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which is the generalization of the earlier Equation (136) To summarize, we have simply redefined Q(s, a ) in the nondeterministic case to be the expected value of its previously defined quantity for the deterministic case Now that we have generalized the definition of Q to accommodate the nondeterministic environment functions r and 6, a new training rule is needed Our earlier training rule derived for the deterministic case (Equation 137) fails to converge in this nondeterministic setting Consider, for example, a nondeterministic reward function r(s, a ) that produces different rewards each time the transition (s,a } is repeated In this case, the training rule will repeatedly alter the values of Q ( S ,a ) , even if we initialize the Q table values to the correct Q function In brief, this training rule does not converge This difficulty can be overcome by modifying the training rule so that it takes a decaying weighted average of the current Q value and the revised estimate Writing Q, to denote the agent's estimate on the nth iteration of the algorithm, the following revised training rule is sufficient to assure convergence of Q to Q: Q ~ ( s ,) -+( 1 - un)Qn-l(s,a ) a where 1 1 visits, ( s ,a ) where s and a here are the state and action updated during the nth iteration, and where visits,(s, a ) is the total number of times this state-action pair has been visited up to and including the nth iteration The key idea in this revised rule is that revisions to Q are made more gradually than in the deterministic case Notice if we were to set a, to 1 in Equation (1310) we would have exactly the training rule for the deterministic case a, With smaller values of a , this term is now averaged in with the current ~ ( s ) to produce the new updated value Notice that the value of a , in Equation (1311) decreases as n increases, so that updates become smaller as training progresses By reducing a at an appropriate rate during training, we can achieve convergence to the correct Q function The choice of a, given above is one of many that satisfy the conditions for convergence, according to the following theorem due to Watkins and Dayan (1992).

a, =

Now for the implementation. You will want the button click to run the calculation in fact, to make it work a bit harder I am going to run the calculation 10 times. Each time the calculation works out the first 90 members of the Fibonacci series of integers: each member of the series is the sum of the two previous members of the series, so it goes 1, 1, 2, 3, 5, 8, 13, 21 and so on. The numbers get to be pretty big, which is why you need to declare the fibValues array holding the series of numbers to be a long long int. After setting the first two members of the series manually, you go into a loop, setting the nth member of the series to be the sum of the (n 2)th and (n 1)th members. Then you take the current member of the array and append that to a growing, comma-separated string actually, since NSStrings are immutable, you are creating a new NSString with the new contents. Each time around the loop, you are putting a space between the lines and adding a number at the beginning of the next calculation. Then finally you put the outputString into a multiline text field. See Listing 12 2.

+ a,[r + y max Q,-~(S',a')]

(1310)

Listing 12 2. Fibonacci_FunAppDelegate.m -(IBAction)startCalculation:(id)sender { NSAutoreleasePool *fibPool = [[NSAutoreleasePool alloc] init]; NSString *outputString; outputString = @"1: "; for (int x=1; x < 11; x++) { long long int fibValues[90]; fibValues[0] = 0; fibValues[1] = 1; for (int i=2; i < 90; i++) { fibValues[i] = fibValues[i-2] + fibValues[i-1]; NSString *thisFibString; thisFibString = [[NSString alloc] initWithFormat:@"%lli, ",fibValues[i]]; outputString = [outputString stringByAppendingFormat:@"%@", thisFibString]; } outputString = [outputString stringByAppendingFormat:@"\n\n%i: ", x+1]; [outputField setStringValue:outputString]; } [fibPool drain]; }

Theorem 132 Convergence of Q learning for nondeterministic Markov decision processes Consider a Q learning agent i n a nondeterministic MDP with bounded rewards (Vs, a)lr(s,a)l 5 c The Q learning agent uses the training rule of Equation (1310), initializes its table ~ ( s ) to arbitrary finite values, and uses a a, discount factor y such that 0 5 y < 1 Let n(i, s, a ) be the iteration corresponding to the ith time that action a is applied to state s If each state-action pair is visited infinitely often, 0 5 a, < 1, and ,

CANDIDATE-ELIMINATION The positive training example generalizes the S boundary, from Trace 3 S to S4 One member of Gg must also be deleted, because it is no longer more general than the S 3 4 boundary

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