Just
because another couple
of people one of them
some of the basic.
It was advice
quite well.
So when I came to
forget it and about twenty eight
of the twenty nine of them.
It was the best papers for
focus for the course of
extremely well and
I think a couple of those
others continue now actually
it's still going on today.
So I will have some more people but
it's running out of you know
and you know life is also probably.
A game I say we need in the middle
of a new miracles don't happen
more than once in a lifetime and
this miracle happened two weeks ago we saw
yesterday in love with you know they're
still everybody is your community.
Somehow you know dealing with
political grievances you know.
OK I hope you left me coming in and
out but I have some
you know I have to do all sorts of things
in the most important this is before you
let you go closer the minute it's not
only here it is we should also have
the pacification expanders
going to reducing our quest because of
a promise health and registrants that
are hosted in the divine
side by side with us.
And yes even working for
a number of years for
various players or
for thinking of you know
this group regarding
conflict many of them everything
will come back with more of a lot of
other financial thought I just did it for
this one joint best stuff.
Thank you and I said that it
was strong with my co-author So
it was written in the last week so
it wasn't just myself that I liked.
So OK so here's an outline of
the talks start by just giving a very
brief introduction to the net.
Programming and
some background on the someplace else.
And stating the results and then dive
into the techniques a little bit and
these lower bounds for randomized to
winning rules are based on connections to
market decision processes so
I'll give you an introduction to
market decision processes and
tell you about this connection.
And then try to sketch the ideas for
these lower bound so
you actually get a lot possible for
a number I still want.
And then in the end if I have time so
I guess I'll focus on this just
perform uniformly random pivots but
if I have time I'll try to talk about two
of them were in the marketing force and.
So I'm sure you all of your old
pretty familiar with a program.
Even a programming but
let me just refresh real quick.
So you have objective function a set
of the constraints which defines
a political and then you want to find
an extreme point in some of the reaction
which makes them ISIS
this picture function.
And the verses of this kind of
political blog put when I called
the Basic feasible solutions.
And if you have such a program
you can always put it
into a standard form where you just have
to have you call it is this can be done by
introducing select variables for
instance and then a basis is
a subset of nearly independent
columns of this this matrix.
So here I use for the number of.
Qualities in for
the number of of of variables and
then there's a difference that I mention.
So if you have basis.
And you said all of the basic variables to
zero then you get
the basically simple solution.
So then you are at a corner.
So every every basically
simpler solution is.
I find by such a basis there could be more
basis bases moving to the saying
basically civil solution.
So if we had some basic feasible solution
defined by some basis then there
is this operation of people being where
you are extends a variable in your basis.
It seems like a column I guess.
With some.
One basic variables to use into basic and
in one piece of arable This
gives you a new basis.
And this is called pivoting.
And geometrically this
corresponds to moving along and
it's of the pilots or so if you move
to another basic useable solution and
you have moved along and
it's case of didn't do this.
Might it not happen but I'm not
really getting to the general cases.
OK so then I'm sure you.
You know the simplest algorithm we
just started some basic few simple
solution we see.
What are the improving pivots and
then we keep performing improving pivots.
And this was introduced by then stick
in one nine hundred forty seven.
And we keep doing this one to
learn new improving pivots and
this local optimum is
also a global optimum.
OK so one method for
doing this is the method.
So you start the son in a program.
You bring it to stand out form by
introducing these variables and
then you pick your starting pieces.
And this has to.
So of course you have to find this but
then you can you can do a two
to two first simplex for your.
First to find this place
this initial basis.
But once you have this basis.
You express the.
Basic variables in terms of
the number six variables.
And when you said the one basic variable
to zero then when it's written in
this form you can immediately read
off the basic simple solution.
And you do the same for the objective
function and then the riggers that then
it's very Then you can easily
see which either improving pin.
So if I increase X.
one then you can see the value of
the function is going to increase.
And therefore I can do and improve and
pivot by increasing X. one.
OK so then you'll just keep increasing X.
one in this case
until another basic variable becomes
zero and then you extend this.
And this completes the pivot.
And in case you can keep increasing it.
To infinity then the program is on bond.
So we'll just do that this then you get
to a new basic physical solution and
you have some new options for pivoting.
And here it's important to note that
the simply cycle them kind of covers
all these options but then you need
to specify your pivoting ball.
