1 00:00:06,960 --> 00:00:14,880 OK, so today we are pleased to welcome a unanimous withdrawal from a University of Warwick, where he's a postdoc. 2 00:00:14,880 --> 00:00:21,750 So he got his PSG in 2019. And I did and say in Paris. 3 00:00:21,750 --> 00:00:29,400 So his main research interests lie in numerical approximations for statistical models and in particular, 4 00:00:29,400 --> 00:00:33,060 sampling algorithms on the Monte Carlo method. 5 00:00:33,060 --> 00:00:38,750 And today we'll talk about it properly suggested long month project or so. 6 00:00:38,750 --> 00:00:43,200 You know, thank you very much better on interaction. 7 00:00:43,200 --> 00:00:51,150 Thank you very much for the invitation, and I'm very happy to talk today in the seminar for attending. 8 00:00:51,150 --> 00:01:01,500 So, uh, my talk today would be about having done the Monte Carlo and some. 9 00:01:01,500 --> 00:01:14,520 Tuning properties of this something algorithm and tuning programmes that we can face within when using this this sampler, and I'm going to. 10 00:01:14,520 --> 00:01:22,230 Start straight away. This is a joint work with another product from the new week whose name is going. 11 00:01:22,230 --> 00:01:27,560 And I'm going to speak my presentation into three parts. 12 00:01:27,560 --> 00:01:33,680 That's what I'm going to explain a bit what is going on in Monte-Carlo do some presents. 13 00:01:33,680 --> 00:01:43,970 What is the method? How it works and to also introduce some tuning problems for both smart metres. 14 00:01:43,970 --> 00:01:51,760 We are going to look at some suggestions that have been proposed in the literature to tackle these problems. 15 00:01:51,760 --> 00:01:54,530 And we are going to motivate you to take on the question, 16 00:01:54,530 --> 00:01:59,520 the problem we are going to study and in particular, we are going to introduce diffusion process. 17 00:01:59,520 --> 00:02:06,470 That's not a diffusion. So here the word normal diffusion should be understood as it does in several names of the ratio. 18 00:02:06,470 --> 00:02:11,330 The one we are going to look at is also named Alabi pronounced the diffusion. 19 00:02:11,330 --> 00:02:18,410 Or can you take several names? We just use this nose, run them all of these, 20 00:02:18,410 --> 00:02:28,850 and we are going to connect with passion of HSC that relies on one amazing stuff into integration time, known as organisation. 21 00:02:28,850 --> 00:02:33,770 And we are going to compare some exponential mixing rates between the two. 22 00:02:33,770 --> 00:02:40,760 This will motivate our study and our construction of an algorithm because he suggested last month 23 00:02:40,760 --> 00:02:49,430 trajectories that we will present as a robust alternative to agency and try to convince you what sense it is. 24 00:02:49,430 --> 00:02:57,260 Progress here. What business means? Yeah, robustness refers to robustness of tuning of its metres. 25 00:02:57,260 --> 00:03:07,220 So how how easy it is to calibrate the barometers of a particular intuition time? 26 00:03:07,220 --> 00:03:12,230 And I might we call it a robust technology to HMRC, 27 00:03:12,230 --> 00:03:19,850 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. 28 00:03:19,850 --> 00:03:27,500 OK. So if you have any questions, I would make some some pauses between between these sections. 29 00:03:27,500 --> 00:03:31,790 But if you want to interrupt me, there's also possible online as well. 30 00:03:31,790 --> 00:03:39,840 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. 31 00:03:39,840 --> 00:03:43,490 So here they are. All right. 32 00:03:43,490 --> 00:03:51,890 And he has the he has the goal in this talk, we are going to focus on the following objective, 33 00:03:51,890 --> 00:03:58,080 which is to stop at approximately from the target distribution. We set a density of spectral Ebix measure or not. 34 00:03:58,080 --> 00:04:02,150 So the first line here, we only assume that is a distinction by that. 35 00:04:02,150 --> 00:04:16,680 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. 36 00:04:16,680 --> 00:04:28,360 So this becomes a function, we assume that it is if you smooth. 37 00:04:28,360 --> 00:04:38,200 And uh, and there's also. Multiculturalism, CMC something, right? 38 00:04:38,200 --> 00:04:42,760 And somehow, we are not going to refer to any particular statistical model here. 39 00:04:42,760 --> 00:04:52,630 Just think about this staggering submission by us, the distribution of high dimensional Bayesian statistical model, for instance. 40 00:04:52,630 --> 00:05:02,440 And we we will study the properties of of several mcms categories, including this HSC, 41 00:05:02,440 --> 00:05:09,970 something which is built upon internal dynamics that are defined here. And so what's what do they mean these dynamics? 42 00:05:09,970 --> 00:05:18,880 Well. You see, they are defined. Here is a system of function equation which describes the motion of both Position 43 00:05:18,880 --> 00:05:24,040 X and the TV so that if you start from any location in the state space X0. 44 00:05:24,040 --> 00:05:37,570 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. 45 00:05:37,570 --> 00:05:41,370 And that's it. OK. So these are not mix. 46 00:05:41,370 --> 00:05:47,760 They have one property, they leave this distribution by starting violence in the sense that if you start 47 00:05:47,760 --> 00:05:52,830 from zero zero that are distributed according to by starting from for any team, 48 00:05:52,830 --> 00:05:59,490 activity will also be to be according to buy. So in that sense, they leave by starting violent now. 49 00:05:59,490 --> 00:06:08,050 Of course, these these are not exactly deterministic, which means that they cannot yield negative process. 50 00:06:08,050 --> 00:06:11,410 Simply because if you start from location x zero zero, 51 00:06:11,410 --> 00:06:17,160 behaviour would be completely deterministic and you can report the conversion distribution to was based on the. 52 00:06:17,160 --> 00:06:26,260 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. 53 00:06:26,260 --> 00:06:34,150 But before that, we are going to, uh, to present one method to approximate these dynamics because in most general situations, 54 00:06:34,150 --> 00:06:42,880 we cannot have custom solutions for these benefits. So what we do instead is we approximate these dynamics with a timely situation. 55 00:06:42,880 --> 00:06:55,120 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, 56 00:06:55,120 --> 00:06:58,400 also known as the storm of any updates, which is defined as follows. 57 00:06:58,400 --> 00:07:04,150 So what is its principle? Well, it's built upon the splitting of damage and dynamics into two parts. 