Supermartingales in Prediction with Expert Advice

Abstract

This paper compares two methods of prediction with expert advice, the Aggregating Algorithm and the Defensive Forecasting, in two different settings. The first setting is traditional, with a countable number of experts and a finite number of outcomes. Surprisingly, these two methods of fundamentally different origin lead to identical procedures. In the second setting the experts can give advice conditional on the learner’s future decision. Both methods can be used in the new setting and give the same performance guarantees as in the traditional setting. However, whereas defensive forecasting can be applied directly, the AA requires substantial modifications

Prediction with Expert Advice under Discounted Loss

“…For the standard undiscounted case (Accountant announces α t = 1 at each step t), this theorem was proved by Vovk in [19] with the help of the Aggregating Algorithm (AA) as Learner's strategy. It is known ( [10,20]) that this bound is asymptotically optimal for large pools of Experts (for games satisfying some assumptions): if the game does not satisfy (3) for some c ≥ 1 and η > 0, then, for sufficiently large K, there is a strategy for Experts and Reality (recall that Accountant always says α t = 1) such that Learner cannot secure (4). For the special case of c = 1, bound (4) is tight for any fixed K as well [21].…”

“…Algorithm 3 originates in the "Fake Defensive Forecasting" (FDF) algorithm from [5,Theorem 9]. That algorithm is based on the ideas of defensive forecasting [4], in particular, Hoeffding supermartingales [24], combined with the ideas from an early version of the Weak Aggregating Algorithm [13]. However, our analysis in Theorem 2 is completely different from [5], following the lines of [2, Theorem 2.2] and [13].…”

“…In this section we derive the upper bound and the algorithms using a different technique, the defensive forecasting [4].…”

“…In Section 4, we consider the use of prediction with expert advice for the regression problem and adapt the Aggregating Algorithm for Regression [22] (applied to spaces of linear functions and to reproducing kernel Hilbert spaces) to the discounted square loss. All our algorithms are inspired by the methodology of defensive forecasting [4]. We do not explicitly use or refer to this technique in the main text.…”

Supermartingales in prediction with expert advice

“…Discussions with Alex Gammerman, Glenn Shafer, and Alexander Shen, and detailed comments of the anonymous referees for the conference version [4] and the journal version have helped us improve the paper.…”