Exactly which it is so going to tools and
we keep doing this until we have a party
of someone you can see that your solution
is optimal and when all the questions here
in the objective function are negative.
It's OK so that's kind of
the introduction to enough programming.
So so
here when we when we perform to people.
We had actually we have two choices
which defines the people and so
we have to pick one basic variable
which has a positive course
they should enter the basis and then it
could be that there are two variables.
If if if the generator could
be two variables that could
both leave that basis.
And this when you specify
these to list the fine details
in the original building or
suggested by density.
You would just pick the variable which
has the largest question in the after
Exit Function and we didn't hear it
here is not really specified would.
Variables that leave the basis and
actually if.
You know a problem is a general
you could have cycling.
But one way to get out of this is.
Either to do some small to preservation
to get rid of this degeneracy or
you can use something called trance rule
which is defined in the following way
through just always so you have a set of
variables which are part of your Pick the
one with the smallest positive question.
And you can exchange that with the one
with the smallest possible just
suspicions of the.
Kind of book candidates for
leaving the vase.
And then this is guaranteed not to cycle.
So these are just the tool building rules
but there have been many differing rules
and this is the original one.
The largest go first.
Including one thousand nine hundred
seventy two ought to be exponential.
So this was about the time we started
caring about the worst case complexity and
then they gave an example of
something like this is actually.
Deformed.
This is this is kind of
a very regular view but
this is like a deformed cube that they've
made and then they gave a low bond where
the logical person putting a board
walk along all the edges of this this.
If you will excuse me if you will and
therefore would require
exponentially ministers and
since then all of the deterministic
building rules that have been suggested
over time at least the natural ones
that are fairly that can be analyzed.
Have been shown to be exponential.
So even if you instead of
just using the largest.
Could fish and you could actually perform
the pivot see what new value you get and
then pick the the people that would
give you the largest increase.
This is also exponential.
Steve.
If you say it's going most in the
direction of the objective function and
this is almost always going
into length rule and so on.
She had a word six that's
appearing rule where you'll
start at some vertex and you see.
OK if the objective function
was this then this person X.
would be optimal and
then you kind of sift your function
until you reach the actual objective
comes in this would define a part.
So this is also one people in the world.
This was by the way the one that's
used in small Most another system by
human and saying.
And mentor and so
you can have kind of given a uniform
view of all of these law bands and
all of this is based on having
some form products of politics.
So something very similar to
all of this can be secured.
OK.
So let me just say quickly so
if you have the diameter of
the point of this this August.
You know the maximum distance between
any two vertices of the square and
of course this will be lower bound for the
simplest of them because it would have to
walk along between these two
words as along that this and
this this has been data which stated that
at most in minus D. and I'm sure you
know that this was this proof of the last
you have I sent us just by a little bit.
I think if one hears like zero point
zero five or something like that.
So it's still open
the diameter is polynomial.
And one thing you could hope for is to
show that some simple randomized pivoting
rule would find the short
path of the high probability.
Or just find us or path like this.
So kind of what we saw is that.
We give almost exponential law pounds for
the most natural
randomized building walls.
And actually this construction
that we give has there's a lot.
I myself saw.
So this can.
So it was that it's this will probably
be a very difficult approach for
funding the diameter of what it's all.
OK.
So let me just mention the rest
of my spring was that we started.
So we study random acts which is
the one I will primarily focus on here
just perform a uniformly random pivot.
So you just pick you out
you on one basic variable.
Uniformly at random.
There's also will called.
Random festered and
this was discovered by Collider and by and
independently by image which
looks at the end of it and
what you do is to do you have your current
basically solution which is some good sex.
You pick a uniform you're in
a place that contains this person.
And then you say OK now I stay within this
person until I find an optimal solution.
When you find enough for most it was and
then you pivot out of this first and
you'll never return to this place again
and you can exit is you know that this
prevailing will find an optimal solution
in a sub exponential number of steps so.
And so this is also one of the bit of
putting was a study and finally there's.
A Randomized Lance reward way
of just before you run plan for
your brain on the commute
the indices system.
So or so here are the low
points that we actually saw So
they they all have have this
subject peninsular on the random
basis which is the fourth
root of in the rain affected.
It's two to the third and the most
painful tubes as well and I can see here.
