Statistical physics of inference: thresholds and algorithms

By Lenka Zdeborov'a, Florent Krzakala. 2015

501 citations recorded by the source corpus.

Many questions of fundamental interest in today's science can be formulated as inference problems: some partial, or noisy, observations are performed over a set of variables and the goal is to recover, or infer, the values of the variables based on the indirect information contained in the measurements. For such problems, the central scientific questions are: Under what conditions is the information contained in the measurements sufficient for a satisfactory inference to be possible? What are the most efficient algorithms for this task? A growing body of work has shown that often we can understand and locate these fundamental barriers by thinking of them as phase transitions in the sense of statistical physics. Moreover, it turned out that we can use the gained physical insight to develop new promising algorithms. The connection between inference and statistical physics is currently witnessing an impressive renaissance and we review here the current state-of-the-art, with a pedagogical focus on the Ising model which, formulated as an inference problem, we call the planted spin glass. In terms of applications we review two classes of problems: (i) inference of clusters on graphs and networks, with community detection as a special case and (ii) estimating a signal from its noisy linear measurements, with compressed sensing as a case of sparse estimation. Our goal is to provide a pedagogical review for researchers in physics and other fields interested in this fascinating topic.

Subfields: Condensed Matter Physics, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Continual Learning in the Teacher-Student Setup: Impact of Task Similarity (2021, background): “…the dynamics of learning in neural network models (Gardner & Derrida, 1989; Seung et al., 1992; Watkin et al., 1993; Engel & Van den Broeck, 2001; Zdeborová & Krzakala, 2016) and has recently experienced a surge of activity in the machine learning community (Zimmer et al., 2014; Zhong et al.,…”
Modelling the influence of data structure on learning in neural networks (2019, background): “This lack of tractable models prevents us from analysing the impact of data sets on the training of neural networks and their ability to generalise from examples, which remains an open problem both in statistical learning theory [1, 2], and in analysing the average-case behaviour of algorithms in sy”
Dynamics of stochastic gradient descent for two-layer neural networks in the teacher–student setup (2019, background): “[23] Lenka Zdeborová and Florent Krzakala.”
Asymptotic learning curves of kernel methods: empirical data v.s. Teacher-Student paradigm (2019, methodology): “The Teacher-Student paradigm has been broadly used to study supervised learning (Saad and Solla, 1995; Monasson and Zecchina, 1995; Opper and Saad, 2001; Engel and Van den Broeck, 2001; Zdeborová and Krzakala, 2016; Barbier et al., 2019; Gabrié et al., 2018; Aubin et al., 2018; Franz et al., 2018).”
Inverse statistical problems: from the inverse Ising problem to data science (2017, background): “The expression of a gene is tightly controlled to ensure that the right amounts of proteins are produced at the right time.”
Comparing dynamics: deep neural networks versus glassy systems (2018, background): “This scenario has tantalizing similarities with the one found in several combinatorial optimization problems in which easy, hard and impossible algorithmic phases have been found, see e.g. (Monasson et al., 1999; M´ezard et al., 2002; Krzakała et al., 2007; Zdeborov´a & Krzakala, 2016; Achlioptas & ”
Capturing the learning curves of generic features maps for realistic data sets with a teacher-student model (2021, background): “Teacher-student models are a popular framework to study the high-dimensional asymptotic performance of machine learning problems with synthetic data, and have been the subject of intense investigations spanning three decades [1, 2, 3, 4, 5, 6, 7].”
Machine learning and the physical sciences (2019, background): “This rich learning dynamics and curves can appear in many machine learning problems, as was shown in various models, see e.g. more recent review (Zdeborová and Krzakala, 2016).”
Extreme value statistics of correlated random variables: A pedagogical review (2019, unknown): “[278] L.”
Beyond neural scaling laws: beating power law scaling via data pruning (2022, unknown): “[24] Lenka Zdeborová and Florent Krzakala.”
The committee machine: computational to statistical gaps in learning a two-layers neural network (2018, methodology): “The AMP algorithm can be obtained as a Taylor expansion of loopy belief-propagation (see [33]) and also originates in earlier statistical physics works [36, 37, 38, 39, 40, 26].”
Optimal errors and phase transitions in high-dimensional generalized linear models (2017, unknown): “1 n2 E[‖X*X*−EP(x|Y,Φ)[xx T ]‖(2)F]→ ρ (2)− q*(α)(2) [13]”