### abstract ###
The games of prediction with expert advice are considered in this paper
We present some modification of Kalai and Vempala algorithm of following the perturbed leader for the case of unrestrictedly large one-step gains
We show that in general case the cumulative gain of any probabilistic prediction algorithm can be much worse than the gain of some expert of the pool
Nevertheless, we give the lower bound for this cumulative gain in general case and construct a universal algorithm which has the optimal performance; we also prove that in case when one-step gains of experts of the pool have ``limited deviations'' the performance of our algorithm is close to the performance of the best expert
### introduction ###
Experts algorithms are used for online prediction or repeated decision making or repeated game playing
Any such algorithm is based on a ``pool of experts''
At any step  SYMBOL , each expert gives its recommendation
From this, a ``master decision'' is performed
After that, losses (or rewards)  SYMBOL  are assigned to each expert  SYMBOL  by the environment (or adversary)
The master algorithm also receives some loss or reward depending on the master decision
The goal of the master algorithm is to perform almost as well as the best expert in hindsight in the long run
Prediction with Expert Advice considered in this paper proceeds as follows
We are asked to perform sequential actions at times  SYMBOL
At each time step  SYMBOL , we observe results of actions of experts in the form of their gains and losses on steps  SYMBOL
After that, at the beginning of the step  SYMBOL   Learner  makes a decision to follow one of these experts, say Expert  SYMBOL
At the end of step  SYMBOL  Learner receives the same gain or loss as Expert  SYMBOL  at step  SYMBOL
We use notations and definitions from~ CITATION  and~ CITATION
Let  SYMBOL  be the cumulative loss of Expert  SYMBOL  at time  SYMBOL
Given  SYMBOL ,  SYMBOL , at time  SYMBOL , a natural idea to solve the expert problem is ``to follow the leader'', i e to select the expert  SYMBOL  which performed best in the past
The following simple example from Kalai and Vempala~ CITATION  shows that Learner can perform much worse than each expert: let the current losses of two experts on steps  SYMBOL  be  SYMBOL  and  SYMBOL
The ``Follow Leader'' algorithm always chooses the wrong prediction
The method of following the perturbed leader was discovered by Hannan~ CITATION
Kalai and Vempala~ CITATION  rediscovered this method and published a simple proof of the main result of Hannan
They called the algorithm of this type FPL (Following the Perturbed Leader)
Hutter and Poland~ CITATION  presented a further developments of the FPL algorithm for countable class of experts, arbitrary weights and adaptive learning rate
The FPL algorithm outputs prediction of an expert  SYMBOL  which minimizes  SYMBOL  SYMBOL \xi_t^i SYMBOL i=1,m SYMBOL t=1,2,SYMBOL p(t)=e^{-t} SYMBOL SYMBOL  SYMBOL  SYMBOL SYMBOL n SYMBOL i SYMBOL t SYMBOL 0s^i_t1 SYMBOL t SYMBOL B_t SYMBOL s_tB_t SYMBOL t SYMBOL s^i_t SYMBOL i SYMBOL 0s^i_t1$ seems to be too restrictive
In Appendix~ we consider some applications of results of Sections~- of this paper
We define two financial experts learning the fractional Brownian motion whose one-step gains at any step can not be restricted in advance
This application is at the bottom of our special interest in zero-sum games with unbounded gains in Section~
In this paper we present some modification of Kalai and Vempala algorithm for the case of unrestrictedly large one-step gains not bounded in advance
We show that in general case, the cumulative gain of any probabilistic prediction algorithm can be much worse than the gain of some expert of the pool
Nevertheless, we give the lower bound for cumulative gain of any probabilistic algorithm in general case and prove that our universal algorithm has optimal performance; we also prove that in case when one-step gains of experts of the pool have ``limited deviations'' (in particular, when they are bounded) the performance of our algorithm is close to the performance of the best expert
This result is some improvement of results mentioned above