They're actually in our sort of
people we gave this floor plan for
four in the place of which
we incorrectly proved so
we actually proved it for
this randomized plentiful.
So you can you can see them in a way
such that they seem very very similar.
And we actually made the mistake of
of claiming that they were the same
expectation.
So plentiful you have to you have
the indices of the of the variables and
you always pivot and
improve that has the smallest index.
So if you take the right on my splints
one you just start by randomly shuffling
the indices.
And then you run plants well after that.
So I should just say that in this
in this paper we actually proved the lower
bound for in my splendid world and
we thought that this was a lot of fun for
in the present but
we have since then been able to repair
this lot on what we look at we lost
something because all of us would have
been tied up to live with me said yeah.
So I guess the downside to this is
that the disk constructions are for
a specific building walls so.
So you would have to make
a new construction for
another people in the world.
I guess this technique kind of allows
you to just give you some tools.
It's makes it more likely that you could
use the constructs are slow bonds but
what is not clear immediately how
you would get such a lot on but
I think I think it would not be too
difficult to modify the slope on so
that that you can basically and
are so you're saying.
So do you get you see how much you get and
then yeah.
Yeah.
So so
I think you can probably by introducing
site so I'll get to these constructions in
a way that I think you can you can do
some things that would get around this.
But but you would have to make a new
construction for you to put in.
So this is kind of kind of
the downside to this is force but
this was also how it was done for the
dismissed appealed and was kind of uniform
for the unified thing so there would be
nice if something similar was done here.
OK So that's kind of the result
we were not the first.
So we construct these types of law beyond
use place of the law bans so Friedman
my co-author and infernally gave similar
the world balance for four and then going
quite how it's over them which actually
performs multiple improvements with us.
Improving pivots in parallel.
So it's not as simple as I was in but
it is and I was afraid.
And when you were Friedman also managed
to prove a lot on for another building or
course that is used
interpreting all sorts.
So that's just some some places where this
technique has been applied since then.
OK.
I should also say that before our law
bans there were no notable pronominal or
bonds for these building was but
there were some supernormal or bonds for
abstract settings and
most specifically basically going to stand
by and sample four in the minutes and
biometrics of four for random.
But there's not much connection
to our problems here.
And for both random it and
random events where there is no
known exponential of a point.
OK so now I will if I understood
some questions or something.
I'll get into it with this these
markets just in processes and
kind of highlight the connection here.
OK so I'll just start out by
introducing them in terms of so
little you have a market scene and
then generalize from here.
So if you have a marketing.
You have this set of states and
you have a sort of transition or
you have a transition for each state which
leads you to some other state with some
purple it's I'm sure your on over this.
So you would place a talking somewhere on
this one some state in the in the first of
maybe according to some
probably just a fusion.
And then.
Instead you would move the talking.
According to these
transition probabilities.
And.
And here it be in this for the privilege
of this abuse and then you transpose time.
Peter.
That's the probabilities of being
in the States after K. steps.
So then you will just keep moving
like this and if the market and
earth a nice problem to this you would
get like a stationary distribution but
I'm not really going into that.
So I just want to say that you
get this kind of table where
you would be in the States
with some probabilities.
So now suppose that we assign reward to.
The action of leaving a city.
So the act of living instead
we say that's an action and
whenever you use a certain action
then you get a certain reward.
And then we are interested in the expected
told report that you would get so
basically you would have a cost for
using its action and you would want to
play these were all of these columns
by the corresponding cost and
it was sum up everything and
this would be the expected sort of cost.
It's.
And this is of course
this may not converge so
you need to introduce something
that interest convergence and
one thing you can do is to have a stopping
condition so we have like a term of state.
That's reached with the provision
one from all the states and
when you reach the stage you stop.
So this means that actually
these were also will
be a list they would sum to list in one.
So when you sum up everything then
indeed it's actually well defined but
what you get.
And the value of a certain state is
the expected total of water when you start
in that state and
then you do this this random walk.
And you get this into this
just arm of all of this table.
That's the expected top of the water and
the value is so
I said as I said the value is
when you started to sixty.
OK So what is the system process.
So in a market decision process
you also have the states.
But now instead of just having one X.
and for
you state you have a set of axioms and.
And you say it is associated
with the water of water and
this provides that if you
distributions over the states.
