Loss functions, complexities, and the Legendre transformation

Kalnishkan, Yuri; Vyugin, Michael V.; Vovk, Vladimir

doi:10.1016/j.tcs.2003.11.005

Cited by 11 publications

(17 citation statements)

References 8 publications

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“…(16)). It is worth mentioning that one can reconstruct the superprediction set Σ Λ from the generalized entropy of the game, and also from the predictive complexity of the game (see [18] for the definitions and proofs in the case of binary games).…”

Section: Remark 16mentioning

confidence: 99%

Supermartingales in prediction with expert advice

Chernov

Kalnishkan

Zhdanov

et al. 2010

Theoretical Computer Science

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We apply the method of defensive forecasting, based on the use of game-theoretic supermartingales, to prediction with expert advice. In the traditional setting of a countable number of experts and a finite number of outcomes, the Defensive Forecasting Algorithm is very close to the well-known Aggregating Algorithm. Not only the performance guarantees but also the predictions are the same for these two methods of fundamentally different nature. We discuss also a new setting where the experts can give advice conditional on the learner's future decision. Both the algorithms can be adapted to the new setting and give the same performance guarantees as in the traditional setting. Finally, we outline an application of defensive forecasting to a setting with several loss functions

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Section: Remark 16mentioning

confidence: 99%

Supermartingales in prediction with expert advice

Chernov

Kalnishkan

Zhdanov

et al. 2010

Theoretical Computer Science

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“…4 describes the connections between predictive complexity and generalised entropy, Sect. 5 surveys negative results on the existence of complexity (after [KVV04b] and [KVV04a]) and Sect. 6 proves the uniqueness theorem from [KVV04b].…”

Section: Introductionmentioning

confidence: 99%

“…5 surveys negative results on the existence of complexity (after [KVV04b] and [KVV04a]) and Sect. 6 proves the uniqueness theorem from [KVV04b]. Section 7 discusses linear inequalities between different predictive complexities after [Kal02].…”

Section: Introductionmentioning

confidence: 99%

Predictive Complexity for Games with Finite Outcome Spaces

Kalnishkan

2015

Measures of Complexity

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“…We consider the online prediction question studied by [15], [16], [10], [7], [9] in the setting of a stationary stochastic process. In this setting, we have a sequence of outcomes x 0 , x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Predictive Complexity and Generalized Entropy Rate of Stationary Ergodic Processes

Ghosh

Nandakumar

2012

Lecture Notes in Computer Science

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Abstract. In the online prediction framework, we use generalized entropy of to study the loss rate of predictors when outcomes are drawn according to stationary ergodic distributions over the binary alphabet. We show that the notion of generalized entropy of a regular game [10] is well-defined for stationary ergodic distributions. In proving this, we obtain new game-theoretic proofs of some classical information theoretic inequalities. Using Birkhoff's ergodic theorem and convergence properties of conditional distributions, we prove that a classical Shannon-McMillanBreiman theorem holds for a restricted class of regular games, when no computational constraints are imposed on the prediction strategies. If a game is mixable, then there is an optimal aggregating strategy which loses at most an additive constant when compared to any other lower semicomputable strategy. The loss incurred by this algorithm on an infinite sequence of outcomes is called its predictive complexity. We use our version of Shannon-McMillan-Breiman theorem to prove that when a restriced regular game has a predictive complexity, the predictive complexity converges to the generalized entropy of the game almost everywhere with respect to the stationary ergodic distribution.

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Loss functions, complexities, and the Legendre transformation

Cited by 11 publications

References 8 publications

Supermartingales in prediction with expert advice

Supermartingales in prediction with expert advice

Predictive Complexity for Games with Finite Outcome Spaces

Predictive Complexity and Generalized Entropy Rate of Stationary Ergodic Processes

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