# 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.

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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