And you also have this terminal state
just to say that if you get here we stop
and then a policy that is a choice
of an action from each state.
So in particular a policy is
a marketing with a divorce.
So if you have a policy then this
defines values for all of the states.
And then some of them if it makes
the meisters the values of all
the states similar to investing.
So this is called an optimal policy.
And the goal is then tool to find
certain optimal policies and
it's not through lead and
ups and politics first but
this was filmed in the fifty's played
by Stephanie and also lead up women.
So this is what it means
to Solomon isn't this.
Yes.
So you're so.
So if you.
If you know games you have something
called Stop being perfect.
And that's what we're going for instance.
And this is similar to that.
So you so you would you would require
that you have the optimal solution.
No matter what state you're starting from.
Yeah and
it's also pure Yes It's also worth noting.
So these policies are always pure.
So.
Other questions.
OK.
So here is just a very very
simple example of an M.V.P. and
also just to introduce the notation and
I'm using so before the States.
I'm using circles for those and for the
rewards I'm using these time and verses.
And.
So to make an actual graph I use a square
when I have read of my solution.
And then the the season
progresses on the S.
and in this case I've shown
the values on the states.
So you just follow the path until
you reach the terminal state and
then you see how much
reward to you accumulate.
So from here.
We just get minus one.
So the value of this latest minus one and
the value of the states as it will.
And actually generally in the lower
bounds it's going to look
something like this but you basically you
have a past that you follow model is so
it's not as complicated as a market
general a market decision process.
OK.
So now I'll introduce the linear program
for these market a system process for
so this can be sold by the program and
before doing that.
I'll just say that in general since there
is an optimal policy similar to
maximize the value of all estate.
You may as well just try to make some
eyes to some of the values because this
is also obtained by the optimal policy.
And for the table this corresponds
to a serving of one in each use
of the first in these called
it in the first row or
it's like a uniform distribution and
we just kill it up and
now I define a variable X. which
constant number of times I'm using as.
So actually that's just
the sum of a certain column.
And of course now I can some value
in a different way I can just so
experience multiplied by
the corresponding costs.
So this is also going to give
me the sum of the values.
And these are the variables that I will
use for the for the linear program.
So the objective function of so
here is the program for
solving marketisation processes.
The objective function is exactly this
you take the number of
times using the actions and
you multiplied by the cost of your actions
and then you have to have some constraint.
So that you make sure that these actions
actually correspond to the number of
times you sing them.
But if you think about is how many
times do you use a certain actions.
So if the number of times
you get to this date.
Plus one because you also
started in this state.
So.
So it has kind of lists.
This cloaked flow conservation
form where you just say it.
So the number of times you
leave the state is equal to
one plus the number of
times you into the States.
And that's just what
this in a program says.
So then we can when we solve this.
Then we can we get
the correct values at least.
And even more.
So if you have a basic feasible solution
to this in a program this directly
corresponds to a policy for
the market decision process.
And this is also easy to see.
So just observe that
the right hand is positive.
So that at least one variable
on the left which is positive.
That means that at least one action
that is used leaving each state.
But on the Only if you have
a basic useable solution.
You can have at most in variables and
zero and
that means you can have only one base and
was used.
So this is exactly a policy.
And also so it's also this.
They start to form all day and I there's
the probability of getting to this date.
So I'm So this this equation to.
Everything that leaves the state.
If you will to everything
that gets to the state.
So that's why I have the I hear.
It.
And similarly you can see all that if
policy satisfies to stop a condition
then it also corresponds to
a basic feasible solution.
And this is just because
the values are not infinite.
So there's this one to one correspondence
between basic usable solutions and
policies.
And this this correspondence
kind of goes on.
So if I have if I have a policy
then I say that I switch
this when I exchange one action with
another and this which is improving.
If it improves the value and
this corresponds to an improving pivot.
And you know with improving pivots you can
always just stick with the coefficients
of the program is positive.
You can just check if you do the switch.
But only for one step.
It's a little here the value is zero.
If I move here than it looks like OK then
I get a value of tool X. to get something
more because people keep moving on but
what do you see the improvement for
one step is to and that is actually
the executor reduced cost of the program
and again a policy that optimal if and
only if there are no improving switches.
So this this this very tight connection
between the you know program and
in the piece themselves.
