OK, so today we are pleased to welcome a unanimous withdrawal from a University of Warwick, where he's a postdoc.
So he got his PSG in 2019. And I did and say in Paris.
So his main research interests lie in numerical approximations for statistical models and in particular,
sampling algorithms on the Monte Carlo method.
And today we'll talk about it properly suggested long month project or so.
You know, thank you very much better on interaction.
Thank you very much for the invitation, and I'm very happy to talk today in the seminar for attending.
So, uh, my talk today would be about having done the Monte Carlo and some.
Tuning properties of this something algorithm and tuning programmes that we can face within when using this this sampler, and I'm going to.
Start straight away. This is a joint work with another product from the new week whose name is going.
And I'm going to speak my presentation into three parts.
That's what I'm going to explain a bit what is going on in Monte-Carlo do some presents.
What is the method? How it works and to also introduce some tuning problems for both smart metres.
We are going to look at some suggestions that have been proposed in the literature to tackle these problems.
And we are going to motivate you to take on the question,
the problem we are going to study and in particular, we are going to introduce diffusion process.
That's not a diffusion. So here the word normal diffusion should be understood as it does in several names of the ratio.
The one we are going to look at is also named Alabi pronounced the diffusion.
Or can you take several names? We just use this nose, run them all of these,
and we are going to connect with passion of HSC that relies on one amazing stuff into integration time, known as organisation.
And we are going to compare some exponential mixing rates between the two.
This will motivate our study and our construction of an algorithm because he suggested last month
trajectories that we will present as a robust alternative to agency and try to convince you what sense it is.
Progress here. What business means? Yeah, robustness refers to robustness of tuning of its metres.
So how how easy it is to calibrate the barometers of a particular intuition time?
And I might we call it a robust technology to HMRC,
but can also be seen simply as a robust extension in the sense that you can recover the sample of a particular case of this.
OK. So if you have any questions, I would make some some pauses between between these sections.
But if you want to interrupt me, there's also possible online as well.
I have a much. So let's go I start with a few notations just to define the metrics, I'm going to use this.
So here they are. All right.
And he has the he has the goal in this talk, we are going to focus on the following objective,
which is to stop at approximately from the target distribution. We set a density of spectral Ebix measure or not.
So the first line here, we only assume that is a distinction by that.
The medium density positive everywhere Onazi and Will denotes Phi to be minus log density up to an additive constant if we call with unselfconscious.
So this becomes a function, we assume that it is if you smooth.
And uh, and there's also. Multiculturalism, CMC something, right?
And somehow, we are not going to refer to any particular statistical model here.
Just think about this staggering submission by us, the distribution of high dimensional Bayesian statistical model, for instance.
And we we will study the properties of of several mcms categories, including this HSC,
something which is built upon internal dynamics that are defined here. And so what's what do they mean these dynamics?
Well. You see, they are defined. Here is a system of function equation which describes the motion of both Position
X and the TV so that if you start from any location in the state space X0.
Zero zero wealth, you will follow some contours here of an extended target base to define freeze density that as a product of by and goes.
And that's it. OK. So these are not mix.
They have one property, they leave this distribution by starting violence in the sense that if you start
from zero zero that are distributed according to by starting from for any team,
activity will also be to be according to buy. So in that sense, they leave by starting violent now.
Of course, these these are not exactly deterministic, which means that they cannot yield negative process.
Simply because if you start from location x zero zero,
behaviour would be completely deterministic and you can report the conversion distribution to was based on the.
So to be able to achieve that, we are going to need to do something on the messenger algorithm, which we will do in two sites.
But before that, we are going to, uh, to present one method to approximate these dynamics because in most general situations,
we cannot have custom solutions for these benefits. So what we do instead is we approximate these dynamics with a timely situation.
Here, I present the most probably one of the most famous additions for form for the I'm going to be known as the leapfrog integrator,
also known as the storm of any updates, which is defined as follows.
So what is its principle? Well, it's built upon the splitting of damage and dynamics into two parts.
First, we start with the motion of the velocity for half a step for each of the two,
but then we will follow ease following the dynamics of the position.
