Universal prediction of individual sequences

Abstract

The problem of predicting the next outcome of an individual binary sequence using finite memory, is considered. The finite-state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite-state (FS) predictor. It is proved that this FS predictability can be attained by universal sequential prediction schemes. Specifically, an efficient prediction procedure based on the incremental parsing procedure of the Lempel-Ziv data compression algorithm is shown to achieve asymptotically the FS predictability. Finally, some relations between compressibility and predictability are pointed out, and the predictability is proposed as an additional measure of the complexity of a sequence.

Conditional Universal Consistency

“…Foremost is the literature on universal consistency, beginning with Blackwell [1956] and Hannan [1957], and continuing through Banos [1968], Megiddo [1980], Fudenberg and Levine [1995] and Auer, Cesa-Bianchi, Freund and Schapire [1995]. This work is closely connected to work in the computer science literature by Vovk [1990], Desantis, Markowski and Wegman [1992], Feder, Mehrav and Gutman [1992], Kivinen and Warmuth [1993], Chung [1994], Littlestone and Warmuth [1994] and Freund and Schapire [1996] on "worst case analysis. "…”

Regret in the On-Line Decision Problem

“…Feder, Mehrav and Gutman [10] and Vovk [28] at about the same time. Vovk [28] shows how the exponential weighted algorithm can be used to prove Theorem 2 for In the case where the state of the world in each period is not binary, Littlestone and Warmuth [25] and Kivinen and Warmuth [24] show that Theorem 2 holds, but only for particular loss function.…”

“…In particular they extend Hannan's theorem to the case where the row player knows only the payoff from the strategy played in each round, thus providing for an on-line version of the classical bandit problem. 11 10 We note that the important ingredients for a proof of Hannan's theorem can also be found in [9]. That paper does not contain an explicit statement of the theorem or proof.…”

Untitled

“…A well established approach in such cases (Vovk, 1990;Littlestone & Warmuth, 1994;Littlestone, 1989;Feder, Merhav, & Gutman, 1992;Merhav & Feder, 1993;Cesa-Bianchi et al, 1997;Cesa-Bianchi et al, 1996) is to assume nothing about the (x t , y t ) pairs, and instead, for a given F, to give bounds on the number of mistakes made by a given learning algorithm in terms of the minimum over f ∈ F of the number η of trials t for which f (x t ) = y t . Learning models like this are often referred to as agnostic learning models 4 (Kearns, Schapire, & Sellie, 1994).…”