Download Prediction, Learning, and Games by Nicolo Cesa-Bianchi, Gabor Lugosi PDF

By Nicolo Cesa-Bianchi, Gabor Lugosi

This crucial new textual content and reference for researchers and scholars in computer studying, video game conception, facts and data concept bargains the 1st accomplished remedy of the matter of predicting person sequences. not like general statistical techniques to forecasting, prediction of person sequences doesn't impose any probabilistic assumption at the data-generating mechanism. but, prediction algorithms will be built that paintings good for all attainable sequences, within the feel that their functionality is often as good because the most sensible forecasting technique in a given reference classification. The crucial subject matter is the version of prediction utilizing professional suggestion, a normal framework in which many comparable difficulties might be forged and mentioned. Repeated video game enjoying, adaptive info compression, sequential funding within the inventory industry, sequential development research, and several difficulties are considered as situations of the specialists' framework and analyzed from a typical nonstochastic perspective that frequently finds new and exciting connections. previous and new forecasting equipment are defined in a mathematically exact manner on the way to represent their theoretical barriers and probabilities.

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Extra resources for Prediction, Learning, and Games

Example text

To do this we can exploit simple geometrical properties exhibited by the potential when combined with specific loss functions. In this chapter we develop several techniques of this kind and use them to derive tighter regret bounds for various loss functions. 3 we consider a basic property of a loss function, exp-concavity, which ensures a bound of (ln N )/η for the exponentially weighted average forecaster, where η must be smaller than a critical value depending on the specific exp-concave loss. 4 we take a more extreme approach by considering a forecaster that chooses predictions minimizing the worst-case increase of the potential.

For brevity, write 2 C= √ 2−1 . 5 (with, say, η1 = η2 ) Hn∗ − Hn ≤ 2 − ηn+1 1 η1 n ln N + ≤ 2 max 2M ln N , t=1 1 C 1 ln E eηt (X t −EX t ) ηt n Vn ln N + t=1 1 ln E eηt (X t −EX t ) . ηt Since ηt ≤ 1/(2M), ηt (X t − E X t ) ≤ 1 and we may apply the inequality e x ≤ 1 + x + (e − 2)x 2 for all x ≤ 1. We thus find that Hn∗ − Hn ≤ 2 max 2M ln N , 1 C n Vn ln N + (e − 2) ηt var(X t ). t=1 Now denote by T the first time step t when Vt > M 2 . Using ηt ≤ 1/(2M) for all t and VT ≤ 2M 2 , we get n n ηt var(X t ) ≤ M + t=1 ηt var(X t ).

That is, we let the parameter η of the exponential potential depend on the round number t. As the best nonuniform bounds for √ the exponential potential are obtained by choosing η = 8(ln N )/n, a natural choice for √ a time-varying exponential potential is thus ηt = 8(ln N )/t. 2, we obtain for this choice of ηt a regret bound whose main term √ is 2 (n/2) ln N and is therefore better than the doubling trick bound. More precisely, we prove the following result. 3. Assume that the loss function is convex in its first argument and takes values in [0, 1].

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