X4 Footstep Agent, we recompose again with equations of motion for the velocity adjusted.
And so together they yield this leapfrog in the weight, or they will do some of it dates.
But there are some, some properties when it comes to playing a bit of this correction and b the simpler after I just eat something out of it.
And amongst these properties are two proteins that are particularly useful are
the fact that the preserving first and second that they are time reversible.
So the second second notion simply means that somehow the forward move here.
Can be related to the back one move up to a flip of the velocity issue, we first sign of a velocity somehow related to backward moving to the one.
And we will call you, Michael. I mean, I think we all just think, though here we are a composition of the number of steps.
Right. So in particular, if we choose ED to be proportional to two for each.
And we will approximate using this if I can, Rachel.
I'm kind of trajectory off.
So built upon this tiny segregation, the standards Hamilton and Monte Carlo algorithm was introduced in nineteen eighty eight.
And its principle is the following.
So it requires choosing something by step, intuition, time see, and the number of steps will be personnel to operate.
And at the start of the trajectory, we will simply draw fresh and noise.
The Prime. And from zero and we prime we will propose.
And you may like Al Harrington and trajectory for any progress that we are going to accept with stunning probability.
That corresponds to the ratio of densities. Between the output of the trajectory and the initial position, velocity X0.
And. I thought to be able to yield.
Mac of Canada believes it's such an ATC mission, if you just look at the three last points, if you want to have a backup candidate.
There's no distribution balance. You need a technical, just technical step that's insists on flipping the momentum.
Whenever there is a rejection, so you sign just minus in velocity, when neither do you.
Of course, this is of very little importance here because at every step of recovery, we just refresh free the momentum,
which means that today could just erase this last step, and it would not have changed anything in the algorithm.
So, so if the momentum keeps erased by four momentum refreshments, why talking about this for two minutes when you have the answer?
Because if you have two years later, a transition of this algorithm was introduced known as a it, and it's built on the following principle.
It's the same equation, except it involves now an additional parameter alpha known as assistance barbiturate.
So when that fight is equal to zero, we recover exactly the HMRC algorithm that they just presented.
Now, as soon as Alpha is strictly positive, the momentum refreshment becomes passion,
right, and the closest allies to one, the more partial the refreshment vehicle.
OK, so one motivation for for inducing persistence these successive trajectories is that somehow to yield efficient something,
we want to be able to explore efficiently the safe space to be able to reuse the corporation as much as possible.
So in that sense, it's it's a good motivation for it.
Now there is a drawback here, which is that now the last step. Involve some women, something that will be only partially raised as soon as advice,
really positive, and when that fact goes to white, there would be less and less.
So in particular, this this decision, because of these momentum, Phillips received,
I think, limited interest compared to the do not start out of HFC Sumpter,
although there are quite a lot of work that have been devoted to trying to reduce the number of momentum flips by metallurgical innovations,
including, for instance, built upon delayed reaction methods, for instance, where we increase a bit decompression time to be able to avoid rejection.
Because when these when these momentum piece occurs,
somehow it does really the opposite of what we would like because it always will backtrack upon the rejection.
So when the crystals drips, somehow the beat the dynamics of X are guided by velocity.
So if you just flick the velocity v, then then you will just go backwards, OK?
Right, so this is actually quite frustrating for this choice of alpha,
especially because one of the original goal of the paper in 1991 by your wits was to suggest to use Alpha out close
to one to be able to mimic a difficult process known as Los Angeles and possibly introducing a few sites and.
And the point is that this choice actually leads to a lot of money,
people that has to be controlled somehow by choosing banks that we're going to talk a bit more about this and you few slides.
But before that, I'm going to talk a bit more about the two of the family jewels, about the choice of the times that each and the integration.
So one the question about these algorithms is how to choose its family does,
and we are going to talk a bit about about solutions that have been subject in literature.
To tackle this point.
So one first solution concerns the choice of the time stamp each and is built upon a study of the scaling limits of the HMRC algorithm.
And there are these assumptions here, a tool that assumes that the potential function right as a sum of your potential is right.
So this assumption actually is very strong and very, very realistic because it simply boils down to assuming that distribution is a product for.
