The paper presents numerical experiments and some theoretical developments in prediction with expert advice (PEA). One experiment deals with predicting electricity consumption depending on temperature and uses real da...
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The paper presents numerical experiments and some theoretical developments in prediction with expert advice (PEA). One experiment deals with predicting electricity consumption depending on temperature and uses real data. As the pattern of dependence can change with season and time of the day, the domain naturally admits PEA formulation with experts having different "areas of expertise". We consider the case where several competing methods produce online predictions in the form of probability distribution functions. The dissimilarity between a probability forecast and an outcome is measured by a loss function (scoring rule). A popular example of scoring rule for continuous outcomes is Continuous Ranked Probability Score ( CRPS ). In this paper the problem of combining probabilistic forecasts is considered in the PEA framework. We show that CRPS is a mixable loss function and then the time-independent upper bound for the regret of the Vovk aggregating algorithm using CRPS as a loss function can be obtained. Also, we incorporate a "smooth" version of the method of specialized experts in this scheme which allows us to combine the probabilistic predictions of the specialized experts with overlapping domains of their competence. (c) 2021 Elsevier Ltd. All rights reserved.
We consider the problem of dynamically apportioning resources among a set of options in a worst-case online framework. The model we investigate is a generalization of the well studied online learning model. In particu...
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We consider the problem of dynamically apportioning resources among a set of options in a worst-case online framework. The model we investigate is a generalization of the well studied online learning model. In particular, we allow the learner to see as additional information how high the risk of each option is. This assumption is natural in many applications like horse-race betting, where gamblers know odds for all options before placing bets. We apply Vovk's aggregating algorithm to this problem and give a tight performance bound. The results support our intuition that it is safe to bet more on low-risk options. Surprisingly, the loss bound of the algorithm does not depend on the values of relatively small risks.
Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are multiclass classification and class probability estimation. Central to their definition is the choice of loss f...
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Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend, existing results on M-sums of convex sets. Third, the perspective leads to a natural theory of "polar" loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm.
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