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In detail, the algorithm is as follows:

  1. Randomly sample m states s ( 1 ) , s ( 2 ) , ... s ( m ) S .
  2. Initialize θ : = 0 .
  3. Repeat {
    1. For i = 1 , ... , m {
      1. For each action a A {
        1. Sample s 1 ' , ... , s k ' P s ( i ) a (using a model of the MDP).
        2. Set q ( a ) = 1 k j = 1 k R ( s ( i ) ) + γ V ( s j ' )
        3. / / Hence, q ( a ) is an estimate of R ( s ( i ) ) + γ E s ' P s ( i ) a [ V ( s ' ) ] .
      2. }
      3. Set y ( i ) = max a q ( a ) .
      4. / / Hence, y ( i ) is an estimate of R ( s ( i ) ) + γ max a E s ' P s ( i ) a [ V ( s ' ) ] .
    2. }
    3. / / In the original value iteration algorithm (over discrete states)
    4. / / we updated the value function according to V ( s ( i ) ) : = y ( i ) .
    5. / / In this algorithm, we want V ( s ( i ) ) y ( i ) , which we'll achieve
    6. / / using supervised learning (linear regression).
    7. Set θ : = arg min θ 1 2 i = 1 m θ T φ ( s ( i ) ) - y ( i ) 2
  4. }

Above, we had written out fitted value iteration using linear regression as the algorithm to try to make V ( s ( i ) ) close to y ( i ) . That step of the algorithm is completely analogous to a standard supervised learning (regression) problem in which we have a training set ( x ( 1 ) , y ( 1 ) ) , ( x ( 2 ) , y ( 2 ) ) , ... , ( x ( m ) , y ( m ) ) , and want to learn a function mapping from x to y ; the only difference is that here s plays the role of x . Eventhough our description above used linear regression, clearly other regression algorithms (such as locally weighted linear regression) can also be used.

Unlike value iteration over a discrete set of states, fitted value iteration cannot be proved to always to converge. However, in practice, it often does converge (or approximately converge), and works well for many problems.Note also that if we are using a deterministic simulator/model of the MDP, then fitted value iteration can be simplified by setting k = 1 in the algorithm. This is because the expectation in Equation  [link] becomes an expectation over a deterministic distribution, and so a single example is sufficient to exactly compute that expectation.Otherwise, in the algorithm above, we had to draw k samples, and average to try to approximate that expectation (see the definition of q ( a ) , in the algorithm pseudo-code).

Finally, fitted value iteration outputs V , which is an approximation to V * . This implicitly defines our policy. Specifically,when our system is in some state s , and we need to choose an action, we would like to choose the action

arg max a E s ' P s a [ V ( s ' ) ]

The process for computing/approximating this is similar to the inner-loop of fitted value iteration, where for each action, we sample s 1 ' , ... , s k ' P s a to approximate the expectation. (And again, if the simulator is deterministic, we can set k = 1 .)

In practice, there're often other ways to approximate this step as well. For example, one very common case is if thesimulator is of the form s t + 1 = f ( s t , a t ) + ϵ t , where f is some determinstic function of the states (such as f ( s t , a t ) = A s t + B a t ), and ϵ is zero-mean Gaussian noise. In this case, we can pick the action given by

arg max a V ( f ( s , a ) ) .

In other words, here we are just setting ϵ t = 0 (i.e., ignoring the noise in the simulator), and setting k = 1 . Equivalently, this can be derived from Equation  [link] using the approximation

E s ' [ V ( s ' ) ] V ( E s ' [ s ' ] ) = V ( f ( s , a ) ) ,

where here the expection is over the random s ' P s a . So long as the noise terms ϵ t are small, this will usually be a reasonable approximation.

However, for problems that don't lend themselves to such approximations, having to sample k | A | states using the model, in order to approximate the expectation above, can be computationally expensive.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Machine learning. OpenStax CNX. Oct 14, 2013 Download for free at http://cnx.org/content/col11500/1.4
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