58 00:07:04,150 --> 00:07:08,890 First, we start with the motion of the velocity for half a step for each of the two, 59 00:07:08,890 --> 00:07:14,810 but then we will follow ease following the dynamics of the position. 60 00:07:14,810 --> 00:07:22,030 X4 Footstep Agent, we recompose again with equations of motion for the velocity adjusted. 61 00:07:22,030 --> 00:07:29,080 And so together they yield this leapfrog in the weight, or they will do some of it dates. 62 00:07:29,080 --> 00:07:38,710 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. 63 00:07:38,710 --> 00:07:43,330 And amongst these properties are two proteins that are particularly useful are 64 00:07:43,330 --> 00:07:48,190 the fact that the preserving first and second that they are time reversible. 65 00:07:48,190 --> 00:07:54,650 So the second second notion simply means that somehow the forward move here. 66 00:07:54,650 --> 00:08:05,900 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. 67 00:08:05,900 --> 00:08:14,670 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. 68 00:08:14,670 --> 00:08:20,330 Right. So in particular, if we choose ED to be proportional to two for each. 69 00:08:20,330 --> 00:08:25,460 And we will approximate using this if I can, Rachel. 70 00:08:25,460 --> 00:08:30,770 I'm kind of trajectory off. 71 00:08:30,770 --> 00:08:40,730 So built upon this tiny segregation, the standards Hamilton and Monte Carlo algorithm was introduced in nineteen eighty eight. 72 00:08:40,730 --> 00:08:43,220 And its principle is the following. 73 00:08:43,220 --> 00:08:53,120 So it requires choosing something by step, intuition, time see, and the number of steps will be personnel to operate. 74 00:08:53,120 --> 00:08:59,120 And at the start of the trajectory, we will simply draw fresh and noise. 75 00:08:59,120 --> 00:09:06,480 The Prime. And from zero and we prime we will propose. 76 00:09:06,480 --> 00:09:13,300 And you may like Al Harrington and trajectory for any progress that we are going to accept with stunning probability. 77 00:09:13,300 --> 00:09:22,850 That corresponds to the ratio of densities. Between the output of the trajectory and the initial position, velocity X0. 78 00:09:22,850 --> 00:09:29,780 And. I thought to be able to yield. 79 00:09:29,780 --> 00:09:35,090 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. 80 00:09:35,090 --> 00:09:43,130 There's no distribution balance. You need a technical, just technical step that's insists on flipping the momentum. 81 00:09:43,130 --> 00:09:51,050 Whenever there is a rejection, so you sign just minus in velocity, when neither do you. 82 00:09:51,050 --> 00:09:56,480 Of course, this is of very little importance here because at every step of recovery, we just refresh free the momentum, 83 00:09:56,480 --> 00:10:02,690 which means that today could just erase this last step, and it would not have changed anything in the algorithm. 84 00:10:02,690 --> 00:10:12,530 So, so if the momentum keeps erased by four momentum refreshments, why talking about this for two minutes when you have the answer? 85 00:10:12,530 --> 00:10:25,310 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. 86 00:10:25,310 --> 00:10:32,420 It's the same equation, except it involves now an additional parameter alpha known as assistance barbiturate. 87 00:10:32,420 --> 00:10:39,810 So when that fight is equal to zero, we recover exactly the HMRC algorithm that they just presented. 88 00:10:39,810 --> 00:10:45,600 Now, as soon as Alpha is strictly positive, the momentum refreshment becomes passion, 89 00:10:45,600 --> 00:10:50,710 right, and the closest allies to one, the more partial the refreshment vehicle. 90 00:10:50,710 --> 00:11:04,010 OK, so one motivation for for inducing persistence these successive trajectories is that somehow to yield efficient something, 91 00:11:04,010 --> 00:11:09,360 we want to be able to explore efficiently the safe space to be able to reuse the corporation as much as possible. 92 00:11:09,360 --> 00:11:13,920 So in that sense, it's it's a good motivation for it. 93 00:11:13,920 --> 00:11:22,220 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, 94 00:11:22,220 --> 00:11:29,980 really positive, and when that fact goes to white, there would be less and less. 95 00:11:29,980 --> 00:11:37,450 So in particular, this this decision, because of these momentum, Phillips received, 96 00:11:37,450 --> 00:11:44,500 I think, limited interest compared to the do not start out of HFC Sumpter, 97 00:11:44,500 --> 00:11:56,290 although there are quite a lot of work that have been devoted to trying to reduce the number of momentum flips by metallurgical innovations, 98 00:11:56,290 --> 00:12:03,910 including, for instance, built upon delayed reaction methods, for instance, where we increase a bit decompression time to be able to avoid rejection. 99 00:12:03,910 --> 00:12:06,910 Because when these when these momentum piece occurs, 100 00:12:06,910 --> 00:12:13,990 somehow it does really the opposite of what we would like because it always will backtrack upon the rejection. 101 00:12:13,990 --> 00:12:20,600 So when the crystals drips, somehow the beat the dynamics of X are guided by velocity. 102 00:12:20,600 --> 00:12:35,690 So if you just flick the velocity v, then then you will just go backwards, OK? 103 00:12:35,690 --> 00:12:39,920 Right, so this is actually quite frustrating for this choice of alpha, 104 00:12:39,920 --> 00:12:46,850 especially because one of the original goal of the paper in 1991 by your wits was to suggest to use Alpha out close 105 00:12:46,850 --> 00:12:55,570 to one to be able to mimic a difficult process known as Los Angeles and possibly introducing a few sites and. 106 00:12:55,570 --> 00:12:59,560 And the point is that this choice actually leads to a lot of money, 107 00:12:59,560 --> 00:13:07,210 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. 108 00:13:07,210 --> 00:13:15,070 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. 109 00:13:15,070 --> 00:13:18,460 So one the question about these algorithms is how to choose its family does, 110 00:13:18,460 --> 00:13:24,670 and we are going to talk a bit about about solutions that have been subject in literature. 111 00:13:24,670 --> 00:13:28,980 To tackle this point. 112 00:13:28,980 --> 00:13:44,120 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. 113 00:13:44,120 --> 00:13:52,460 And there are these assumptions here, a tool that assumes that the potential function right as a sum of your potential is right. 114 00:13:52,460 --> 00:14:02,730 So this assumption actually is very strong and very, very realistic because it simply boils down to assuming that distribution is a product for. 115 00:14:02,730 --> 00:14:08,200 Oh. Which means that all the components are in the balance here. 116 00:14:08,200 --> 00:14:16,480 And so if all the components in Iran, then we can simply if we can simply solve each of the something problem individually. 