Oh. Which means that all the components are in the balance here.
And so if all the components in Iran, then we can simply if we can simply solve each of the something problem individually.
So this is not really about how realistic this assumption is here,
but it's more convenient mathematical framework to be able to study the scanning limit and especially the behaviour
of the acceptance rates as dimension goes to infinity by when choosing a time step that decays when dimension.
So that's really the goal here is to understand how to accept reject mechanism works in high
dimension on the simple mathematical model that helps us do something that solves the equation,
I guess. And so so this these works known as optimal scanning.
Actually, I've been, uh, first introduced for one, a much simpler then having studied for my algorithm.
And then in 2013, I've been the writer so far, the HSC,
something that I basically sent in some quarters and the results that they obtain is the following if you choose the time step to be.
Sine function here that's that goes to zero when they mention goes to this, the minus one four,
then I think particularly with the acceptance rate, which converts to a non-trivial balance between going one.
The first reason that this rate that it manages to gets Mexican state that is strictly between zero and one, I think particularly.
And then the mathematical framework is actually meaty enough to be able to enable
the optimisation excuse the optimisation of several measures of efficiency,
including the equation distance, for instance, that yields a very simple, puny ruler.
And that's the best choice of time step.
I seem to think that you can get here these two two time centres that the X excellence rates will converge to sixty five percent.
So this gives us more some information about the number of steps and therefore the number of green evaluations that we are going
to have to compute to be able to reach a certain amount of integration time should be invested proportional to the times.
And this is also a very simple tuning rule for choosing each.
That is simply built on monitoring out of out of mcms outputs.
The acceptance rates, global acceptance rates of the CMC change and to tune the times typical.
OK. Let's say that, so it's quite convenient rule for tuning,
we simply highlighted that it's also a partial answer to the question we just asked because it's tuning of a time step h.
Conditionally on a fixed physical time and for anyone, value of advice could reasonably assume that they have free refreshments.
Now we are going to talk a bit more about the choice of the intuition and.
In particular, we are going to hear simply briefly talk about screening solutions that have been proposed, known as the Newtown sampler.
Let's listen. And somehow, I think it's fair to say that the choice of the integration empty is really one big question for the tuning of HSC.
And there have been a lot of lot of works that if tuning for these, but me too, I just present one idea here.
But there are many others, and I think these works received quite a lot of interest because they are useful to provide
the black box something that's functioning that require really minimal tuning from users.
Also, you can calibrate automatically that is the value of T.
And in the case, often you get on simpler without really detailing the algorithm.
The ID is built upon the principle that the one should try to follow the dynamics until this stuff is met.
While I'm showing that some of the right to such an idea division, of course.
And why? Why this?
Well, because when we look at the equity social media, the squares some distance between zero and we derive these dynamics through the internet,
where we get that whenever these these products is positive, then this distance would be increasing when it becomes negative.
And then we would reduce the distance somehow, if he wants to be able to explore that space efficiently, we want this distance to be to be large.
And therefore these Nets sampler provides automatic tuning algorithm whose aim is to maximise the distance.
By having said that.
So if you are happy with this, with this measure of efficiency and you are happy with this, go well, that's definitely fine, right?
I mean, that's you. You should choose this measure of efficiency.
Somehow, it gives you guarantees about the ability and metrics, but it cannot give you any decent guarantees for a fixed component.
So in other words, it might be that if you maximise the distance you have good mixing for, especially for for large components.
Compressors have lost gains, but you really don't say anything uniformly, especially for components that might have small scale.
So this we are going to highlight in the following. He's not really particularly effective of the 20s or 30s, if not me, where they say,
I'm really highlighting the fact that's, for instance, you want to choose a fixed value of tea.
Actually, you can find very simple examples where this can become very problematic.
And actually, you don't have to go very far. Actually already on the go and target you with inteligente of scales.
So we suppose we have different sigma eyes here that different ones we just have from viruses.
So you might want to see my decon pronouncement that different. And and we look at the the problem of choosing one value of integration and that
we'll be able to control all the correlations at once over the components one.
OK, so we defined this with the correlation function or fruit of tea for component.
