# 2.12 Machine learning lecture 12 course notes  (Page 4/8)

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Both value iteration and policy iteration are standard algorithms for solving MDPs, and there isn't currently universal agreement over which algorithm is better.For small MDPs, policy iteration is often very fast and converges with very few iterations. However, for MDPs with largestate spaces, solving for ${V}^{\pi }$ explicitly would involve solving a large system of linear equations, and could be difficult. In these problems,value iteration may be preferred. For this reason, in practice value iteration seems to be used more often than policy iteration.

## Learning a model for an mdp

So far, we have discussed MDPs and algorithms for MDPs assuming that the state transition probabilities and rewards are known. In many realistic problems,we are not given state transition probabilities and rewards explicitly, but must instead estimate them from data. (Usually, $S,A$ and $\gamma$ are known.)

For example, suppose that, for the inverted pendulum problem (see problem set 4), we had a number of trials in the MDP, that proceeded as follows:

$\begin{array}{cc}& {s}_{0}^{\left(1\right)}\stackrel{{a}_{0}^{\left(1\right)}}{\to }{s}_{1}^{\left(1\right)}\stackrel{{a}_{1}^{\left(1\right)}}{\to }{s}_{2}^{\left(1\right)}\stackrel{{a}_{2}^{\left(1\right)}}{\to }{s}_{3}^{\left(1\right)}\stackrel{{a}_{3}^{\left(1\right)}}{\to }...\\ & {s}_{0}^{\left(2\right)}\stackrel{{a}_{0}^{\left(2\right)}}{\to }{s}_{1}^{\left(2\right)}\stackrel{{a}_{1}^{\left(2\right)}}{\to }{s}_{2}^{\left(2\right)}\stackrel{{a}_{2}^{\left(2\right)}}{\to }{s}_{3}^{\left(2\right)}\stackrel{{a}_{3}^{\left(2\right)}}{\to }...\\ & ...\end{array}$

Here, ${s}_{i}^{\left(j\right)}$ is the state we were at time $i$ of trial $j$ , and ${a}_{i}^{\left(j\right)}$ is the corresponding action that was taken from that state. In practice, each of the trials above might be run until the MDP terminates(such as if the pole falls over in the inverted pendulum problem), or it might be run for some large but finite number of timesteps.

Given this “experience” in the MDP consisting of a number of trials, we can then easily derive the maximum likelihood estimates for the statetransition probabilities:

${P}_{sa}\text{(}s\text{′}\text{)}\frac{\text{#times took we action}\phantom{\rule{1pt}{0ex}}a\text{in state}s\text{and got to}s\text{′}}{\text{#times we took action}\phantom{\rule{1pt}{0ex}}a\text{in state}s}$

Or, if the ratio above is “0/0”—corresponding to the case of never having taken action $a$ in state $s$ before—the we might simply estimate ${P}_{sa}\left({s}^{\text{'}}\right)$ to be $1/|S|$ . (I.e., estimate ${P}_{sa}$ to be the uniform distribution over all states.)

Note that, if we gain more experience (observe more trials) in the MDP, there is an efficient way to update our estimated state transition probabilities usingthe new experience. Specifically, if we keep around the counts for both the numerator anddenominator terms of  [link] , then as we observe more trials, we can simply keep accumulating those counts. Computing the ratio of these countsthen given our estimate of ${P}_{sa}$ .

Using a similar procedure, if $R$ is unknown, we can also pick our estimate of the expected immediate reward $R\left(s\right)$ in state $s$ to be the average reward observed in state $s$ .

Having learned a model for the MDP, we can then use either value iteration or policy iteration to solve the MDP using the estimated transition probabilitiesand rewards. For example, putting together model learning and value iteration, here is one possible algorithm for learning in an MDP with unknown state transitionprobabilities:

1. Initialize $\pi$ randomly.
2. Repeat $\left\{$
1. Execute $\pi$ in the MDP for some number of trials.
2. Using the accumulated experience in the MDP, update our estimates for ${P}_{sa}$ (and $R$ , if applicable).
3. Apply value iteration with the estimated state transition probabilities and rewards to get a new estimated value function $V$ .
4. Update $\pi$ to be the greedy policy with respect to $V$ .
3. $\right\}$

where we get a research paper on Nano chemistry....?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
nanocopper obvius
Alexandre
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
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
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
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