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In these lecture notes we present different methods and concepts developed in statistical physics to analyze gradient descent dynamics in high-dimensional non-convex landscapes.
Characteristic vectors of bordered matrices with infinite dimensions
Eugene P. Wigner · 1955
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Integral Equations
F. G. Tricomi · 1957
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Random matrices in physics
Eugene P. Wigner · 1967
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Distribution of Eigenvalues for Some Sets of Random Matrices
V.A. Marcenko and L.A. Pastur · 1967
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The eigenvalue spectrum of a large symmetric random matrix
Samuel F Edwards and Raymund C Jones · 1976
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Dynamic theory of the spin-glass phase
H. Sompolinsky and Annette Zippelius · 1981
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Relaxational dynamics of the edwards-anderson model and the mean-field theory of spin-glasses
H. Sompolinsky and Annette Zippelius · 1982
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Grassmann variables and supersymmetry in the theory of disordered systems
E. Brézin · 1985
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Spin glass theory and beyond
Marc Mézard, Giorgio Parisi, and Miguel A. Virasoro · 1987
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Correlated lattice fermions in d = ∞ d=\infty dimensions
Walter Metzner and Dieter Vollhardt · 1989
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Three unfinished works on the optimal storage capacity of networks
Elizabeth Gardner and Bernard Derrida · 1989
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Limit laws for random matrices and free products
Dan Voiculescu · 1991
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The sphericalp-spin interaction spin glass model: the statics
Andrea Crisanti and H-J Sommers · 1992
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Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model
L. F. Cugliandolo and J. Kurchan · 1993
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Mean-field monte carlo approach to the sherrington-kirkpatrick model with asymmetric couplings
H Eissfeller and M Opper · 1994
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Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions
Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J Rozenberg · 1996
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Symmetric langevin spin glass dynamics
G Ben Arous and Alice Guionnet · 1997
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Random-matrix theories in quantum physics: common concepts
Thomas Guhr, Axel Müller-Groeling, and Hans A Weidenmüller · 1998
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Stationary points of the thouless-anderson-palmer free energy
Andrea Cavagna, Irene Giardina, and Giorgio Parisi · 1998
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Aging of spherical spin glasses
G Ben Arous, Amir Dembo, and Alice Guionnet · 2001
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Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices
Yan V. Fyodorov · 2004
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Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
Jinho Baik, Gérard Ben Arous, and Sandrine Péché · 2005
Cited alongside, same era.
Cugliandolo-kurchan equations for dynamics of spin-glasses
Gérard Ben Arous, Amir Dembo, and Alice Guionnet · 2006
Cited alongside, same era.
Financial applications of random matrix theory: a short review, 2009
J. P. Bouchaud and M. Potters · 2009
Cited alongside, same era.
Phase transitions in the distribution of bipartite entanglement of a random pure state
Celine Nadal, Satya N Majumdar, and Massimo Vergassola · 2010
Cited alongside, same era.
Universality of wigner random matrices: a survey of recent results
László Erdős · 2011
Cited alongside, same era.
Theoretical perspective on the glass transition and amorphous materials
Ludovic Berthier and Giulio Biroli · 2011
Landscape complexity for the empirical risk of generalized linear models
Antoine Maillard, Gérard Ben Arous, and Giulio Biroli · 2020
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Marvels and pitfalls of the langevin algorithm in noisy high-dimensional inference
Stefano Sarao Mannelli, Giulio Biroli, Chiara Cammarota, Florent Krzakala, Pierfrancesco Urbani, and Lenka Zdeborová · 2020
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Dynamical mean-field theory for stochastic gradient descent in gaussian mixture classification
Francesca Mignacco, Florent Krzakala, Pierfrancesco Urbani, and Lenka Zdeborová · 2020
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Bad global minima exist and sgd can reach them
Shengchao Liu, Dimitris Papailiopoulos, and Dimitris Achlioptas · 2020
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Algorithmic thresholds for tensor pca
Gerard Ben Arous, Reza Gheissari, and Aukosh Jagannath · 2020
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Concentration of the complexity of spherical pure p-spin models at arbitrary energies
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Cited alongside, same era.
Random matrices and complexity of spin glasses
Antonio Auffinger, Gérard Ben Arous, and Jiří Černý · 2013
Cited alongside, same era.
Topological and dynamical complexity of random neural networks
Gilles Wainrib and Jonathan Touboul · 2013
Cited alongside, same era.
A statistical model for tensor pca
Emile Richard and Andrea Montanari · 2014
Cited alongside, same era.
The complexity of spherical p p -spin models—a second moment approach
Eliran Subag · 2017
Cited alongside, same era.
Statistical and computational phase transitions in spiked tensor estimation
Thibault Lesieur, Leo Miolane, Marc Lelarge, Florent Krzakala, and Lenka Zdeborova · 2017
Cited alongside, same era.
Out-of-equilibrium dynamical mean-field equations for the perceptron model
Elisabeth Agoritsas, Giulio Biroli, Pierfrancesco Urbani, and Francesco Zamponi · 2018
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Eliran Subag and Ofer Zeitouni · 2021
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Counting equilibria of large complex systems by instability index
Gérard Ben Arous, Yan V Fyodorov, and Boris A Khoruzhenko · 2021
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Dynamics of liquids in the large-dimensional limit
Chen Liu, Giulio Biroli, David R Reichman, and Grzegorz Szamel · 2021
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The high-dimensional asymptotics of first order methods with random data
Michael Celentano, Chen Cheng, and Andrea Montanari · 2021
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Analytical study of momentum-based acceleration methods in paradigmatic high-dimensional non-convex problems
Stefano Sarao Mannelli and Pierfrancesco Urbani · 2021
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Entropic barriers as a reason for hardness in both classical and quantum algorithms
Matteo Bellitti, Federico Ricci-Tersenghi, and Antonello Scardicchio · 2021
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Random Matrix Methods for Machine Learning
Romain Couillet and Zhenyu Liao · 2022
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Optimal learning rate schedules in high-dimensional non-convex optimization problems, 2022
Stéphane d’Ascoli, Maria Refinetti, and Giulio Biroli · 2022
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Optimization of random high-dimensional functions: Structure and algorithms, 2022
Antonio Auffinger, Andrea Montanari, and Eliran Subag · 2022
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The high-d landscapes paradigm: spin-glasses, and beyond
Valentina Ros and Yan V Fyodorov · 2022
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Phase retrieval: From computational imaging to machine learning
Jonathan Dong, Lorenzo Valzania, Antoine Maillard, Thanh-an Pham, Sylvain Gigan, and Michael Unser · 2022
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Rigorous dynamical mean field theory for stochastic gradient descent methods
Cedric Gerbelot, Emanuele Troiani, Francesca Mignacco, Florent Krzakala, and Lenka Zdeborova · 2022
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High-dimensional limit theorems for sgd: Effective dynamics and critical scaling
Gerard Ben Arous, Reza Gheissari, and Aukosh Jagannath · 2022
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Generalized lotka-volterra equations with random, nonreciprocal interactions: The typical number of equilibria
Valentina Ros, Felix Roy, Giulio Biroli, Guy Bunin, and Ari M Turner · 2023
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The star-shaped space of solutions of the spherical negative perceptron
Brandon Livio Annesi, Clarissa Lauditi, Carlo Lucibello, Enrico M Malatesta, Gabriele Perugini, Fabrizio Pittorino, and Luca Saglietti · 2023
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