117 00:14:16,480 --> 00:14:21,190 So this is not really about how realistic this assumption is here, 118 00:14:21,190 --> 00:14:28,000 but it's more convenient mathematical framework to be able to study the scanning limit and especially the behaviour 119 00:14:28,000 --> 00:14:35,710 of the acceptance rates as dimension goes to infinity by when choosing a time step that decays when dimension. 120 00:14:35,710 --> 00:14:40,630 So that's really the goal here is to understand how to accept reject mechanism works in high 121 00:14:40,630 --> 00:14:49,660 dimension on the simple mathematical model that helps us do something that solves the equation, 122 00:14:49,660 --> 00:14:55,270 I guess. And so so this these works known as optimal scanning. 123 00:14:55,270 --> 00:15:04,540 Actually, I've been, uh, first introduced for one, a much simpler then having studied for my algorithm. 124 00:15:04,540 --> 00:15:08,260 And then in 2013, I've been the writer so far, the HSC, 125 00:15:08,260 --> 00:15:18,280 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. 126 00:15:18,280 --> 00:15:23,440 Sine function here that's that goes to zero when they mention goes to this, the minus one four, 127 00:15:23,440 --> 00:15:29,140 then I think particularly with the acceptance rate, which converts to a non-trivial balance between going one. 128 00:15:29,140 --> 00:15:36,220 The first reason that this rate that it manages to gets Mexican state that is strictly between zero and one, I think particularly. 129 00:15:36,220 --> 00:15:44,500 And then the mathematical framework is actually meaty enough to be able to enable 130 00:15:44,500 --> 00:15:50,110 the optimisation excuse the optimisation of several measures of efficiency, 131 00:15:50,110 --> 00:15:57,880 including the equation distance, for instance, that yields a very simple, puny ruler. 132 00:15:57,880 --> 00:16:01,000 And that's the best choice of time step. 133 00:16:01,000 --> 00:16:09,770 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. 134 00:16:09,770 --> 00:16:17,420 So this gives us more some information about the number of steps and therefore the number of green evaluations that we are going 135 00:16:17,420 --> 00:16:26,930 to have to compute to be able to reach a certain amount of integration time should be invested proportional to the times. 136 00:16:26,930 --> 00:16:32,250 And this is also a very simple tuning rule for choosing each. 137 00:16:32,250 --> 00:16:37,590 That is simply built on monitoring out of out of mcms outputs. 138 00:16:37,590 --> 00:16:45,830 The acceptance rates, global acceptance rates of the CMC change and to tune the times typical. 139 00:16:45,830 --> 00:16:55,570 OK. Let's say that, so it's quite convenient rule for tuning, 140 00:16:55,570 --> 00:17:04,680 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. 141 00:17:04,680 --> 00:17:15,790 Conditionally on a fixed physical time and for anyone, value of advice could reasonably assume that they have free refreshments. 142 00:17:15,790 --> 00:17:20,320 Now we are going to talk a bit more about the choice of the intuition and. 143 00:17:20,320 --> 00:17:32,350 In particular, we are going to hear simply briefly talk about screening solutions that have been proposed, known as the Newtown sampler. 144 00:17:32,350 --> 00:17:45,430 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. 145 00:17:45,430 --> 00:17:52,360 And there have been a lot of lot of works that if tuning for these, but me too, I just present one idea here. 146 00:17:52,360 --> 00:18:00,190 But there are many others, and I think these works received quite a lot of interest because they are useful to provide 147 00:18:00,190 --> 00:18:05,840 the black box something that's functioning that require really minimal tuning from users. 148 00:18:05,840 --> 00:18:09,610 Also, you can calibrate automatically that is the value of T. 149 00:18:09,610 --> 00:18:15,910 And in the case, often you get on simpler without really detailing the algorithm. 150 00:18:15,910 --> 00:18:25,450 The ID is built upon the principle that the one should try to follow the dynamics until this stuff is met. 151 00:18:25,450 --> 00:18:29,470 While I'm showing that some of the right to such an idea division, of course. 152 00:18:29,470 --> 00:18:31,270 And why? Why this? 153 00:18:31,270 --> 00:18:45,700 Well, because when we look at the equity social media, the squares some distance between zero and we derive these dynamics through the internet, 154 00:18:45,700 --> 00:18:53,740 where we get that whenever these these products is positive, then this distance would be increasing when it becomes negative. 155 00:18:53,740 --> 00:19:02,050 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. 156 00:19:02,050 --> 00:19:11,650 And therefore these Nets sampler provides automatic tuning algorithm whose aim is to maximise the distance. 157 00:19:11,650 --> 00:19:13,160 By having said that. 158 00:19:13,160 --> 00:19:19,280 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? 159 00:19:19,280 --> 00:19:23,000 I mean, that's you. You should choose this measure of efficiency. 160 00:19:23,000 --> 00:19:30,520 Somehow, it gives you guarantees about the ability and metrics, but it cannot give you any decent guarantees for a fixed component. 161 00:19:30,520 --> 00:19:41,660 So in other words, it might be that if you maximise the distance you have good mixing for, especially for for large components. 162 00:19:41,660 --> 00:19:51,800 Compressors have lost gains, but you really don't say anything uniformly, especially for components that might have small scale. 163 00:19:51,800 --> 00:19:59,020 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, 164 00:19:59,020 --> 00:20:04,060 I'm really highlighting the fact that's, for instance, you want to choose a fixed value of tea. 165 00:20:04,060 --> 00:20:09,070 Actually, you can find very simple examples where this can become very problematic. 166 00:20:09,070 --> 00:20:14,920 And actually, you don't have to go very far. Actually already on the go and target you with inteligente of scales. 167 00:20:14,920 --> 00:20:20,150 So we suppose we have different sigma eyes here that different ones we just have from viruses. 168 00:20:20,150 --> 00:20:31,450 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 169 00:20:31,450 --> 00:20:36,190 we'll be able to control all the correlations at once over the components one. 170 00:20:36,190 --> 00:20:40,990 OK, so we defined this with the correlation function or fruit of tea for component. 171 00:20:40,990 --> 00:20:45,430 I had to be just a coalition of tea and excited zero. 172 00:20:45,430 --> 00:20:49,480 And here on the cushion case, virtually everything is explicit. 