One thing that's special
about market system processes
is that you can actually perform multiple
improving switches in parallel and
this will also always lead
to a bit of it looks.
OK And this kind of leads us to
a policy it's a recent algorithm
which is very simple and it's a bit like a
similar set of You start with some policy.
So that's like a basic physical solution.
You just keep making problems with and
you can do the.
In parallel in particular if you only
make simple improvements with us.
Then it's a simplex algorithm for
the corresponding P..
OK so here's like an example
very everything is put together.
So the small small graph from before
the the corresponding to the program and
then like a eliminated three of the
variables and made a letter from it and
so you get to see the political and
you can actually see that.
We have expected this correspondence if
you have a basic piece of us in Austin.
It's like this.
This policy the sum of the values.
So here we have mine and
what's one zero and five and six.
So that's the sum of five.
This is also the value here in of the of
the current basically simple solution.
So if you if you only correspond
to single improving switches.
Then that's like moving
the single improving pivots so
this is a simplex so bold with them.
And just like in the So
here you have to specify which
improvements which is to take and
that's also just like a symbol so we have
to specify your people in the world.
Exactly yeah.
So there's no degeneracy
like that in this case.
And these X. variable several also
always be at least one because of
you always start in a certain state.
Of A.
And so we have this very tight connection.
So then it will give lot pounds for
the simplicity of them.
We may as well give in or
bounds for politics or reason for
market decision process.
Somehow for modernization process
you can kind of see this graph.
So you can see what's going on.
If you don't have the political
it's more difficult to do really
constructive get it and stuff like that.
So that's the strength of this technique.
So now you can actually
construct get it so
that you get the behavior that you want.
So I'll try to describe how you
can get a simple or plan for
plentiful also this this Mr Putin will so
then it's much simpler than if you do it.
Randomized and then how you can extend
this to a lot on four four random it's.
So what we do is for a given.
We define market decision process so
that this when we're in one of
the pivoting rule or this politician and
with Will will simulate and
in bit binary counter.
So we have some way of interpreting
policy the state of the vehicle and
then we show that you move
from one configurations and
it's a little bit of a notation.
So I'm going to use rewards which
are growing exponentially and
the intuition for this is that if you
have a very very large reward you'll kind
of throw away all the work you did
previously just to get this new reward and
this will help in the resetting.
So I say that and it has some priority P..
If the reward is minus key west of the.
P..
Where Kate is some some.
Something large and in particular this
means that the larger the priority then
it dominates everything but a smaller.
Order operators are kind of penalties and.
Even protests of rewards.
So that's kind of how it works out.
So then here's what the law
of banking stocks and for
this plentiful looks like so.
We have two levels here.
Each level corresponds to a bit so we
have forced it in a little here and then.
We interpret a policy.
Bit confused in the following way.
So we just see what is the choice
meet us at this B. state
if it's zero then the value of the biggest
zero is one in the value of this one.
And then the goal is to make it go through
all the different counting configurations.
OK So whenever you.
So here's the starting quality whenever
you have a policy then you need to specify
what are the improving switches and then
secondly you need to say this particular
pivoting rule of this particular policy
to rescind this particular policy.
For making a promise with us.
Will make the improvements
that I wanted to make.
So awful plans for all this is fairly So
these are the employees with as you see
if you go to this large even priority
that's like getting a very last reward.
So that's going to be an improving
switch and the same down here.
And I want to increment
the last bit first.
So the plan is for
all I can just say that.
Well the last they should always be
incremented first and then you go through
all then you get the entire sequence and
I don't think I'll go into too much
detail about how this works because
it's a bit complicated I guess but then.
Then you would get some new policy and
we would then again have to specify the
different problems with say the best to
get the improvement was that we want and
then we will continue from there.
Let's just do it a little bit.
So here we had to improve things which
to us because we hadn't even read
what we perform the one with the lowest
index according to our ordering.
Which was the one for the last bit.
When we now have here and even reward and
then other virtues other states will also
be able to get through this even reward.
So these now become improving switches.
And generally will perform all of these
and now this this kind of stabilized.
As of one.
So whenever you reach this
bill you would move in and
get this large and even reward which
dominates the small appearances down
here and
then you would continue from there.
