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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

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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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