173 00:20:49,480 --> 00:20:53,750 The calculations are very straightforward and easy. These are two corporations. 174 00:20:53,750 --> 00:21:00,150 They simply boil down to cosine is function with different elements. OK. 175 00:21:00,150 --> 00:21:06,060 So we can see that's on the graph here, if you sit, oppose all these other operations for this, 176 00:21:06,060 --> 00:21:13,260 these functions with given bandwidth and you have a bunch of them with different skills, 177 00:21:13,260 --> 00:21:23,730 then the task of choosing one that you see that will that will control all the conditions at once is quite hard, right? 178 00:21:23,730 --> 00:21:26,610 And sometimes it's just into other networks. 179 00:21:26,610 --> 00:21:36,100 If you look at the maximum value of these operation functions as a function of where these functionality can be actually arbitrary, 180 00:21:36,100 --> 00:21:42,800 arrest you close to one. So one main solution. 181 00:21:42,800 --> 00:21:54,900 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. 182 00:21:54,900 --> 00:21:57,260 So I think we will draw the amount of interest. 183 00:21:57,260 --> 00:22:05,990 I'm from a certain distribution here in state how they consider the choice of the exponential distribution with rate number. 184 00:22:05,990 --> 00:22:11,900 And from from this, the rest is the same as before. 185 00:22:11,900 --> 00:22:15,740 And the purpose of this is to be able somehow to induce a smoothing effect on 186 00:22:15,740 --> 00:22:24,360 the correlations so drawing at random and making these repeated repeated loops. 187 00:22:24,360 --> 00:22:33,240 The goal is to somehow average the corporations of HMRC and negotiate case the conditions are that as when we can see here, 188 00:22:33,240 --> 00:22:41,490 some of the conditions of this randomise approach actually boils down to the corruption of the HMRC for 189 00:22:41,490 --> 00:22:48,450 components and that we obtain something here that can be uniformly controlled by the CFO of the larger scale. 190 00:22:48,450 --> 00:22:59,050 There is a uniform down that will be monotony was like to no minus one thing that would control 191 00:22:59,050 --> 00:23:03,330 that once all your operations and recall that this number here is the rate of the commission. 192 00:23:03,330 --> 00:23:08,220 The submission, which means that the universe of lambda is the expected integration. 193 00:23:08,220 --> 00:23:11,550 So when the expected integration time goes to infinity, 194 00:23:11,550 --> 00:23:26,890 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. 195 00:23:26,890 --> 00:23:32,680 So in the next section, I'm going to motivate bits. 196 00:23:32,680 --> 00:23:39,100 The problem is that we are going to study particular, I'm going to introduce this loss of an efficient process. 197 00:23:39,100 --> 00:23:49,300 So if you have any questions before, please don't hesitate. All right. 198 00:23:49,300 --> 00:23:58,600 All right, so here we are going to introduce a 1990s way to introduce some randomness in the current trajectories. 199 00:23:58,600 --> 00:24:05,140 And the process that we are going through to do this here is known as the launch of our diffusion process, 200 00:24:05,140 --> 00:24:11,920 and there are several of the names, including an approach to asking you to pronounce its initial announcement as a lot of gotcha. 201 00:24:11,920 --> 00:24:22,570 Depending on you, I think you will look at it should not be confused with what it's highly of an upper limit. 202 00:24:22,570 --> 00:24:29,320 And so here just note that it involves still both the position and the Velocity V, 203 00:24:29,320 --> 00:24:37,780 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. 204 00:24:37,780 --> 00:24:46,450 We just speak about internet dynamics and when Islamic gamma is tricky because it didn't reek of a stochastic differential equation here, 205 00:24:46,450 --> 00:24:57,440 that is somehow. Somehow, same assumption, and I don't but but see some momentum refreshment at that continuously induced by your brain motion. 206 00:24:57,440 --> 00:25:05,080 OK, so this this gamma is known in the region either as dumping damage or the friction comet. 207 00:25:05,080 --> 00:25:13,800 And you can see some out here, one property of this of process is that it leaves the stationary distribution by store. 208 00:25:13,800 --> 00:25:20,770 She's just probably beaten by an in violence, just as dynamics do. 209 00:25:20,770 --> 00:25:25,150 How are they? One difference with the confident because as soon as gamma is pretty positive. 210 00:25:25,150 --> 00:25:29,020 Well, now the randomness induced can can yield equity. 211 00:25:29,020 --> 00:25:44,110 So I think there is because you will be able somehow to convert distribution based on starting from zero zero and suitable assumptions. 212 00:25:44,110 --> 00:25:48,010 So what we try to is this this diffusion process has a negative way in which we run them, 213 00:25:48,010 --> 00:25:54,130 this trajectory that is not built upon randomised integration dynamics. 214 00:25:54,130 --> 00:26:03,070 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 215 00:26:03,070 --> 00:26:12,460 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? 216 00:26:12,460 --> 00:26:17,290 OK. So we consider the same same question example here we have this, 217 00:26:17,290 --> 00:26:22,990 and I'm going to ask you to believe me that we can compute explicitly this with the quotation 218 00:26:22,990 --> 00:26:27,750 functions for the diffusion and then push in case they bow down to these equations. 219 00:26:27,750 --> 00:26:31,990 It's also better known equations, I think it's better simply to plot what happens to these functionality. 220 00:26:31,990 --> 00:26:40,650 OK. So he's the first ballots in the first part, we will consider that there is only one 6th scale by April one. 221 00:26:40,650 --> 00:26:49,580 And we will let us do that. The friction parameter value that's several values of frequency do, in fact. 222 00:26:49,580 --> 00:26:53,390 And we can see here that the choice going backwards zero. 223 00:26:53,390 --> 00:27:03,620 We recover this sort of grey line that simply boils down to the question is functionality the creation of agency spirit? 