Periodic Reporting for period 1 - ELISA (Exploration for Large Interacting Systems of Agents)
Période du rapport: 2022-09-01 au 2025-02-28
The everyday environment in which we live is made up of encounters. Some are recurrent: every day, or almost every day, we share a meal with family members or work colleagues; we regularly greet our neighbors. And then there are those encounters that are purely coincidental, like the motorist we pass at a red light, or even beyond the visible, when all of us, inhabitants of the same planet, consume the same energy, the same food or share the same eco-system. In physics, these encounters could be called interactions. How can we describe them mathematically when they occur on a large scale? How can we account for the fact that, on a day-to-day basis, each of us seeks to improve our daily lives, based on quantities (such as the price of oil or wheat) that depend on the daily lives of billions of other similar people? How can we predict the evolution of such patterns, and above all, what happens when this gigantic network through which we are all connected begins to oscillate?
In mathematics and physics, models that reproduce the behavior of a multitude of individuals who only see each other through aggregated quantities (also known as macroscopic quantities), the result of each individual's actions, are called mean-field models. For example, the number of people visiting cafés and restaurants on a Friday or Saturday evening is a collective indicator resulting from individual decisions. This collective indicator simply summarizes overall behavior, without knowing the behavior of each individual. It simply aims to describe the statistical behavior of the population.
Mathematically, mean-field phenomena are precisely those in which an individual sees others only in terms of statistical quantities, summarizing the overall state of the population. These models become particularly subtle when each individual making up the gigantic population in which she or he evolves can make choices. If everyone decides to go to a restaurant on a Saturday evening, available tables will certainly run out and menu prices will soar. In fact, a balance is formed between those who insist on going out (for example, because it is an occasion not to be missed) and those who prefer to stay at home. This equilibrium is the basis for restaurant visits. In game theory, this equilibrium is known as a Nash equilibrium.
Mean-field game theory was introduced almost twenty years ago by P.L. Lions and J.M. Lasry, with the aim of answering the following questions: when can an equilibrium be found, and how does it vary with population size? Other questions remain, however. For example, can several equilibria be found in the mathematical sense, and if so, does one make more practical sense than another? Mathematically, the existence of several equilibria corresponds to the presence of abrupt changes in the population's state when one of the model's parameters is moved. One of the hypotheses of the ERC ELISA project is that these abrupt state changes can be erased by forcing the entire population to oscillate under the effect of small random pulses, and then averaging the observed states over each of these small pulses. To validate this hypothesis, we need to construct suitable random pulses. The difficulty lies in the fact that these impulses aim to modify the statistical state of the population as a whole: in effect, we are talking about moving a very large object.
In mathematics and physics, models that reproduce the behavior of a multitude of individuals who only see each other through aggregated quantities (also known as macroscopic quantities), the result of each individual's actions, are called mean-field models. For example, the number of people visiting cafés and restaurants on a Friday or Saturday evening is a collective indicator resulting from individual decisions. This collective indicator simply summarizes overall behavior, without knowing the behavior of each individual. It simply aims to describe the statistical behavior of the population.
Mathematically, mean-field phenomena are precisely those in which an individual sees others only in terms of statistical quantities, summarizing the overall state of the population. These models become particularly subtle when each individual making up the gigantic population in which she or he evolves can make choices. If everyone decides to go to a restaurant on a Saturday evening, available tables will certainly run out and menu prices will soar. In fact, a balance is formed between those who insist on going out (for example, because it is an occasion not to be missed) and those who prefer to stay at home. This equilibrium is the basis for restaurant visits. In game theory, this equilibrium is known as a Nash equilibrium.
Mean-field game theory was introduced almost twenty years ago by P.L. Lions and J.M. Lasry, with the aim of answering the following questions: when can an equilibrium be found, and how does it vary with population size? Other questions remain, however. For example, can several equilibria be found in the mathematical sense, and if so, does one make more practical sense than another? Mathematically, the existence of several equilibria corresponds to the presence of abrupt changes in the population's state when one of the model's parameters is moved. One of the hypotheses of the ERC ELISA project is that these abrupt state changes can be erased by forcing the entire population to oscillate under the effect of small random pulses, and then averaging the observed states over each of these small pulses. To validate this hypothesis, we need to construct suitable random pulses. The difficulty lies in the fact that these impulses aim to modify the statistical state of the population as a whole: in effect, we are talking about moving a very large object.
We have studied several examples of smoothing by `randomization'. In the first example, we propose an explicit construction that amounts to defining a continuous (continuous-time) random walk with values in the set of all possible one-dimensional statistical laws. This construction is obtained by using the geometry of the set of statistical laws (also called space of probability measures). In short, we move in a Gaussian random fashion in the space of random variables and then return to probability measures by reordering. We show that this approach has the expected effects, forcing, for example, the uniqueness of equilibria in mean-field games or the convergence of gradient methods in optimization problems. In parallel, we are studying other methods, based on other examples of random walks in the space of probability measures.
On the other hand, we have studied the impact of random impulses acting only on a few statistical data in the population, such as the mean. In some cases, the smoothing effects are limited but sufficient to significantly change the behavior of the mean-field model, for example its long-time behavior. Similar techniques enable us to gain a finer understanding of models in which a multitude of small individuals are not only in competition with each other, but also with major players. For example, consumers of raw materials act not only according to the consumption of other individuals, but also according to the policies of producers.
These techniques may find application in the understanding of statistical learning mechanisms. Here, the set of artificial neurons constitutes a population that needs to be controlled in order to perform a regression task. In this regard, we have studied the algorithm used to adjust the neurons under another form of regularization, called entropy regularization. We prove that, for an infinite number of features and neurons and for many statistical feature distributions, gradient descent performs well when initialized close to the optimizer.
On the other hand, we have studied the impact of random impulses acting only on a few statistical data in the population, such as the mean. In some cases, the smoothing effects are limited but sufficient to significantly change the behavior of the mean-field model, for example its long-time behavior. Similar techniques enable us to gain a finer understanding of models in which a multitude of small individuals are not only in competition with each other, but also with major players. For example, consumers of raw materials act not only according to the consumption of other individuals, but also according to the policies of producers.
These techniques may find application in the understanding of statistical learning mechanisms. Here, the set of artificial neurons constitutes a population that needs to be controlled in order to perform a regression task. In this regard, we have studied the algorithm used to adjust the neurons under another form of regularization, called entropy regularization. We prove that, for an infinite number of features and neurons and for many statistical feature distributions, gradient descent performs well when initialized close to the optimizer.
One of our aims is to demonstrate the relevance of randomization approaches and thus develop a theory of diffusion equations on all probability measures. This direction of research is still in its infancy, but on the basis of the first results obtained, the aim is precisely to demonstrate its relevance and systematize its operation as far as possible.
In application terms, random impulses, as described above, can be regarded as a way to explore all the possible statistical states for a population. One of the next steps is to understand how this exploration phenomenon can help learning, i.e. the discovery of an equilibrium through data acquisition. So far, we have proofs of concept in some preliminary examples but this is our objective to generalize these results to more general examples.
In application terms, random impulses, as described above, can be regarded as a way to explore all the possible statistical states for a population. One of the next steps is to understand how this exploration phenomenon can help learning, i.e. the discovery of an equilibrium through data acquisition. So far, we have proofs of concept in some preliminary examples but this is our objective to generalize these results to more general examples.