I had to be just a coalition of tea and excited zero.
And here on the cushion case, virtually everything is explicit.
The calculations are very straightforward and easy. These are two corporations.
They simply boil down to cosine is function with different elements. OK.
So we can see that's on the graph here, if you sit, oppose all these other operations for this,
these functions with given bandwidth and you have a bunch of them with different skills,
then the task of choosing one that you see that will that will control all the conditions at once is quite hard, right?
And sometimes it's just into other networks.
If you look at the maximum value of these operation functions as a function of where these functionality can be actually arbitrary,
arrest you close to one. So one main solution.
So just even a little retro to tackle this problem is to rely on choosing at random the amount of time each iteration and so each before each starts.
So I think we will draw the amount of interest.
I'm from a certain distribution here in state how they consider the choice of the exponential distribution with rate number.
And from from this, the rest is the same as before.
And the purpose of this is to be able somehow to induce a smoothing effect on
the correlations so drawing at random and making these repeated repeated loops.
The goal is to somehow average the corporations of HMRC and negotiate case the conditions are that as when we can see here,
some of the conditions of this randomise approach actually boils down to the corruption of the HMRC for
components and that we obtain something here that can be uniformly controlled by the CFO of the larger scale.
There is a uniform down that will be monotony was like to no minus one thing that would control
that once all your operations and recall that this number here is the rate of the commission.
The submission, which means that the universe of lambda is the expected integration.
So when the expected integration time goes to infinity,
somehow this is bound vanishes and we are able to control all these smooth correlations at once by choosing a number minus one sufficiently large.
So in the next section, I'm going to motivate bits.
The problem is that we are going to study particular, I'm going to introduce this loss of an efficient process.
So if you have any questions before, please don't hesitate. All right.
All right, so here we are going to introduce a 1990s way to introduce some randomness in the current trajectories.
And the process that we are going through to do this here is known as the launch of our diffusion process,
and there are several of the names, including an approach to asking you to pronounce its initial announcement as a lot of gotcha.
Depending on you, I think you will look at it should not be confused with what it's highly of an upper limit.
And so here just note that it involves still both the position and the Velocity V,
and it's very it's a you see that it's it's very related to an and that it makes in the sense that when these permits are gamma is equal to zero.
We just speak about internet dynamics and when Islamic gamma is tricky because it didn't reek of a stochastic differential equation here,
that is somehow. Somehow, same assumption, and I don't but but see some momentum refreshment at that continuously induced by your brain motion.
OK, so this this gamma is known in the region either as dumping damage or the friction comet.
And you can see some out here, one property of this of process is that it leaves the stationary distribution by store.
She's just probably beaten by an in violence, just as dynamics do.
How are they? One difference with the confident because as soon as gamma is pretty positive.
Well, now the randomness induced can can yield equity.
So I think there is because you will be able somehow to convert distribution based on starting from zero zero and suitable assumptions.
So what we try to is this this diffusion process has a negative way in which we run them,
this trajectory that is not built upon randomised integration dynamics.
And so there is an actual question that here we are investigating these can we can we achieve some robustness opportunity with activity in the
sense that can we achieve some some unified control over the U.S. using this division process in a similar way as we had for randomisation?
OK. So we consider the same same question example here we have this,
and I'm going to ask you to believe me that we can compute explicitly this with the quotation
functions for the diffusion and then push in case they bow down to these equations.
It's also better known equations, I think it's better simply to plot what happens to these functionality.
OK. So he's the first ballots in the first part, we will consider that there is only one 6th scale by April one.
And we will let us do that. The friction parameter value that's several values of frequency do, in fact.
And we can see here that the choice going backwards zero.
We recover this sort of grey line that simply boils down to the question is functionality the creation of agency spirit?
And as soon as I saw positive, all the conditions will converge to zero.
I still go to the first point. Now, if you compare these choices of Ghana, well,
the choice can make or zero is the one that will make these coalitions reach zero faster than any or the choice of Ghanaians,
there is no right to do so. Somebody who if you're able to to to use to use tool to mimic exactly the dynamics and you get this to equal pay over two,
then you just get to independence ethics samples with a C.