224 00:27:03,620 --> 00:27:08,070 And as soon as I saw positive, all the conditions will converge to zero. 225 00:27:08,070 --> 00:27:16,600 I still go to the first point. Now, if you compare these choices of Ghana, well, 226 00:27:16,600 --> 00:27:24,970 the choice can make or zero is the one that will make these coalitions reach zero faster than any or the choice of Ghanaians, 227 00:27:24,970 --> 00:27:38,350 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, 228 00:27:38,350 --> 00:27:43,210 then you just get to independence ethics samples with a C. 229 00:27:43,210 --> 00:27:50,440 And there's no hope by somehow inducing some momentum refreshment that you will go faster than this. 230 00:27:50,440 --> 00:27:56,300 So in that case, this means that if you want to sample from a standard, if done there, 231 00:27:56,300 --> 00:28:03,670 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. 232 00:28:03,670 --> 00:28:08,890 Now we are going to to you just right and those are affect the next time. 233 00:28:08,890 --> 00:28:16,730 But just before that, I just want to mention that you can see the behaviour when when Gamma one and two here is 234 00:28:16,730 --> 00:28:26,770 three and above a certain threshold equal to then they become they decay monotonic here 235 00:28:26,770 --> 00:28:32,230 and somehow there's a phase transition gamma equal to that also coincides with the choice 236 00:28:32,230 --> 00:28:38,160 of gamma that will that will optimise the exponential rate of convergence to zero. 237 00:28:38,160 --> 00:28:43,870 OK. So any value of galaxies would be below the threshold, 238 00:28:43,870 --> 00:28:51,220 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, 239 00:28:51,220 --> 00:28:54,820 we will call them over three. 240 00:28:54,820 --> 00:29:02,350 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. 241 00:29:02,350 --> 00:29:05,800 Here they are. All right. So which was two different values of government gamma or zero? 242 00:29:05,800 --> 00:29:23,520 And this face One Nation two here. The fate of. 243 00:29:23,520 --> 00:29:31,410 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. 244 00:29:31,410 --> 00:29:36,750 And also maybe, maybe I should use another word we can discuss. 245 00:29:36,750 --> 00:29:48,370 OK, thank. So, so here we are going to fall to two different venues have come up. 246 00:29:48,370 --> 00:30:01,310 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. 247 00:30:01,310 --> 00:30:08,570 So these bombings are here. Well, we can see that there there's different views, 248 00:30:08,570 --> 00:30:13,000 depending on whether stigma is higher than the referent, higher or lower than the reference states. 249 00:30:13,000 --> 00:30:20,630 Michael, sorry, blue line corresponds to a reference can you can see that particular every scale at a lower than than this reference scale. 250 00:30:20,630 --> 00:30:25,430 They they have conditions that will decay in there would be oscillatory, 251 00:30:25,430 --> 00:30:30,470 but they will all decay to zero and they will all be uniformly dominated by this. 252 00:30:30,470 --> 00:30:33,310 By this quotation corresponding to see Mike or what? 253 00:30:33,310 --> 00:30:42,230 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. 254 00:30:42,230 --> 00:30:50,720 The following if you choose some of the friction Panopto to be equal to the two divided by the not just the larger scale. 255 00:30:50,720 --> 00:30:54,140 Well, you can control all the corporations. 256 00:30:54,140 --> 00:31:03,380 Here by by the way, the coalition responding to this larger scale, which decays exponentially fast, we wish them well somehow. 257 00:31:03,380 --> 00:31:09,170 This is a similar property as what we would observe from an opposition team. 258 00:31:09,170 --> 00:31:24,560 And in that sense, its neighbours. We argue that that's a positive choice of friction allow for better business opportunity. 259 00:31:24,560 --> 00:31:29,780 So now I'm going to connect this notion diffusion to the randomisation process. 260 00:31:29,780 --> 00:31:34,850 And here you don't have to bother too much about the first half. If you don't want some of this, 261 00:31:34,850 --> 00:31:40,520 first half is simply to express that stochastic process is we are going to study here the 262 00:31:40,520 --> 00:31:44,330 notion that diffusion randomisation process can be characterised through their infinite, 263 00:31:44,330 --> 00:31:52,220 infinitesimal generators. So this quantity here that simply corresponds to the derivative of the expen expected function. 264 00:31:52,220 --> 00:31:57,780 If something from sentiment and we are going to define the Giteau corresponding to them, 265 00:31:57,780 --> 00:32:08,850 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. 266 00:32:08,850 --> 00:32:15,170 And so the randomisation generator and the generico can be expressed as the sum of to generate those. 267 00:32:15,170 --> 00:32:22,450 And they have one component, which is this. And can I mention one generator? 268 00:32:22,450 --> 00:32:30,770 Could I thought it was one and they differ through the the choice of momentum refreshment? 269 00:32:30,770 --> 00:32:41,800 So I wonder what the randomisation? The refreshments are in just the discrete way by a question process with an intensity that. 270 00:32:41,800 --> 00:32:47,410 So the higher that is, the more often refreshments, OK? 271 00:32:47,410 --> 00:32:53,320 OK. And the higher alpha is spammy to hear. 272 00:32:53,320 --> 00:33:00,490 The more partial the refreshment becomes assistant for me was how partial the refreshments are, right? 273 00:33:00,490 --> 00:33:09,490 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. 274 00:33:09,490 --> 00:33:12,400 Nicotine was way slower than motion. 275 00:33:12,400 --> 00:33:23,330 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, 276 00:33:23,330 --> 00:33:27,040 that goes to one, the spike in these particular rates for this value of gamma. 277 00:33:27,040 --> 00:33:32,970 Then what happens is that the generator from the mothership CMC converges to the generator of the stratosphere. 278 00:33:32,970 --> 00:33:41,080 OK, so is really what does it mean when it means that the more passion and frequency the refreshment becomes A.? 279 00:33:41,080 --> 00:33:52,300 Well, the closer we get from continuous refreshment induced by the emission and therefore the closer optimisation C becomes from the most efficient. 280 00:33:52,300 --> 00:34:03,530 So we will highlight this this result, together with another result that we have found which studies quantitative mixing rates for organisations. 281 00:34:03,530 --> 00:34:12,780 OK. So here we do not appear to be just transition, budgetary transition, 282 00:34:12,780 --> 00:34:20,250 kind of a fundamental issue with arbitrary parameters in density London and Bessie Smith Alpha. 283 00:34:20,250 --> 00:34:28,590 And we are going to try to states some convergence reasons about the new stamp process under these assumptions, a street year. 284 00:34:28,590 --> 00:34:35,670 So these assumptions A3 boils down to assuming that the distribution is truly concave, 285 00:34:35,670 --> 00:34:44,610 which is the same as assuming that the issue of the potential is lower bounded by positive constants. 