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Resilient propagation (RPROP) [727, 728] performs a direct adaptation of the weight step using local gradient information Weight adjustments are implemented in the E E form of a reward or punishment, as follows: If the partial derivative, vji (or wkj ), of weight vji (or wkj ) changes its sign, the weight update value, ji ( kj ), is decreased by the factor, The reason for this penalty is because the last weight update was too large, causing the algorithm to jump over a local minimum On the other hand, if the derivative retains its sign, the update value is increased by factor + to accelerate convergence For each weight, vji (and wkj ), the change in ji (t) vji (t) = + ji (t) 0 where weight is determined as
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E if vji (t) > 0 E if vji (t) < 0 otherwise
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E E + ji (t 1) if v (t 1) v (t) > 0 ji ji E E ji (t) = ji (t 1) if vji (t 1) vji (t) < 0 (t) otherwise ji vji (t + 1) = vji (t) + vji (t)
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RPROP is summarized in Algorithm 61 The value of 0 indicates the rst weight step, and is chosen as a small value, eg 0 = 01 [728] It is shown in [728] that the performance of RPROP is insensitive to the value of 0 Parameters max and min respectively specify upper and lower limits on update step sizes It is suggested in [728] that = 05 and + = 12 Algorithm 61 RPROP Neural Network Training Algorithm Initialize NN weights to small random values; Set ji = kj = 0 , i = 1, , I + 1, j = 1, , J + 1, k = 1, , K; Let t = 0; while stopping condition(s) not true do for each wkj , j = 1, , J + 1, k = 1, , K do E E if wkj (t 1) wkj (t) > 0 then kj (t) = min{ kj (t 1) + , max };
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E wkj (t) = sign wkj (t) kj (t); wkj (t + 1) = wkj (t) + wkj (t); E E else if wkj (t 1) wkj (t) < 0 then kj (t) = max{ kj (t 1) , min }; wkj (t + 1) = wkj (t) wkj (t 1); E wkj = 0;
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E wkj (t) = sign wkj (t) kj (t); wkj (t + 1) = wkj (t) + wkj (t);
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end Repeat the above for each vji weight, j = 1, , J, i = 1, , I + 1; end
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For problems where only the immediate reward is maximized (ie there is no value function, only a reward function), Williams [911] proposed weight update rules that perform a gradient descent on the expected reward These rules are then integrated with back-propagation Weights are updated as follows: wkj = kj (rp k )ekj (614)
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where kj is a non-negative learning rate, rp is the reinforcement associated with pattern zp , k is the reinforcement threshold value, and ekj is the eligibility of weight wkj , given as [ln(gj )] (615) ekj = wkj where (616) gj = P (ok,p = tk,p |wk , zp )
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is the probability density function used to randomly generate actions, based on whether the target was correctly predicted or not Thus, this NN reinforcement learning rule computes a GD in probability space Similar update equations are used for the vji weights
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Connectionist Q-Learning
Neural networks have been used to learn the Q-function in Q-learning [527, 891, 745] The NN is used to approximate the mapping between states and actions, and even to generalize between states The input to the NN is the current state of the environment, and the output represents the action to execute If there are na actions, then either one NN with na output units can be used [825], or na NNs, one for each of the actions, can be used [527, 891, 745] Assuming that one NN is used per action, Lin [527] used the Q-learning in equation (610) to update weights as follows: w(t) = [r(t) + max Q(t 1) Q(t)] w Q(t)
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where Q(t) is used as shorthand notation for Q(s(t), a(t)) and w Q(t) is a vector of the output gradients, Q (t), which are calculated by means of back-propagation w Similar equations are used for the vj weights Watkins [891] proposed a combination of Q-learning with TD( )-learning, in which case, w(t) = [r(t) + max Q(t 1) Q(t)]