Spin Glass Theory And Beyond: An Introduction To The Replica Method And Its Applications

By M. M'ezard, Giorgio Parisi, M. Virasoro. 1986

Subfields: Network Science & Complex Systems

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Citations in the corpus

The jamming transition as a paradigm to understand the loss landscape of deep neural networks (2018, background): “Calculations in infinite dimensions [43, 55] showed that marginal stability is associated with a hierarchical organization of minima of the energy (a phenomenon referred to as replica symmetry breaking [56]), and exponents were found to follow γ = 0.”
Hyperuniform states of matter (2018, background): “The discovered exotic disordered spin ground states, which are distinctly different from spin glasses [343], are the spin analogs of disordered stealthy hyperuniform many-particle ground states [62, 63, 65] that have been shown to be endowed with novel bandgap and wave characteristics [82–85, 87, 93”
Deep relaxation: partial differential equations for optimizing deep neural networks (2017, background): “While stochastic gradient descent converges slower than gradient descent, i.e., without any stochasticity [59], in practice, deep networks are trained with SGD and obtain superior generalization performance only with small mini-batches [15,28].”
Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks (2020, methodology): “• We present another way to arrive at the same analytical expressions using the replica method of statistical physics and a saddle-point approximation (Sherrington & Kirkpatrick, 1975; Mézard et al., 1987).”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, methodology): “[14, 23] rely on the replica method [63], while those in Refs.”
Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks (2020, methodology): “The calculation proceeds by a replica symmetric ansätz 33 , evaluating the saddle point equations and taking the β → ∞ limit.”
An introduction to spontaneous symmetry breaking (2019, unknown): “[105–107].”
Shaping the learning landscape in neural networks around wide flat minima (2019, methodology): “[18] Marc Mézard, Giorgio Parisi, and Miguel Virasoro.”
Optimal errors and phase transitions in high-dimensional generalized linear models (2017, methodology): “This “decoupling property” results from the mean- eld approach in statistical physics, used through in the replica method to perform a formal calculation of the free entropy of the model [49,74].”
Capturing the learning curves of generic features maps for realistic data sets with a teacher-student model (2021, methodology): “We refer to [21, 3, 22] for an introduction to this remarkable heuristic (but seemingly never failing) approach.”
Entropic gradient descent algorithms and wide flat minima (2020, background): “[32] Marc Mézard, Giorgio Parisi, and Miguel Virasoro.”
Numerical implementation of dynamical mean field theory for disordered systems: application to the Lotka–Volterra model of ecosystems (2019, methodology): “It can be shown by perturbation theory in the strength of interactions that χij is of order S −1/2 for i 6= j [17] (see Appendix A).”
Iterated Local Search: Framework and Applications (2018, background): “This has been referred to in many different ways: “Massif Central” phenomenon [31], principle of proximate optimality [36], and replica symmetry [66].”
The committee machine: computational to statistical gaps in learning a two-layers neural network (2018, methodology): “These works relied on non-rigorous heuristic approaches, such as the replica and cavity methods [13, 14].”