need the following generalisation of the concept of a game (see [VW98]) A triple ( Gamma; Sigma) is called a game with signals. The set Sigma is called the space of possible signals. The set Gamma and the function bear the same names and satisfy the same assumptions as above. An ordinary game may be considered as a game with signals with the space of signals ....

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V. Vovk and C. J. H. C. Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory, pages 12--23, 1998.

.... class of games are connected with nance; the simplest game of this kind is Cover s game: 0; 1) K ; f(p 1 ; p K ) 2 [0; 1) K j p 1 p K = 1g; y 1 ; y K ) p 1 ; p K ) ln K X k=1 y k p k ; the nancial meaning of this game is described in, eg, [17, 56]. Like the Kullback Leibler game, Cover s game is also a generalization of the logloss game (with a nite sample space) it suces to consider degenerate 2 of the form (0; 0; 1; 0; 0) Such degenerate can be regarded as crisp events ; they can be generalized to fuzzy ....

....to fuzzy events = y 1 ; y K ) 2 [0; 1] K , where y k is interpreted as the weight of evidence in favour of k occurring. Cover s universal portfolio algorithm [16, 17] is the AA applied to this game and a particular decision pool, the constant rebalanced portfolios (cf [56]) however, the AA and Cover s algorithm were found independently of each other (the AA was obtained by bridging the Weighted Majority Algorithm and the Bayesian mixing rule) Cover s game ignores transaction costs and the possibility of short selling in nancial markets. The long short game, ....

[Article contains additional citation context not shown here]

Volodya Vovk and Chris J H C Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory, pages 12-23, New York, 1998. Assoc Comput Mach.

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V. G. Vovk and C. Watkins. Universal portfolio selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory (COLT-98), pages 12--23, New York, 1998. ACM Press.

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V. Vovk and C.J.H.C. Watkins. Universal Portfolio Selection. Proceedings of the 11th Annual Conference on Computational Learning Theory, 12-23, 1998. 18

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V. Vovk and C. Watkins. Universal Portfolio Selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory, 12-23, 1998.

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V. Vovk and C. Watkins. Universal Portfolio Selection. In Proceedings of the 11th Annual Conference on Computational Learning Theory, 12-23, 1998.

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