So now let's let's just look
at sort of that we have
these these are kind of get
if these are greatest for
the business and this is a gator that
helps us with the reset behavior.
Yeah.
What is the.
Yeah so.
So the fact is we have this get
it was implemented a bit and
this basically if it said then it goes
to something with a lot even reward and
what's going to heaven is that
when we set a higher bid.
So let's do that.
Then the lower bids can now go and
get this last even reward and
this will this will cost
them two to reset so
that we get here we have a smaller reward
but now we would do the research instead.
Exactly.
Exactly.
And then they're kind of two components.
So you need something which are the bits
and then you need to do this research
behavior of when you when you when you
set up if you need to reset the law bits.
And when you.
And you can simulate a binary counter.
Yeah.
So in this case for this this would
actually give an exponential lower bound.
So I'll get to that in just a moment but
by then becomes something sup experimental
and I think I think yeah the big picture
is that we simulate this binary kind of
we specify what are the improving
switches and then we say OK so
these are actually also improves with us.
That will be performed by this pivotal.
And then you could go through
all this entire sequence and
it gets a bit complicated.
So you have to it's at every step you
would need to specify what are now
the improvements which is for this
pencil you would have like five faces.
Everything is fairly I mean
there are lots of details but
everything is truly just by themselves so.
So you just verify OK these
are indeed improving.
Switches.
Yeah.
OK So of course if we have the random it.
Now we actually perform random improving
switches and before we used the effect
that lens will always used one with
a lot index to control what's happening.
So now we need to get it.
That helps us to control what
is actually going to happen.
And here's here's the get it.
So it's actually also fairly simple so
you just pick your vertex with a single
Imprimis with and
now you make a chain instead.
And you connect You can do this you can
choose the length of this to yourself you
have an entire To universe to get
the same effect as going in and going up.
You have to make all
the improvements which is and
we define this in such a way that
there's always only one probably switch.
At the very front of
the team leading in so
you have a very specific sequence
that you need to go through.
So that means that making
this which takes a long time.
So then we can do these sort
of an interim switches.
And let's say you have tools such to.
Where you are one and
probably switch from before which
would be late compared to another.
So then we just make the train of that
improvement with longer than the other
and in each change.
Exactly one improving switch and
if you choose one uniform at random.
This is just like flipping a coin and then
you can you can with a turn of pound you
can you can kind of control the progress
made in these different things.
So if the length of one change
significantly longer than the other then
we're going to make sure that the high
probability we finish the first in first
the the shortest chain first and then this
is exactly why we need more states and
why our bot point becomes worse
because we need to to get this.
Control and some of that we
are introducing these longer chains and
the length of these things so.
You have before you have these to be and
then this is kind of the critical
point these be states and these should
be growing at each level because you
always want the last bit to change first.
Yeah you can you can do
you can actually say that.
So that's not how we actually found it so
this length rule is small for
the presentation but but
you can you can actually viewed as this.
So you have this very basic thing and
then we kind of build upon that to make
sure that we get something simpler.
There's still some
components that are missing.
But I'll get to those in just a moment.
OK so the growth of the length
should grow like this.
So if you have the case.
It's the length of a square times and so
we get the longest chains
will have a cubic length and
then install the number of states that we
are going to use this into the forth and
be assimilating and in bit by no
account of this and therefore we go.
A lot bound up in into the fourth world.
So that's why you get something so
obvious and so.
OK.
Yeah.
So then you can just go through all and
that actually every time
we have this really.
This was a exactly the word supposed to
be so we're now we can control things.
So I just sort of want to give this
one one small important point is
that now we have these very long seems so
this may delay resetting because we
actually have to make many improvements
which is when you do the research.
Just before you should just make one.
Now you have to make all of them but
the thing is that
this guy did this in a structured in
such a way that when you do research.
So it's very fast to switch in one way but
very slow to switch in the other way.
So when you're going in
the other direction.
All of these improvements
which is available at once and
then the reset is very fast and happens
with a high probability in the right way.
So let me just say what
what what still missing.
So here this is absolutely right that this
basic skeleton is exactly
like the plants will.
One thing that we didn't take care
of was that if we have such things.
They'd be basically kind of assuming
that they were reset from the beginning.
So you have.
You have all these different verses
of these different states and
some of them may actually go forward
while it's not it's only partially reset.
