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Will the algorithm of Table 131 converge toward a Q equal to the true Q function The answer is yes, under certain conditions First, we must assume the system is a deterministic MDP Second, we must assume the immediate reward values are bounded; that is, there exists some positive constant c such that for all states s and actions a , Ir(s, a)l < c Third, we assume the agent selects actions in such a fashion that it visits every possible state-action pair infinitely often By this third condition we mean that if action a is a legal action from state s, then over time the agent must execute action a from state s repeatedly and with nonzero frequency as the length of its action sequence approaches infinity Note these conditions are in some ways quite general and in others fairly restrictive They describe a more general setting than illustrated by the example in the previous section, because they allow for environments with arbitrary positive or negative rewards, and for environments where any number of state-action transitions may produce nonzero rewards The conditions are also restrictive in that they require the agent to visit every distinct state-action transition infinitely often This is a very strong assumption in large (or continuous!) domains We will discuss stronger
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convergence results later However, the result described in this section provides the basic intuition for understanding why Q learning works The key idea underlying the proof of convergence is that the table entry ~ ( s a) with the largest error must have its error reduced by a factor of y whenever , it is updated The reason is that its new value depends only in part on error-prone Q estimates, with the remainder depending on the error-free observed immediate reward r
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Theorem 131 Convergence of Q learning for deterministic Markov decision processes Consider a Q learning agent in a deterministic MDP with bounded rewards (Vs,a )lr(s, a ) [ 5 c The* Q learning agent uses the training rule of Equation (137), initializes its table Q(s,a ) to arbitrary finite values, and uses a discount a factor y such that 0 y < 1 Let Q,(s, a ) denote the agent's hypothesis ~ ( s ), following the nth update If each state-action pair is visited infinitely often, then Q,(s, a ) converges to Q(s, a ) as n + oo, for all s, a
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Proof Since each state-action transition occurs infinitely often, consider consecutive intervals during which each state-action transition occurs at least once The proof consists of showing that the maximum error over all entries in the Q table is reduced by at least a factor of y during each such interval Q, is the agent's table of estimated Q values after n updates Let An be the maximum error in Q,; that is
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Below we use s' to denote S(s, a ) Now for any table entry (in@, ) that is updated a on iteration n + 1, the magnitude of the error in the revised estimate Q , + ~ ( S , a ) is
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I Q , + I ( S ,) - Q(s, all = I(r a
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+ y max Qn(s',a')) - (r + y m x Q ( d ,a'))]
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a ' a a a
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= y I m y Qn(st, - m y Q(s1, I a') a')
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5 y max IQn(s1, - ~ ( s ' , I a') a')
5 Y m I Q , (s",a') - Q W , a')I y
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IQn+i (s, a ) - Q(s, all 5 Y An
The third line above follows from the second line because for any two functions fi and f2 the following inequality holds
In going from the third line to the fourth line above, note we introduce a new variable s" over which the maximization is performed This is legitimate because the maximum value will be at least as great when we allow this additional variable to vary Note that by introducing this variable we obtain an expression that matches the definition of A, Thus, the updated Q , + ~ ( S , a ) for any s, a is at most y times the maximum error in the Q,, table, A, The largest error in the initial table, Ao, is bounded because values of ~ ~ a ) s , Q(s, a ) are bounded for all s, a Now after the first interval ( and
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