And there's no hope by somehow inducing some momentum refreshment that you will go faster than this.
So in that case, this means that if you want to sample from a standard, if done there,
that no matter if you want to sample from a isotopic distribution, well amongst the stars of gamma, gamma equals zero is the best for sure.
Now we are going to to you just right and those are affect the next time.
But just before that, I just want to mention that you can see the behaviour when when Gamma one and two here is
three and above a certain threshold equal to then they become they decay monotonic here
and somehow there's a phase transition gamma equal to that also coincides with the choice
of gamma that will that will optimise the exponential rate of convergence to zero.
OK. So any value of galaxies would be below the threshold,
we will refer to these values as the end of the arms regime and every bad you have come out at the higher than this threshold,
we will call them over three.
So now let's plot to the same with operations back in at least one framework, we are going now to fix some values of gamma and let the scales back.
Here they are. All right. So which was two different values of government gamma or zero?
And this face One Nation two here. The fate of.
Well, the only thing I wanted to to, uh, to say here is that some of the deaths are two to three different behaviours above.
And also maybe, maybe I should use another word we can discuss.
OK, thank. So, so here we are going to fall to two different venues have come up.
We are going to plot different, different states. So on the lessons, I can see that same grass as a few slides earlier for AMC and when get equal to.
So these bombings are here. Well, we can see that there there's different views,
depending on whether stigma is higher than the referent, higher or lower than the reference states.
Michael, sorry, blue line corresponds to a reference can you can see that particular every scale at a lower than than this reference scale.
They they have conditions that will decay in there would be oscillatory,
but they will all decay to zero and they will all be uniformly dominated by this.
By this quotation corresponding to see Mike or what?
OK, so actually this this actually is quite interesting and it is quite simple to rule here if you want to control all the quotations at once.
The following if you choose some of the friction Panopto to be equal to the two divided by the not just the larger scale.
Well, you can control all the corporations.
Here by by the way, the coalition responding to this larger scale, which decays exponentially fast, we wish them well somehow.
This is a similar property as what we would observe from an opposition team.
And in that sense, its neighbours. We argue that that's a positive choice of friction allow for better business opportunity.
So now I'm going to connect this notion diffusion to the randomisation process.
And here you don't have to bother too much about the first half. If you don't want some of this,
first half is simply to express that stochastic process is we are going to study here the
notion that diffusion randomisation process can be characterised through their infinite,
infinitesimal generators. So this quantity here that simply corresponds to the derivative of the expen expected function.
If something from sentiment and we are going to define the Giteau corresponding to them,
then this one and we are going to define two different types of refreshments, one built upon Poisson process and the other one beat upon Winbush.
And so the randomisation generator and the generico can be expressed as the sum of to generate those.
And they have one component, which is this. And can I mention one generator?
Could I thought it was one and they differ through the the choice of momentum refreshment?
So I wonder what the randomisation? The refreshments are in just the discrete way by a question process with an intensity that.
So the higher that is, the more often refreshments, OK?
OK. And the higher alpha is spammy to hear.
The more partial the refreshment becomes assistant for me was how partial the refreshments are, right?
And on the second process, there's this family to get on a friction barbecue that controls how how fast we are going to refresh the momentum.
Nicotine was way slower than motion.
And so there is one reason that we present here is that when you choose Alpha, that goes to one while the intensity number goes to infinity,
that goes to one, the spike in these particular rates for this value of gamma.
Then what happens is that the generator from the mothership CMC converges to the generator of the stratosphere.
OK, so is really what does it mean when it means that the more passion and frequency the refreshment becomes A.?
Well, the closer we get from continuous refreshment induced by the emission and therefore the closer optimisation C becomes from the most efficient.
So we will highlight this this result, together with another result that we have found which studies quantitative mixing rates for organisations.
OK. So here we do not appear to be just transition, budgetary transition,
kind of a fundamental issue with arbitrary parameters in density London and Bessie Smith Alpha.
And we are going to try to states some convergence reasons about the new stamp process under these assumptions, a street year.
So these assumptions A3 boils down to assuming that the distribution is truly concave,
which is the same as assuming that the issue of the potential is lower bounded by positive constants.
