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can make it run much more quickly. Specifically, in the inner loop of the algorithm where we apply value iteration, if instead of initializing valueiteration with V = 0 , we initialize it with the solution found during the previous iteration of our algorithm, then that will provide value iteration witha much better initial starting point and make it converge more quickly.

Continuous state mdps

So far, we've focused our attention on MDPs with a finite number of states. We now discuss algorithms for MDPs that may have an infinite number of states. For example, for a car,we might represent the state as ( x , y , θ , x ˙ , y ˙ , θ ˙ ) , comprising its position ( x , y ) ; orientation θ ; velocity in the x and y directions x ˙ and y ˙ ; and angular velocity θ ˙ . Hence, S = R 6 is an infinite set of states, because there is an infinite number of possible positionsand orientations for the car. Technically, θ is an orientation and so the range of θ is better written θ [ - π , π ) than θ R ; but for our purposes, this distinction is not important. Similarly, the inverted pendulum you saw in PS4 has states ( x , θ , x ˙ , θ ˙ ) , where θ is the angle of the pole. And, a helicopter flying in 3d space has states of the form ( x , y , z , φ , θ , ψ , x ˙ , y ˙ , z ˙ , φ ˙ , θ ˙ , ψ ˙ ) , where here the roll φ , pitch θ , and yaw ψ angles specify the 3d orientation of the helicopter.

In this section, we will consider settings where the state space is S = R n , and describe ways for solving such MDPs.


Perhaps the simplest way to solve a continuous-state MDP is to discretize thestate space, and then to use an algorithm like value iteration or policy iteration, as described previously.

For example, if we have 2d states ( s 1 , s 2 ) , we can use a grid to discretize the state space:

a normal grid

Here, each grid cell represents a separate discrete state s ¯ . We can then approximate the continuous-state MDP via a discrete-state one ( S ¯ , A , { P s ¯ a } , γ , R ) , where S ¯ is the set of discrete states, { P s ¯ a } are our state transition probabilities over the discrete states, and so on. We can then use value iteration or policy iterationto solve for the V * ( s ¯ ) and π * ( s ¯ ) in the discrete state MDP ( S ¯ , A , { P s ¯ a } , γ , R ) . When our actual system is in some continuous-valued state s S and we need to pick an action to execute, we compute the corresponding discretized state s ¯ , and execute action π * ( s ¯ ) .

two downsides. First, it uses a fairly naive representation for V * (and π * ). Specifically, it assumes that the value function is takes a constant value over each of the discretization intervals(i.e., that the value function is piecewise constant in each of the gridcells).

To better understand the limitations of such a representation, consider a supervised learning problem of fitting a function to this dataset:

graph. roughly x=y

Clearly, linear regression would do fine on this problem. However, if we instead discretize the x -axis, and then use a representation that is piecewise constant in eachof the discretization intervals, then our fit to the data would look like this:

the above data set, with a stepwise line added

This piecewise constant representation just isn't a good representation for many smooth functions. It results in little smoothing over the inputs, and nogeneralization over the different grid cells. Using this sort of representation, we would also need a very fine discretization (very small grid cells) to get a good approximation.

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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 meant by 'nano scale'?
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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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
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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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