Lectures on Choquet’s Theorem

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Lectures on Choquet's theorem English. Phelps, Robert R. New search for: Phelps, Robert R. A fundamental tradeoff in online learning is the exploration-exploitation tradeoff, where the learner needs to pull the suboptimal arm sufficiently many times to identify the best arm with better precision, and also needs to exploit the current knowledge to focus on the arm which seems to be the best. In this section we aim to prove minimax lower bounds of the regrets in the two-armed bandits.

DA2PL'2014 - Preference modeling with Choquet integral - Michel Grabisch

The two-point method suggests that we may find two worlds such that. Using the above arguments and the inequality. To show this stronger lower bound, we need a more delicate construction of the two points.

ISBN 13: 9780442075972

Hence, applying the previous arguments, we have. Note that these two bounds exactly characterize the tradeoff between exploitation and exploration. Combining these bounds, we have. Both bounds are tight and there exist policies attaining these bounds; we refer to bibliographic notes for details. Interestingly, the two-point method can also give the correct minimax lower bound of the regret for general multi-armed bandits in later lectures, we will provide a proof using multiple points. However, the following application of the two-point method differs from all previous ones.

In previous examples, we fix two points and prove that all estimators or policies cannot achieve small risk or regret under both points.

We first define the multi-armed bandit problem. This bound is tight for multi-armed bandits. The idea of constructing points based on given approaches is widely used in problems where several rounds of adaptivity are available.


Lectures on Choquet's Theorem - Robert R. Phelps - Google Könyvek

One future lecture will be devoted exclusively to this topic. The inequalities between f -divergences and the two-point method can be mostly found in the excellent textbook Tsabykov For the examples, the first lower bound of entropy estimation is taken from Wu and Yang For the robust estimators achieving the lower bound, see Chen, Gao and Ren and Ilias et al.

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The lower bounds of bandit problems are taken from Voger and Lai and Robbins For more on bandit problems such as the upper bounds, we refer to the book Bubeck and Cesa-Bianchi This site uses Akismet to reduce spam. Learn how your comment data is processed. Example 1 1. Both inequalities are tight. TV vs. None of these bounds are tight.

This bound is tight. KL vs. Hence, this choice fails for some parameters n,k. On the estimation of an infinite-dimensional parameter in Gaussian white noise.

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Doklady Akademii Nauk. Russian Academy of Sciences, Lucien Le Cam. A method for extending the representation theorems: Caps. A different method for extending the representation theorems.

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  • Orderings and dilations of measures. Additional Topics. Back Matter Pages About this book Introduction A well written, readable and easily accessible introduction to "Choquet theory", which treats the representation of elements of a compact convex set as integral averages over extreme points of the set.

    The interest in this material arises both from its appealing geometrical nature as well as its extraordinarily wide range of application to areas ranging from approximation theory to ergodic theory. Many of these applications are treated in this book.