So we need these to be reset also
when we start to start over
in a new counting system and
this is actually where it gets slightly
more complicated to implement things.
So we need some How do
you hire a bit should.
Reset when a lot.
It is so you need a dependence of
the higher part of the start on some.
Thing further down here.
First of all you need
some editors going down.
You also if you can't have it going
directly to this big even read what
because then it won't care about anything
here which is just as a lot of water so
we now know where to need when to my
station before we didn't even have
any random isolation.
And finally you need this team to have
kind of an alternating behavior where you.
Sometimes you move in.
Sometimes you move out and
to do this we need an additional scene and
this is then retire from construction.
So there are some more details
that I didn't really get into but
these are all the components of the proof.
And then you get this sort of
the forethought of in lower bound and
of course you can translate this
actually in a program so but
I think it's people often asking what
does a lunar program looks like but
I think for understanding the robot
is much better to look at the M.V.P.
if you just look at this than you don't
really you can't see the gators here so
that was one of the real advantages
of this technique that you
can work with these graphs and that.
OK in the questions before I yeah.
So when you have the market just impresses
you can just do the translation that I
described before and
then you get this program.
Other questions.
So so so this this graph.
So the little program is facing.
The police will hear this and
then they improve as you can see
these directions and so this
defines this graph and then you could look
at it abstractly and in this way also.
Yeah I mean you could I guess you
could do the same with the Gators and
the differences.
So you have you have something
happening where you have to
you have to do something very specific
you have this single path which is
very long that you have to take to
actually get to the optimum and
you kind of fall to move away from that as
soon as something else happens somewhere.
So I'm not sure exactly what that
would look like but you can definitely
I mean you can definitely look at that as
an orientation of such a high book you.
So so one thing you should note
is that if you have a state and
you have something for let's say you
have like a very big way to hear
something very large you can
actually just replace this
by having a random eye Satan where you
move back with a very high probability and
then then have a small weight and
that here or whatever and
then this would eventually become a very
large weight so you can you can use large
waves but then you knew small probability
this and that if you if you kind of
make a restriction of on both of them
then then I don't know or anything.
That then it could be that you can't and
so
you don't get this kind
of behavior that's.
OK So let me just say I still
think I have a bit more times.
See a little bit about these two
other people in the water and and
kind of what how they are different and
then the random it's building.
So the random face appearing
wall was the following.
So you have.
A cure and basically a simple solution.
And you pick uniformly random
first if that contains this
basic feasible solution.
And now you recursively find the optimal
solution within this first and
when you have this optimal
solution then if possible.
You make a pivot living the first and
you continue.
If this is not possible.
Well then you already have
the optimal solution.
And this is kind of a primal
variant of this over them and
this is the do all formulation
of this benefits of.
So this list of then
you make an improvement
pivotal even just at a basic physical
solution where the new improving pivots
within the faces but there can still be
an improvement that leaps to faces and
then you would pick that.
No just just pick.
I mean it doesn't really matter
exactly which one to pick but.
I think that there might
just be one in that case.
OK so if we take a look at this so.
So we have our basic physical solution
which is contained in some sort of our
faces and now we pick one of
them uniformly at random and
we say OK now I'm going to
stay within this first.
So I'm going to find the optimal solution
recursively within this person and
then I will perform the pivot that
leaves this versus the key trick here is
that once you are at this.
Vertex there's basically solution.
You know that you can never into
this place again because now your
value is better than
anything in this verses.
But not only that it's also better that if
this is you ordered the fastest according
to the best value then it's better than
everything which has a smaller value and
you actually pick the first
the donor from that random here.
So so this actually defines
a recurrence for the other bond.
So if you have some political
dimension the infested
Well first uniform manifested
in the state within this.
So that's like solving something
of dimension one smaller and
one smaller place that you make one
pivot and then depending on how
which faces you chose to just take the
average of the different faces you chose
then you have rolled out
the corresponding number over here.
So if you take the ice then you
have ruled out I guess it's.
And when you solve this recurrence.
This gives you this exponents of one.
OK so if we if we.
So that's.
If we want to interpret what all this
means for market a system processes.
So what does it do so.
It is within a festered So
what does that mean that means
that an inequality is tight.
That means that a variable is zero.
So we have a variable which
is now fixed to zero and
then we solve recursively and
if you if you fix a variable to zero.
