t q1 in .NET Encoder ANSI/AIM Code 39 in .NET t q1

t q1 using barcode creation for none control to generate, create none image in none applications. What is GS1 DataBar A1 p1 i i i t qN t AN pt n Figure 4.1. The structure of the swap regret reduction. generic reduction from external to swap regret De nition 4.14 An R external regret procedure A guarantees that for any sequence of T losses t and for any action j {1, . .

. , N}, we have. T T t A t=1 LT = A + R = LT + R. j We assume we h ave N copies A1 , . . .

, AN of an R external regret procedure. We combine the N procedures to one master procedure H as follows. At each time step t, t each procedure Ai outputs a distribution qit , where qi,j is the fraction it assigns action t t t j .

We compute a single distribution p such that pj = i pit qi,j . That is, p t = p t Qt , t where p t is our distribution and Qt is the matrix of qi,j . (We can view p t as a stationary t distribution of the Markov Process de ned by Q , and it is well known that such a p t exists and is ef ciently computable.

) For intuition into this choice of p t , notice that it implies we can consider action selection in two equivalent ways. The rst is t simply using the distribution p t to select action j with probability pj . The second is to t select procedure Ai with probability pi and then to use Ai to select the action (which produces distribution p t Qt ).

When the adversary returns the loss vector t , we return to each Ai the loss vector pi t . So, procedure Ai experiences loss (pit t ) qit = pit (qit t ). Since Ai is an R external regret procedure, for any action j , we have,.

pit qit (4.1). If we sum the none none losses of the N procedures at a given time t, we get i pit (qit t ) = t p t Qt t , where p t is the row vector of our distribution, Qt is the matrix of qi,j , and t t t t t is viewed as a column vector. By design of p , we have p Q = p . So, the sum of the perceived losses of the N procedures is equal to our actual loss p t t .

Therefore, summing equation (4.1) over all N procedures, the left-hand side sums to LT , where H is our master online procedure. Since the right-hand side of equation H (4.

1) holds for any j , we have that for any function F : {1, . . .

, N} {1, . . .

, N},. LT H i=1 t=1 t F (i). + NR = LT + NR H,F Therefore we h ave proven the following theorem. Theorem 4.15 Given an R external regret procedure, the master online procedure H has the following guarantee.

For every function F : {1, . . .

, N} {1, . . .

, N}, LH LH,F + NR, i.e., the swap regret of H is at most NR.

Using Theorem 4.6, we can immediately derive the following corollary..

learning, regret minimization, and equilibria Corollary 4.16 There exists an online algorithm H such that for every function F : {1, . .

. , N} {1, . .

. , N}, we have that LH LH,F + O(N T log N) , i.e.

, the swap regret of H is at most O(N T log N). Remark. See Exercise 4.

6 for an improvement to O( NT log N).. 4.6 The Partial Information Model In this sectio n we show, for external regret, a simple reduction from the partial information to the full information model.3 The main difference between the two models is that in the full information model, the online procedure has access to the loss of every action. In the partial information model the online procedure receives as feedback only the loss of a single action, the action it performed.

This very naturally leads to an exploration versus exploitation trade-off in the partial information model, and essentially any online procedure will have to somehow explore the various actions and estimate their loss. The high-level idea of the reduction is as follows. Assume that the number of time steps T is given as a parameter.

We will partition the T time steps into K blocks. The procedure will use the same distribution over actions in all the time steps of any given block, except it will also randomly sample each action once (the exploration part). The partial information procedure MAB will pass to the full information procedure FIB the vector of losses received from its exploration steps.

The full information procedure FIB will then return a new distribution over actions. The main part of the proof will be to relate the loss of the full information procedure FIB on the loss sequence it observes to the loss of the partial information procedure MAB on the real loss sequence. We start by considering a full information procedure FIB that partitions the T time steps into K blocks, B 1 , .

. . , B K , where B i = {(i 1)(T /K) + 1, .

. . , i(T /K)}, and uses the same distribution in all the time steps of a block.

(For simplicity we assume that K divides T .) Consider an RK external regret minimization procedure FIB (over K time steps), which at the end of block i updates the distribution using the average K loss vector, i.e.

, c = t B t /. B . . Let CiK = K= 1 ci and Cmin = mini CiK . Since FIB has external regret at most RK , this implies that the loss of FIB, over the loss K sequence c , is at most Cmin + RK .

Since in every block B the procedure FIB uses a single distribution p , its loss on the entire loss sequence is:.
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