286 00:34:44,610 --> 00:34:50,850 OK, so the right hand side of the assumption actually was already assumed the first line of presentation, 287 00:34:50,850 --> 00:34:57,780 because it just boils down to assuming that the greatness in this context and here is the theorem. 288 00:34:57,780 --> 00:35:06,480 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. 289 00:35:06,480 --> 00:35:14,310 If I go to one aspect of this rate for constants here is not the square root of an endless m. 290 00:35:14,310 --> 00:35:18,610 Then for any choice of alpha, we get some exponential convergence. 291 00:35:18,610 --> 00:35:25,670 Can we expect to do to sustain distance in perspective and to go over here as follows? 292 00:35:25,670 --> 00:35:29,550 Right. So so far, I need this. I think for something, it seems up to the first one, right, 293 00:35:29,550 --> 00:35:36,480 so that certain distances this distance between the pointy measures and some after time T two starts from distribution new, 294 00:35:36,480 --> 00:35:44,880 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. 295 00:35:44,880 --> 00:35:49,470 And so then of course, I see here on CS where this can be controlled. 296 00:35:49,470 --> 00:35:55,350 But what is important here to highlight is the rate of convergence up and so. 297 00:35:55,350 --> 00:36:02,520 So these are the bounds. It's true, but somehow this this mathematical framework here allows allows us to to get quite 298 00:36:02,520 --> 00:36:06,570 explicit rates of convergence with respect to the bombing to alpha and bow. 299 00:36:06,570 --> 00:36:11,490 And we also due to this constant slowdown and capital assumption. 300 00:36:11,490 --> 00:36:18,060 OK. And so the rate of convergence here, you can see that it's optimised its increasing function of alpha, 301 00:36:18,060 --> 00:36:22,670 and therefore it's optimised when alpha goes to one. 302 00:36:22,670 --> 00:36:26,480 My solution is the design solution when I was 12. 303 00:36:26,480 --> 00:36:36,750 And so when you combine both both results together and when you you remark it, it actually this this rate of much. 304 00:36:36,750 --> 00:36:43,110 The limit of this rate of emergence was obtained when that was when it was obtained for the nonchalant decision in the previous paper, 305 00:36:43,110 --> 00:36:48,870 when actually you can see these results actually has an interpolation between previous results that you 306 00:36:48,870 --> 00:36:55,200 give you some co-authors and what we have done like early on doing this for the natural diffusion. 307 00:36:55,200 --> 00:37:02,760 And it's some interpret these results and the interpretation and interpretation we present here is the following we might argue 308 00:37:02,760 --> 00:37:16,090 that the natural diffusion can be seen as a limit of SUMMARISATION that achieves the fastest exponential rate amongst song like. 309 00:37:16,090 --> 00:37:26,830 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. 310 00:37:26,830 --> 00:37:36,340 Please don't hesitate. It's the proof. 311 00:37:36,340 --> 00:37:43,380 The proof here is built upon a synchronous closing argument, just as in the paper that I referred to. 312 00:37:43,380 --> 00:37:49,810 And so the choice of coupling is not that complicated is that choice of of coping. 313 00:37:49,810 --> 00:37:59,890 Now here the the technical technical parts relies on choosing a twist of the metric that is 314 00:37:59,890 --> 00:38:04,840 amongst the best tweets of the metric you can to be able to obtain the best rates as possible. 315 00:38:04,840 --> 00:38:12,930 You can also. We're going to get more Panopto and some details. 316 00:38:12,930 --> 00:38:18,620 Good. OK, so let's go. 317 00:38:18,620 --> 00:38:23,870 So here is the time position that we consider for the of division. 318 00:38:23,870 --> 00:38:30,470 So this is finding someone motivated construction of some player directly built on lots of 319 00:38:30,470 --> 00:38:37,010 trajectories instead of as an alternative to to using randomised to try something new. 320 00:38:37,010 --> 00:38:46,810 Sometimes. And we are going here to define setting a vision that is built upon this leapfrogging director that you see in the media. 321 00:38:46,810 --> 00:38:54,010 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. 322 00:38:54,010 --> 00:38:59,860 So here Alpha is no longer a free barbecue is set by the values of friction, gamma and the time step h. 323 00:38:59,860 --> 00:39:05,140 OK. So when, when, when it goes to zero infinity, we can interpret. 324 00:39:05,140 --> 00:39:14,950 These are the Times station of the Najiba dynamics, and you can see then when amazing to see where we just recall the stomach. 325 00:39:14,950 --> 00:39:19,690 So we use these particular Thanksgiving edition because it's enabled to to preserve some, 326 00:39:19,690 --> 00:39:27,550 some useful properties of this stuff, but it dates time variability and present for something useful. 327 00:39:27,550 --> 00:39:36,430 When reading this correction on something, we are going to denote the update category, as he said, they often have to come up. 328 00:39:36,430 --> 00:39:45,350 That refers to the distribution of HIV h given exorbitant. 329 00:39:45,350 --> 00:39:52,060 So Excite is not out going on. So start don't, though, not a special moment of refreshment. 330 00:39:52,060 --> 00:40:02,650 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. 331 00:40:02,650 --> 00:40:13,270 But it's just Eskom composition step of the subject to denote completion of the muck of chew gum. 332 00:40:13,270 --> 00:40:18,400 So here is the algorithm, so we could expect adjusted large event trajectories or models, 333 00:40:18,400 --> 00:40:24,490 and it requires choosing the amount of friction gamma a time separation to wish them. 334 00:40:24,490 --> 00:40:32,800 So at the start of each trajectory, we completely throw a new Gaussian refreshment just as for some other agency. 335 00:40:32,800 --> 00:40:38,140 And then instead of HSC, we have now to propose a on trajectory. 336 00:40:38,140 --> 00:40:44,200 With friction going up, and so when is equal to zero, I know, right, what's really happened, 337 00:40:44,200 --> 00:40:47,710 but when you go to see you actually recover, exactly, do the agency go with them? 338 00:40:47,710 --> 00:40:52,380 Actually, he's done. City-The does not exist that somehow we should just consider the racial difference. 339 00:40:52,380 --> 00:41:00,950 I guess what's what we obtain for the HMRC case and when I get my strictly positive, then then the excellence ratio becomes. 