2.12 Machine learning lecture 12 course notes  (Page 5/8)

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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 $\left(x,y,\theta ,\stackrel{˙}{x},\stackrel{˙}{y},\stackrel{˙}{\theta }\right)$ , comprising its position $\left(x,y\right)$ ; orientation $\theta$ ; velocity in the $x$ and $y$ directions $\stackrel{˙}{x}$ and $\stackrel{˙}{y}$ ; and angular velocity $\stackrel{˙}{\theta }$ . Hence, $S={\mathbb{R}}^{6}$ is an infinite set of states, because there is an infinite number of possible positionsand orientations for the car. Technically, $\theta$ is an orientation and so the range of $\theta$ is better written $\theta \in \left[-\pi ,\pi \right)$ than $\theta \in \mathbb{R}$ ; but for our purposes, this distinction is not important. Similarly, the inverted pendulum you saw in PS4 has states $\left(x,\theta ,\stackrel{˙}{x},\stackrel{˙}{\theta }\right)$ , where $\theta$ is the angle of the pole. And, a helicopter flying in 3d space has states of the form $\left(x,y,z,\phi ,\theta ,\psi ,\stackrel{˙}{x},\stackrel{˙}{y},\stackrel{˙}{z},\stackrel{˙}{\phi },\stackrel{˙}{\theta },\stackrel{˙}{\psi }\right)$ , where here the roll $\phi$ , pitch $\theta$ , and yaw $\psi$ angles specify the 3d orientation of the helicopter.

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

Discretization

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 $\left({s}_{1},{s}_{2}\right)$ , we can use a grid to discretize the state space:

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

two downsides. First, it uses a fairly naive representation for ${V}^{*}$ (and ${\pi }^{*}$ ). 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:

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:

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.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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