OK, so the right hand side of the assumption actually was already assumed the first line of presentation,
because it just boils down to assuming that the greatness in this context and here is the theorem.
If you choose the intensity from that to be same function as we see in the last line, it's due to go to that.
If I go to one aspect of this rate for constants here is not the square root of an endless m.
Then for any choice of alpha, we get some exponential convergence.
Can we expect to do to sustain distance in perspective and to go over here as follows?
Right. So so far, I need this. I think for something, it seems up to the first one, right,
so that certain distances this distance between the pointy measures and some after time T two starts from distribution new,
while the distribution after time t will be closer and closer to two to zero, it will converge to zero exponentially fast with the rates up.
And so then of course, I see here on CS where this can be controlled.
But what is important here to highlight is the rate of convergence up and so.
So these are the bounds. It's true, but somehow this this mathematical framework here allows allows us to to get quite
explicit rates of convergence with respect to the bombing to alpha and bow.
And we also due to this constant slowdown and capital assumption.
OK. And so the rate of convergence here, you can see that it's optimised its increasing function of alpha,
and therefore it's optimised when alpha goes to one.
My solution is the design solution when I was 12.
And so when you combine both both results together and when you you remark it, it actually this this rate of much.
The limit of this rate of emergence was obtained when that was when it was obtained for the nonchalant decision in the previous paper,
when actually you can see these results actually has an interpolation between previous results that you
give you some co-authors and what we have done like early on doing this for the natural diffusion.
And it's some interpret these results and the interpretation and interpretation we present here is the following we might argue
that the natural diffusion can be seen as a limit of SUMMARISATION that achieves the fastest exponential rate amongst song like.
All right, so now I'm going to have to introduce the algorithm that we are studying, if you have any questions before I jump into the item we propose.
Please don't hesitate. It's the proof.
The proof here is built upon a synchronous closing argument, just as in the paper that I referred to.
And so the choice of coupling is not that complicated is that choice of of coping.
Now here the the technical technical parts relies on choosing a twist of the metric that is
amongst the best tweets of the metric you can to be able to obtain the best rates as possible.
You can also. We're going to get more Panopto and some details.
Good. OK, so let's go.
So here is the time position that we consider for the of division.
So this is finding someone motivated construction of some player directly built on lots of
trajectories instead of as an alternative to to using randomised to try something new.
Sometimes. And we are going here to define setting a vision that is built upon this leapfrogging director that you see in the media.
So it's very similar to this to this [INAUDIBLE] thing, Rachel. The difference is that we get some passive momentum refreshment before and after.
So here Alpha is no longer a free barbecue is set by the values of friction, gamma and the time step h.
OK. So when, when, when it goes to zero infinity, we can interpret.
These are the Times station of the Najiba dynamics, and you can see then when amazing to see where we just recall the stomach.
So we use these particular Thanksgiving edition because it's enabled to to preserve some,
some useful properties of this stuff, but it dates time variability and present for something useful.
When reading this correction on something, we are going to denote the update category, as he said, they often have to come up.
That refers to the distribution of HIV h given exorbitant.
So Excite is not out going on. So start don't, though, not a special moment of refreshment.
Keep this in the city and bits of fresh freshness and something we are going to propose, something you might even call them like this.
But it's just Eskom composition step of the subject to denote completion of the muck of chew gum.
So here is the algorithm, so we could expect adjusted large event trajectories or models,
and it requires choosing the amount of friction gamma a time separation to wish them.
So at the start of each trajectory, we completely throw a new Gaussian refreshment just as for some other agency.
And then instead of HSC, we have now to propose a on trajectory.
With friction going up, and so when is equal to zero, I know, right, what's really happened,
but when you go to see you actually recover, exactly, do the agency go with them?
Actually, he's done. City-The does not exist that somehow we should just consider the racial difference.
I guess what's what we obtain for the HMRC case and when I get my strictly positive, then then the excellence ratio becomes.
This ratio of density is where you have a density suspected of racism, and you can interpret this probability in a in several ways.
I just chose this one just to highlight the fact that's somehow on the denominator here.
