Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

• Slava Rychkov (IHES)

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Some intriguing maths in dynamical mean-field for glasses

• Jorge Kurchan (ENS Paris)

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

• Nicholas Read (Yale University)

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models. However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

• Nicholas Read (Yale University)

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models. However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Optimization and sampling from mean-field spin glass models

• Ahmed El Alaoui (Cornell University)

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures: 1- Optimization: Can we efficiently find ground state configurations whenever they exist? I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case. 2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure? I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature. For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

• Slava Rychkov (IHES)

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Some intriguing maths in dynamical mean-field for glasses

• Jorge Kurchan (ENS Paris)

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

• Nicholas Read (Yale University)

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models. However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Some intriguing maths in dynamical mean-field for glasses

• Jorge Kurchan (ENS Paris)

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

Optimization and sampling from mean-field spin glass models

• Ahmed El Alaoui (Cornell University)

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures: 1- Optimization: Can we efficiently find ground state configurations whenever they exist? I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case. 2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure? I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature. For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

• Slava Rychkov (IHES)

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

• Nicholas Read (Yale University)

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models. However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Optimization and sampling from mean-field spin glass models

• Ahmed El Alaoui (Cornell University)

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures: 1- Optimization: Can we efficiently find ground state configurations whenever they exist? I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case. 2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure? I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature. For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

• Slava Rychkov (IHES)

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Optimization and sampling from mean-field spin glass models

• Ahmed El Alaoui (Cornell University)

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures: 1- Optimization: Can we efficiently find ground state configurations whenever they exist? I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case. 2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure? I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature. For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Some intriguing maths in dynamical mean-field for glasses

• Jorge Kurchan (ENS Paris)

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

Statistical physics analysis of artificial neural networks

• Riccardo Zecchina (Bocconi University)

The scope of these lectures will be to discuss the statistical physics approach to the phase diagram and landscape of machine learning problems focusing on the loss landscape of artificial neural networks. We will show how this analysis suggest new powerful training algorithms based on the role of wide flat minima of the corresponding cost functions.

Statistical physics analysis of artificial neural networks

• Riccardo Zecchina (Bocconi University)

The scope of these lectures will be to discuss the statistical physics approach to the phase diagram and landscape of machine learning problems focusing on the loss landscape of artificial neural networks. We will show how this analysis suggest new powerful training algorithms based on the role of wide flat minima of the corresponding cost functions.

On the solution of the SK model

• Giorgio Parisi (Università di Roma La Sapienza)

Statistical physics of inference and learning: from algorithmic strategies to proofs.

• Florent Krzakala (EPFL)

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Statistical physics of inference and learning: from algorithmic strategies to proofs.

• Florent Krzakala (EPFL)

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Rigorous analysis of pure states of spin glass models on sparse random graphs

• Amin Coja-Oghlan (Goethe University Frankfurt)

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs). Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Rigorous analysis of the complexity of the landscape of spin glass models

• Gerard Ben Arous (New York University)

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent). The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

Statistical physics of inference and learning: from algorithmic strategies to proofs.

• Florent Krzakala (EPFL)

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Statistical physics of inference and learning: from algorithmic strategies to proofs.

• Florent Krzakala (EPFL)

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

The low temperature properties of the Gibbs measure in mean field spin glasses

• Antonio Auffinger (Northwestern University)

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Rigorous analysis of pure states of spin glass models on sparse random graphs

• Amin Coja-Oghlan (Goethe University Frankfurt)

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs). Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Rigorous analysis of the complexity of the landscape of spin glass models

• Gerard Ben Arous (New York University)

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent). The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

The low temperature properties of the Gibbs measure in mean field spin glasses

• Antonio Auffinger (Northwestern University)

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

The low temperature properties of the Gibbs measure in mean field spin glasses

• Antonio Auffinger (Northwestern University)

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Rigorous analysis of the complexity of the landscape of spin glass models

• Gerard Ben Arous (New York University)

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent). The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

Rigorous analysis of pure states of spin glass models on sparse random graphs

• Amin Coja-Oghlan (Goethe University Frankfurt)

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs). Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Rigorous analysis of the complexity of the landscape of spin glass models

• Gerard Ben Arous (New York University)

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent). The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

The low temperature properties of the Gibbs measure in mean field spin glasses

• Antonio Auffinger (Northwestern University)

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Rigorous analysis of pure states of spin glass models on sparse random graphs

• Amin Coja-Oghlan (Goethe University Frankfurt)

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs). Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

TBA - Silvio Franz

• Silvio Franz (Université Paris-Saclay, LPTMS)

TBA