# 2.12 Machine learning lecture 12 course notes  (Page 6/8)

 Page 6 / 8

A second downside of this representation is called the curse of dimensionality . Suppose $S={\mathbb{R}}^{n}$ , and we discretize each of the $n$ dimensions of the state into $k$ values. Then the total number of discrete states we have is ${k}^{n}$ . This grows exponentially quickly in the dimension of the state space $n$ , and thus does not scale well to large problems. For example, with a 10d state,if we discretize each state variable into 100 values, we would have ${100}^{10}={10}^{20}$ discrete states, which is far too many to represent even on a modern desktop computer.

As a rule of thumb, discretization usually works extremely well for 1d and 2d problems (and has the advantage of being simple and quick to implement).Perhaps with a little bit of cleverness and some care in choosing the discretization method, it often works well for problems with up to 4d states. Ifyou're extremely clever, and somewhat lucky, you may even get it to work for some 6d problems. But it very rarely works for problems any higherdimensional than that.

## Value function approximation

We now describe an alternative method for finding policies in continuous-state MDPs, in which we approximate ${V}^{*}$ directly, without resorting to discretization. This approach, caled value function approximation, has been successfully applied to many RL problems.

## Using a model or simulator

To develop a value function approximation algorithm, we will assume that we have a model , or simulator , for the MDP. Informally, a simulator is a black-box that takes as input any (continuous-valued) state ${s}_{t}$ and action ${a}_{t}$ , and outputs a next-state ${s}_{t+1}$ sampled according to the state transition probabilities ${P}_{{s}_{t}{a}_{t}}$ :

simulation. For example, the simulator for the inverted pendulum in PS4 was obtained by using the laws of physics to calculate what position andorientation the cart/pole will be in at time $t+1$ , given the current state at time $t$ and the action $a$ taken, assuming that we know all the parameters of the system such as the length of the pole, the mass of the pole, and so on.Alternatively, one can also use an off-the-shelf physics simulation software package which takes as input a complete physical description of a mechanicalsystem, the current state ${s}_{t}$ and action ${a}_{t}$ , and computes the state ${s}_{t+1}$ of the system a small fraction of a second into the future. Open Dynamics Engine (http://www.ode.com) is one example of a free/open-source physics simulator that can be used to simulate systems like theinverted pendulum, and that has been a reasonably popular choice among RL researchers.

An alternative way to get a model is to learn one from data collected in the MDP. For example, suppose we execute $m$ trials in which we repeatedly take actions in an MDP, each trial for $T$ timesteps. This can be done picking actions at random, executing some specific policy, or via some other way of choosing actions. We would then observe $m$ state sequences like the following:

$\begin{array}{cc}& {s}_{0}^{\left(1\right)}\stackrel{{a}_{0}^{\left(1\right)}}{\to }{s}_{1}^{\left(1\right)}\stackrel{{a}_{1}^{\left(1\right)}}{\to }{s}_{2}^{\left(1\right)}\stackrel{{a}_{2}^{\left(1\right)}}{\to }\cdots \stackrel{{a}_{T-1}^{\left(1\right)}}{\to }{s}_{T}^{\left(1\right)}\\ & {s}_{0}^{\left(2\right)}\stackrel{{a}_{0}^{\left(2\right)}}{\to }{s}_{1}^{\left(2\right)}\stackrel{{a}_{1}^{\left(2\right)}}{\to }{s}_{2}^{\left(2\right)}\stackrel{{a}_{2}^{\left(2\right)}}{\to }\cdots \stackrel{{a}_{T-1}^{\left(2\right)}}{\to }{s}_{T}^{\left(2\right)}\\ & \cdots \\ & {s}_{0}^{\left(m\right)}\stackrel{{a}_{0}^{\left(m\right)}}{\to }{s}_{1}^{\left(m\right)}\stackrel{{a}_{1}^{\left(m\right)}}{\to }{s}_{2}^{\left(m\right)}\stackrel{{a}_{2}^{\left(m\right)}}{\to }\cdots \stackrel{{a}_{T-1}^{\left(m\right)}}{\to }{s}_{T}^{\left(m\right)}\end{array}$

We can then apply a learning algorithm to predict ${s}_{t+1}$ as a function of ${s}_{t}$ and ${a}_{t}$ .

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
Do somebody tell me a best nano engineering book for beginners?
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
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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