You can interpret it as density of the forward trajectory from x zero zero to x LDL and
on the right or here we can interpret it as the density of the backward trajectory.
We split more into.
So as before, if you want us to do something, if you want these three steps through to get a mark of candidates leaves the decision you balance,
you need to flip the momentum upon rejection, but since there are some, some full refreshment of the cities here.
Well, actually, this line is completely erased at each HQ where you propose.
So this is this is a difference with the GMC approach for which for any choice of was the assistance Alfa,
then we could not erase those momentum clips and force them out due to a trade off between between
them having not too many freedoms without having a big enough time step to reduce the carnage.
All right, so we argued that this correction yields a neat way to to stabilise the diffusion in
particular compared to previous approaches aiming to mimic this natural division.
Our approach is different in the sense that we aim to metabolise whole trajectories of land.
We ask people's approaches. We're studying the one step, one step correction.
And for instance, I was talking about this Jamestown. And so if you take it James sideways, I mean, you take only one step and one step.
And you have this of refreshment. That is very special, something we don't see very close to one.
You can also make the most. It's usually when the thermostat goes to zero. However, here there is one difference for for, you know, our push.
We have one degree of freedom, which is the sense of the trajectory shooting by the user.
And and that can help to reduce the coalition base by choosing it in a smart way.
So the second property here is that momentum fees can be raised by food refreshments, which is not the case for GHC approach.
And we argue that that this scandal. So this gives them more time goes and gives a robust extension to HMRC.
That's going to last point here because we've seen that somehow using the positive dumping can enable control of the West or the production function.
And achieve similar similar goals as what we would do when drawing us from them, the inspiration empty.
So in particular, also here we highlight the fact that as soon as my strictly positive trajectories, trajectories will be equity,
which means that you cannot really suffer from some problem that you have in HNC when game is equal to zero,
which is that if you let your your intuition t be large,
you will just come back at some point to your original point and wrestle away some national and resources while having a high correlation.
Right, so when when my strictly positive and you look at the trajectory of a Flamsteed somehow its trajectory with conversion distribution.
To to buy stock, and if TI's large enough, some of the conditions would be to zero.
And some of the output of the Georgie will be closer and closer to to an independent sample from from
not exactly from my because it will be a flow decision because these times of the nation's best bet,
it will become less and less dependent from the stock of the. I mean, how to turn the situation.
At different points. So want to have.
I'm not sure what you mean, so for the no yet done you talking about the choice of integration I see right there at work.
Yeah. When he sgamma is very close to zero, you can get close after a certain amount of discretion 90 to your original point,
but somehow when a sufficiently large,
there is no particular reason why you should come back in a deterministic fashion to your original point on the.
That's. But maybe we can discuss the semantics, I suppose.
So we finally support this agreement with the theoretical.
Justification that essentially is quite similar to the one that has been established for HMRC in 2013.
My best guess in some quarters, which is down to a study of the spending limits under this assumption.
So it's not the same assumption, it's the same assumption as what we have for HMRC.
So we thought for that assumption, we can prove that's for any choice of friction.
Gamma and this molten Gotham will have a skating limits that that will enable two other very
explicit optimisation of efficiency measures that's yielded the tuning for continuing rule.
If you choose your step such that the exit acceptance rate converges to 65 percent, some maximises measure of efficiency.
And this is true for any choice of, in particular, its its seats or so.
As a consequence of the fact that we use the default integrator inside this, this demonstration of their entrepreneur vision,
and we wanted to highlight somehow that we can preserve several properties of HMRC by beating down
that we in particular in particular the setting of the location of the accident probability.
I spoke to H when he study, you can see is is quite related to the agency, although although here it was not the approval.
Right, so that's that's some of that's the shooting rule stimulatory rules for see what you need conditional on T and on the friction, right?
And so the friction. So we we look at the use of positive friction to be able to have a robust tuning strategy for T.
And then we simply argue that the tuning of X can be done in a similar way as what we would do for four agents.
Particularly we also recovered one force getting lost, but the damage. I that's I'm going to just summarise contributions here, so first,
we present not division has a limit the fondamentaux agency that receives the first assessment 17C rate for some gift nuggets.