340 00:41:00,950 --> 00:41:08,860 This ratio of density is where you have a density suspected of racism, and you can interpret this probability in a in several ways. 341 00:41:08,860 --> 00:41:13,930 I just chose this one just to highlight the fact that's somehow on the denominator here. 342 00:41:13,930 --> 00:41:21,760 You can interpret it as density of the forward trajectory from x zero zero to x LDL and 343 00:41:21,760 --> 00:41:28,030 on the right or here we can interpret it as the density of the backward trajectory. 344 00:41:28,030 --> 00:41:34,380 We split more into. 345 00:41:34,380 --> 00:41:42,510 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, 346 00:41:42,510 --> 00:41:50,190 you need to flip the momentum upon rejection, but since there are some, some full refreshment of the cities here. 347 00:41:50,190 --> 00:41:58,310 Well, actually, this line is completely erased at each HQ where you propose. 348 00:41:58,310 --> 00:42:06,440 So this is this is a difference with the GMC approach for which for any choice of was the assistance Alfa, 349 00:42:06,440 --> 00:42:13,160 then we could not erase those momentum clips and force them out due to a trade off between between 350 00:42:13,160 --> 00:42:21,250 them having not too many freedoms without having a big enough time step to reduce the carnage. 351 00:42:21,250 --> 00:42:28,270 All right, so we argued that this correction yields a neat way to to stabilise the diffusion in 352 00:42:28,270 --> 00:42:32,650 particular compared to previous approaches aiming to mimic this natural division. 353 00:42:32,650 --> 00:42:39,100 Our approach is different in the sense that we aim to metabolise whole trajectories of land. 354 00:42:39,100 --> 00:42:43,750 We ask people's approaches. We're studying the one step, one step correction. 355 00:42:43,750 --> 00:42:53,920 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. 356 00:42:53,920 --> 00:42:58,990 And you have this of refreshment. That is very special, something we don't see very close to one. 357 00:42:58,990 --> 00:43:06,370 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. 358 00:43:06,370 --> 00:43:12,250 We have one degree of freedom, which is the sense of the trajectory shooting by the user. 359 00:43:12,250 --> 00:43:21,100 And and that can help to reduce the coalition base by choosing it in a smart way. 360 00:43:21,100 --> 00:43:27,460 So the second property here is that momentum fees can be raised by food refreshments, which is not the case for GHC approach. 361 00:43:27,460 --> 00:43:34,000 And we argue that that this scandal. So this gives them more time goes and gives a robust extension to HMRC. 362 00:43:34,000 --> 00:43:43,800 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. 363 00:43:43,800 --> 00:43:51,750 And achieve similar similar goals as what we would do when drawing us from them, the inspiration empty. 364 00:43:51,750 --> 00:43:58,710 So in particular, also here we highlight the fact that as soon as my strictly positive trajectories, trajectories will be equity, 365 00:43:58,710 --> 00:44:05,010 which means that you cannot really suffer from some problem that you have in HNC when game is equal to zero, 366 00:44:05,010 --> 00:44:11,160 which is that if you let your your intuition t be large, 367 00:44:11,160 --> 00:44:29,590 you will just come back at some point to your original point and wrestle away some national and resources while having a high correlation. 368 00:44:29,590 --> 00:44:39,750 Right, so when when my strictly positive and you look at the trajectory of a Flamsteed somehow its trajectory with conversion distribution. 369 00:44:39,750 --> 00:44:45,630 To to buy stock, and if TI's large enough, some of the conditions would be to zero. 370 00:44:45,630 --> 00:44:54,000 And some of the output of the Georgie will be closer and closer to to an independent sample from from 371 00:44:54,000 --> 00:44:58,050 not exactly from my because it will be a flow decision because these times of the nation's best bet, 372 00:44:58,050 --> 00:45:15,740 it will become less and less dependent from the stock of the. I mean, how to turn the situation. 373 00:45:15,740 --> 00:45:28,100 At different points. So want to have. 374 00:45:28,100 --> 00:45:37,030 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. 375 00:45:37,030 --> 00:45:53,990 Yeah. When he sgamma is very close to zero, you can get close after a certain amount of discretion 90 to your original point, 376 00:45:53,990 --> 00:45:56,770 but somehow when a sufficiently large, 377 00:45:56,770 --> 00:46:05,150 there is no particular reason why you should come back in a deterministic fashion to your original point on the. 378 00:46:05,150 --> 00:46:13,490 That's. But maybe we can discuss the semantics, I suppose. 379 00:46:13,490 --> 00:46:19,170 So we finally support this agreement with the theoretical. 380 00:46:19,170 --> 00:46:30,030 Justification that essentially is quite similar to the one that has been established for HMRC in 2013. 381 00:46:30,030 --> 00:46:37,170 My best guess in some quarters, which is down to a study of the spending limits under this assumption. 382 00:46:37,170 --> 00:46:43,140 So it's not the same assumption, it's the same assumption as what we have for HMRC. 383 00:46:43,140 --> 00:46:48,610 So we thought for that assumption, we can prove that's for any choice of friction. 384 00:46:48,610 --> 00:46:57,210 Gamma and this molten Gotham will have a skating limits that that will enable two other very 385 00:46:57,210 --> 00:47:04,140 explicit optimisation of efficiency measures that's yielded the tuning for continuing rule. 386 00:47:04,140 --> 00:47:12,390 If you choose your step such that the exit acceptance rate converges to 65 percent, some maximises measure of efficiency. 387 00:47:12,390 --> 00:47:19,080 And this is true for any choice of, in particular, its its seats or so. 388 00:47:19,080 --> 00:47:26,640 As a consequence of the fact that we use the default integrator inside this, this demonstration of their entrepreneur vision, 389 00:47:26,640 --> 00:47:33,750 and we wanted to highlight somehow that we can preserve several properties of HMRC by beating down 390 00:47:33,750 --> 00:47:42,060 that we in particular in particular the setting of the location of the accident probability. 391 00:47:42,060 --> 00:47:57,370 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. 392 00:47:57,370 --> 00:48:05,830 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? 393 00:48:05,830 --> 00:48:13,030 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. 394 00:48:13,030 --> 00:48:22,000 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. 395 00:48:22,000 --> 00:48:33,180 Particularly we also recovered one force getting lost, but the damage. I that's I'm going to just summarise contributions here, so first, 396 00:48:33,180 --> 00:48:39,960 we present not division has a limit the fondamentaux agency that receives the first assessment 17C rate for some gift nuggets. 