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See [255] for further discussion. use .net code 3/9 drawer toembed 3 of 9 on .net QR Code ISO speicification Control Techniques for Complex Networks Draft copy April 22, 2007. 11.5 Estimating a value function Value functions h ave appeared in a surprising range of contexts in this book. (i) The usual home for value functions is within the eld of optimization. In the setting of this book, this means MDPs.

9 provides many examples, following the introduction for the single server queue presented in 3. (ii) The stability theory for Markov chains and networks in this book is centered around Condition (V3). This is closely related to Poisson s inequality, which is itself a generalization of the average-cost value function.

(iii) Theorem 8.4.1 contains general conditions ensuring that the h-MaxWeight policy is stabilizing.

The essence of the proof is that the function h is an approximation to Poisson s equation under the assumptions of the theorem. (iv) We have just seen how approximate solutions to Poisson s equation can be used to dramatically accelerate simulation. With the exception of (i), each of these techniques is easily applied in a wide range of settings.

The basic reason for this success is that in each of these three cases we are approximating a value function. In the case of (iii) the function h in the h-MaxWeight policy is only a crude approximation to the average-cost dynamic programming equation; The simplicity of this policy is a consequence of this modest goal. In contrast, the curse of dimensionality arises in optimization when we seek an exact solution.

In this nal section of the book we consider methods to construct approximations via simulation. Our goal is to learn the value function based on experiments on the network. Of course, learning brings its own curses.

This is summarized in the following remark from [350]: A large state space presents two major issues. The most obvious one is the storage problem, as it becomes impractical to store the value function (or optimal action) explicitly for each state. The other is the generalization problem, assuming that limited experience does not provide suf cient data for each and every state.

The rst issue is resolved by restricting to a parameterized family of approximate value functions. The learning problem is then reduced to nding the best approximation within this class. If we are lucky, or have some insight on the structure of the value function, then a parameterization can also resolve the second issue.

For example, if it is known that the value function is convex, then the family of approximations can be constructed to share this property. This imposes some continuity so that if a great deal is learned about the value function evaluated at a particular state x0 , then this information is useful for learning the value function at nearby points. In the case of networks, there are natural parameterizations to consider based on results from previous chapters.

. Control Techniques for Complex Networks Draft copy April 22, 2007. (i) The uid valu 3 of 9 for .NET e function J is the natural starting point to approximate the solution to the average-cost value function. In the case of the single server queue, Theorem 3.

0.1 can be applied to conclude that the following parameterized family includes the actual value function, h (x) = 1 J (x) + 2 x, x R+ , R2 ,. where 1 = 1 when h = h solves Poisson s equation. The discussion in Section 3.4.

5 suggests similar approaches to approximate the discounted-cost value function. Example 11.4.

4 illustrates how this approximation technique extends to networks. (ii) The family of all quadratic functions can be regarded as a parameterized family. Linear programming methods were proposed in Section 8.

6 to construct a solution to (V3). (iii) The perturbed value function introduced in Section 4.9 is another example of a parameterized family of functions that can potentially approximate a value function.

For example, given the family of functions {h(x) = h0 ( )} where h0 ranges x over some class, and the perturbation x de ned in (4.93) depends on the parameter > 0, what is the best value of and h0 Each of the parameterizations in (i) (iii) can be used to obtain an approximate value function for control or simulation. How then can we nd the best approximation The evaluation criterion will depend on the context.

In the case of simulation we might choose the approximation so that the resulting asymptotic variance is minimal. For control, the ultimate goal is to optimize performance over the class of policies considered. The algorithms described here can be used to approximate the value function for application in approximate value iteration or policy iteration.

In this case, the metric to evaluate the approximation should re ect our goal to optimize performance. In the remainder of this section we return to the general Markov setting in which X denotes a Markov chain without control on a state space X with transition matrix P , and unique invariant measure . It isn t necessary to assume that X is countable, but we do assume there is a xed state x X satisfying (x ) > 0.

A function c : X R is given, and our goal is to estimate a value function such as the solution to Poisson s equation, or the discounted-cost value function. The basic approach to compute the best approximation is stochastic approximation or one of its variants..

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