Then we look at the use of positive dumping to enable control of the workplace and therefore allowing
for for rubbish tuning respect to the intuition and mate completing all these conditions at once.
Then we introduced this piece I just did not want to go is I go easy and we argue that this gives a
correction for security in particular compared to previous approaches aiming for this notion of process.
Here we metabolise whole. I think there is, which enabled to choose the lens to take the risk,
and we highlight the fact that the momentum is now erased fully by full momentum investments which yield nice meets mathematical framework.
In particular, the back of can off the chain of the position can be shown to be reversible.
Thanks to this property. And finally, the we,
we extend the outstanding results of best course and quarters for any choice of
friction without that assumption and gets therefore this wonderful scanning for most,
just as we do for agency. So we are interested in several, several perspectives for future work,
so I still present some make comparisons because it's actually is almost finished work, it should be an archive within a few days.
So we are still finishing using to make comparisons, which is why I didn't present here.
And so there are several questions we we are quite interested in in working on in the future, particularly,
we are quite interested in this adaptive tuning strategy for choosing a and gamma to be able to yield competitive.
Competitive some players compared to other athletes doing so that have been proposed for HMRC and therefore hopefully can yield some,
some maybe some information, but this gives a flip.
And also, I think there's also quite a lot to do in terms of comparisons with this fundamental agency approach,
either from a full article viewpoint on medical viewpoint. OK, thank you very much for your attention.
If you have any questions, please don't hesitate. Thank you.
Thanks. You say we have tried to stop it already, but.
Spend. I. Doesn't look too bad to prioritise, to change it, to do it.
So I've got some cards, but I want. That's excellent.
Yeah, that's right. OK, thanks, I.
That doesn't look like it would be best for. And so it's promising.
What kind of guidance for expressing acceptance of the fact that some sides.
I wish I wish I had my mind, my working paper here, but so, yeah,
so so so let me try to to explain what we observe, for instance, so I can see simulations.
Let's let's come back to the bush time off because it's true.
It's easy, but it's also quite easy to even. I'd say quite a few things about this.
So, for instance, we have an example in our working paper that somehow compares is more to get them together with randomisation with HMRC.
OK. And I suppose we would have here.
We take the same same framework that they presented, and we take some, some different scale sigma one sigma.
And in the working paper, I think we just don't seem to be to be between the 150 commission members.
Is this 50, which is OK because everybody is not super high?
And actually when you do that and you look not only distance, but you look at,
as you say, sample size, but not the sample size of one component, but the worst.
If if somebody says all the components that you would get, some of you will, you will see that actually for HMRC, the worse.
Yes, as a function of time for this particular model is quite an theoretic function of of.
That actually breaks down when they when commission member increase and when they must them.
And so if you are really if you want somehow to consider this measure of efficiency because you, you think that you, you all components here,
these depths of rest and you want to have a uniform guarantee of all the yeses at once them, well, somehow with these these solutions,
so much, says one of my agency and models where they can, they can probably do that for this particular model we have, we have some quite smooth,
riskiest time that we can draw for both organisations and families that achieves some good performance, even if there are some gains.
So you might say, OK, maybe for MCI can use instead a good recognition that to be able to reduce reduce the attention deficit.
So this is always doable and we didn't talk about it too much. It's presented as a separate problem than than than this particular problem,
because the preconditioning problem is something that you will do would do a purely somehow and and especially for something,
it's quite sometimes hard to find a good.
You have to find a global recognition of some also finding look out for condition in the position something is somehow.
Sometimes it's easy, but so especially for something, it's not so easy because you cannot use a look.
And when you have to use a global one, and there is a lot of situations where you can sometimes reduce the conditioning of the problem,
but you still end up with it sequentially with a standard of 20 or 50 and in that case, actually the HMRC sample we fixed.
So it's not really a particularity of agency itself.
It's the fixed choice of intuition that causes problems always in the Russian framework.
But it can be extended in all the areas where you see similar to that across the board, and you will also see problems when we can do with this.
I think the message has been. You know, we have a run over as speaker of economic.