397 00:48:39,960 --> 00:48:45,630 Then we look at the use of positive dumping to enable control of the workplace and therefore allowing 398 00:48:45,630 --> 00:48:52,020 for for rubbish tuning respect to the intuition and mate completing all these conditions at once. 399 00:48:52,020 --> 00:48:59,610 Then we introduced this piece I just did not want to go is I go easy and we argue that this gives a 400 00:48:59,610 --> 00:49:06,420 correction for security in particular compared to previous approaches aiming for this notion of process. 401 00:49:06,420 --> 00:49:14,640 Here we metabolise whole. I think there is, which enabled to choose the lens to take the risk, 402 00:49:14,640 --> 00:49:23,910 and we highlight the fact that the momentum is now erased fully by full momentum investments which yield nice meets mathematical framework. 403 00:49:23,910 --> 00:49:31,740 In particular, the back of can off the chain of the position can be shown to be reversible. 404 00:49:31,740 --> 00:49:35,700 Thanks to this property. And finally, the we, 405 00:49:35,700 --> 00:49:39,450 we extend the outstanding results of best course and quarters for any choice of 406 00:49:39,450 --> 00:49:44,970 friction without that assumption and gets therefore this wonderful scanning for most, 407 00:49:44,970 --> 00:49:53,000 just as we do for agency. So we are interested in several, several perspectives for future work, 408 00:49:53,000 --> 00:50:00,410 so I still present some make comparisons because it's actually is almost finished work, it should be an archive within a few days. 409 00:50:00,410 --> 00:50:07,250 So we are still finishing using to make comparisons, which is why I didn't present here. 410 00:50:07,250 --> 00:50:14,300 And so there are several questions we we are quite interested in in working on in the future, particularly, 411 00:50:14,300 --> 00:50:22,490 we are quite interested in this adaptive tuning strategy for choosing a and gamma to be able to yield competitive. 412 00:50:22,490 --> 00:50:29,630 Competitive some players compared to other athletes doing so that have been proposed for HMRC and therefore hopefully can yield some, 413 00:50:29,630 --> 00:50:33,510 some maybe some information, but this gives a flip. 414 00:50:33,510 --> 00:50:41,040 And also, I think there's also quite a lot to do in terms of comparisons with this fundamental agency approach, 415 00:50:41,040 --> 00:50:45,900 either from a full article viewpoint on medical viewpoint. OK, thank you very much for your attention. 416 00:50:45,900 --> 00:51:06,420 If you have any questions, please don't hesitate. Thank you. 417 00:51:06,420 --> 00:51:13,860 Thanks. You say we have tried to stop it already, but. 418 00:51:13,860 --> 00:51:24,210 Spend. I. Doesn't look too bad to prioritise, to change it, to do it. 419 00:51:24,210 --> 00:51:33,250 So I've got some cards, but I want. That's excellent. 420 00:51:33,250 --> 00:51:38,830 Yeah, that's right. OK, thanks, I. 421 00:51:38,830 --> 00:51:48,290 That doesn't look like it would be best for. And so it's promising. 422 00:51:48,290 --> 00:51:55,230 What kind of guidance for expressing acceptance of the fact that some sides. 423 00:51:55,230 --> 00:52:02,790 I wish I wish I had my mind, my working paper here, but so, yeah, 424 00:52:02,790 --> 00:52:10,050 so so so let me try to to explain what we observe, for instance, so I can see simulations. 425 00:52:10,050 --> 00:52:13,680 Let's let's come back to the bush time off because it's true. 426 00:52:13,680 --> 00:52:21,120 It's easy, but it's also quite easy to even. I'd say quite a few things about this. 427 00:52:21,120 --> 00:52:33,570 So, for instance, we have an example in our working paper that somehow compares is more to get them together with randomisation with HMRC. 428 00:52:33,570 --> 00:52:36,940 OK. And I suppose we would have here. 429 00:52:36,940 --> 00:52:43,040 We take the same same framework that they presented, and we take some, some different scale sigma one sigma. 430 00:52:43,040 --> 00:52:48,600 And in the working paper, I think we just don't seem to be to be between the 150 commission members. 431 00:52:48,600 --> 00:52:52,230 Is this 50, which is OK because everybody is not super high? 432 00:52:52,230 --> 00:52:58,290 And actually when you do that and you look not only distance, but you look at, 433 00:52:58,290 --> 00:53:01,810 as you say, sample size, but not the sample size of one component, but the worst. 434 00:53:01,810 --> 00:53:08,970 If if somebody says all the components that you would get, some of you will, you will see that actually for HMRC, the worse. 435 00:53:08,970 --> 00:53:14,940 Yes, as a function of time for this particular model is quite an theoretic function of of. 436 00:53:14,940 --> 00:53:24,300 That actually breaks down when they when commission member increase and when they must them. 437 00:53:24,300 --> 00:53:33,090 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, 438 00:53:33,090 --> 00:53:43,890 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, 439 00:53:43,890 --> 00:53:50,880 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, 440 00:53:50,880 --> 00:54:00,750 riskiest time that we can draw for both organisations and families that achieves some good performance, even if there are some gains. 441 00:54:00,750 --> 00:54:06,910 So you might say, OK, maybe for MCI can use instead a good recognition that to be able to reduce reduce the attention deficit. 442 00:54:06,910 --> 00:54:15,390 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, 443 00:54:15,390 --> 00:54:24,330 because the preconditioning problem is something that you will do would do a purely somehow and and especially for something, 444 00:54:24,330 --> 00:54:26,880 it's quite sometimes hard to find a good. 445 00:54:26,880 --> 00:54:32,940 You have to find a global recognition of some also finding look out for condition in the position something is somehow. 446 00:54:32,940 --> 00:54:37,690 Sometimes it's easy, but so especially for something, it's not so easy because you cannot use a look. 447 00:54:37,690 --> 00:54:45,420 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, 448 00:54:45,420 --> 00:54:52,620 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. 449 00:54:52,620 --> 00:54:55,920 So it's not really a particularity of agency itself. 450 00:54:55,920 --> 00:55:01,040 It's the fixed choice of intuition that causes problems always in the Russian framework. 451 00:55:01,040 --> 00:55:12,360 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. 452 00:55:12,360 --> 00:56:00,352 I think the message has been. You know, we have a run over as speaker of economic.