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An example is where all the packets were assigned a common goal, which was to minimize a weighted combination of delay (W ) and loss (L) as G = W + L (16.1)

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A simple approach to adaptation is to respond in the sense of the most recently available data. Here the CP s cognitive memory contains data which is updated from the contents of the node s mailbox. After this update is made, the CP makes the decision which is most advantageous (lowest cost or highest reward) simply based on this information. This approach is referred to as the Bang-Bang algorithm. Instead some other learning paradigms for CPs can be also used like learning feedforward random neural networks (LFRNN) [57, 68] or random neural networks with reinforcement learning (RNNRL) [57]. Adaptive stochastic nite-state machines (ASFSM) [55, 56, 58] are another class of adaptive models which could be used for these purposes. 16.5.2 The random neural networks-based algorithms For the reinforcement learning approach to CP adaptation, as well the feed-forward neural network predictor, the RNN [57] was used in Gelenbe et al. [44]. IT is an analytically tractable model whose mathematical structure is akin to that of queueing networks. It has product form just like many useful queueing network models, although it is based on nonlinear mathematics. The state qi of the ith neuron in the network is the probability that it is excited. Each neuron i is associated with a distinct outgoing link at a node. These quantities satisfy the following system of nonlinear equations: qi = + (i) / r (i) + (i) with + (i) =

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(16.2) q j w + i ji

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(16.3)

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Here wi+ is the rate at which neuron i sends excitation spikes to neuron j when i is excited, j and wi is the rate at which neuron i sends inhibition spikes to neuron j when i is excited j and r (i) is the total ring rate from the neuron i. For an n neuron network, the network parameters are these n by n weight matrices W+ = wi+ and W = wi which need j j to be learned from input data. Various techniques for learning may be applied to the RNN. These include reinforcement learning and gradient-based learning, which are used in the following. Given some goal G that the CP has to achieve as a function to be to be minimized [i.e. transit delay or probability of loss, or a weighted combination of the two as in Equation (16.1)], a reward R is formulated which is simply R = 1/G. Successive measured values of the R are denoted by Rl, l = 1, 2, . . . These are rst used to compute a decision threshold: Tl = aTl 1 + (1 a)Rl (16.4)

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where a is some constant 0 < a < 1, typically close to 1. Now an RNN with (at least) as many nodes as the decision outcomes is constructed. Let the neurons be numbered 1, . . . , n.

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Thus for any decision i, there is some neuron i. Decisions in this RL algorithm with the RNN are taken by selecting the decision j for which the corresponding neuron is the most excited, i.e. the one with has the largest value of q j . Note that the lth decision may not have contributed directly to the lth observed reward because of time delays between cause and effect. Suppose that we have now taken the lth decision which corresponds to neuron j, and that we have measured the lth reward Rl. Let us denote by ri the ring rates of the neurons before the update takes place. We rst determine whether the most recent value of the reward is larger than the previous smoothed value of the reward, which is referred to as the threshold Tl 1 . If that is the case, then we increase very signi cantly the excitatory weights going into the neuron that was the previous winner (in order to reward it for its new success), and make a small increase of the inhibitory weights leading to other neurons. If the new reward is not better than the previously observed smoothed reward (the threshold), then we simply increase moderately all excitatory weights leading to all neurons, except for the previous winner, and increase signi cantly the inhibitory weights leading to the previous winning neuron (in order to punish it for not being very successful this time). This is detailed in the algorithm given below. We compute Tl 1 and then update the network weights as follows for all neurons i = j: Tl 1 Rl w + (i, j) w + (i, j) + Rl w (i, k) w (i, k) + Rl /(n 2), Else w+ (i, j) w + (i, j) + Rl /(n 2), w (i, k) w (i, k) + Rl If

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