# 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 ${a}_{t}$ .

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
how nano science is used for hydrophobicity
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
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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