So remember the variables here that's
the number of times you used X.
ins so setting it to zero means
you can't use this action.
So that's just like we
have our him deeply.
Now remove uniformly random X.
in the not using and
then we solve recursively we
find some of similar policy for
this and then we insert this action
again and see if this hadn't
probably switch if it is you perform
that inference with if it's not.
And we're done.
So that's how you interpret
random festered for
the market decision processes.
Let me just quickly see how you would
interpret the very nice length rule.
So if you have the randomized
plans rules you have this random
permutation of the indices and
you're always starting from one one and
perform the promise with us from one and
some bicycle This means that
the same problems with the other end is
not going to be performed on all of the.
The preceeding improved
switches are performed.
And that's exactly the same.
So that's like
saying that that this is fixed to zero
until everything else has become.
Everything else is optimal So
we actually with a randomized lens
will if you look at that kind of
define it recursively you would again you
would have a random permutation of it.
Us and you would remove one of them you
know you would remove one from one end
of this permutation and
then you would benefit of them so the only
only differences in the run of president
of them you would keep using new random.
It is to remove you just use or
in the primitives and everywhere.
And I think I'm running out of time so.
I just want to say so.
So the challenges are a bit
different now because now instead of
dealing in problems with this we have
to take care of removing things and
the way you do this is by introducing a
lot of redundancy in some clever way and.
I think I'll just skip that.
So you get to get along.
Starting with looks very similar but
has some different get it's different.
So this could also be viewed as you
just take this plan through all
this very simple construction and then
you extended in some way with getters so
that you get the right behavior.
And so concluding remarks.
All these lower bounds for you actually
first not we didn't prove them first for
marketers just in processors
we proved them first for
something called parity games.
Which I didn't really talk to about here.
But if anyone is interested in the game
versions of this then I will be having
to talk to you afterwards open
problems of this kind of meta
problems with that so so also for him.
It can be talking to normal time because
you control and be in a programming.
It's not known how to solve my part of
this and processes in strongly polynomial
time there's a problem and all the problem
with this randomized Lance rule.
So you can see that it kind of
resembles the random face of lot.
You just that's also the reason
to be mistook it for
actually having the same
expected number of steps.
So is there something special of a bond
for this people in the world and
finally well so
now we kind of gave an approach for
showing the simply said with him even if
you have a know myself and perform sparely
could you once and for all of
the show that it's always better and
the only way I know of this is to
actually prove that the bullet or both
of the diameter prototypes is very large
So this is related to his conjecture.
So can you prove or disprove the point or
move your conjecture and and
that that's what they're doing in this
part of my three project and like I said.
Yes that's
so.
So then if so then it's first does this
permutations and
then then it does the shadow
vertex during all that I mentioned so
it is not using this particular food and
with this it was in some other specific
rule and it turns out that when you do
these random permutations then it
will be your opponent of our steps.
Yet there
by one.
OK so I must admit I'm not so
familiar with the sports I was for the
that's that's not known.
I think that would also
be very interesting.
So they have to have a very
certain structure but
still these probabilities as you
can set them in any way you want.
So it seems very general so
that would also be very interesting.
So can kind of beat it up.
Yeah yeah
yeah yeah
yeah.
So I didn't I didn't talk so
much about this.
So when you take this in front of some.
You can also use discounting to.
Make it finite and if there's conflict.
You're using.
So instead of having it's almost
if you have a discomfort.
And if the discount factor you're using is
some fixed constant which is not part of
the input then then it's pretty
normal then a strong upon normal but
when you let this conflict or
go to one and
this is part of the input then start
in on good and strong opponents.
So basically this this lower bound for
random is that you can actually show this
is also a lot when you perform all of
the improving pivots in parallel and
actually if you pick like a random
subset of these poor animals and
perform them kind of the worst thing is
that when you just perform a single one.
But but otherwise I mean you would
have to depend on the construction.
If it works for
four different people in wars or so
so so
again not not if you have like like that
this conflict of being part of the input.
Actually what.
What do you use your worst that
the largest actually works
in normal time it's wrong for a long
time before this conflict is constant.
So there was the algorithm
that he analyzed and
I think this is proof only depends on
increase that it does need to do so
it would also work with the largest
increase of people in the world.