Testing Ising Models

Daskalakis, Constantinos; Dikkala, Nishanth; Kamath, Gautam

doi:10.1109/tit.2019.2932255

Cited by 51 publications

(86 citation statements)

References 60 publications

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“…As in the proof of Theorem 5.1, the first step is to show the concentration of ∇L N (γ). Towards this, following the proof of Lemma 5 shows that Q r (as defined in (30)) is O β (poly(d max )/N )-Lipschitz, for each r ∈ [ℓ]. Therefore, by arguments as in Lemma 5,…”

Section: 1mentioning

confidence: 83%

“…Lemma 5. For r ∈ [ℓ] and s ∈ [d], let Q r and Q r,s be as defined in (30) and (32), respectively. Then for any two vectors X, X ′ ∈ C N differing in at most one coordinate, the following hold:…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…samples from an Ising model. In a related direction, Daskalakis et al [30] studied the related problems of identity and independence testing, and Neykov et al [19,63] considered problems in graph property testing, given access to multiple samples from an Ising model.…”

Section: Introductionmentioning

confidence: 99%

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High Dimensional Logistic Regression Under Network Dependence

Mukherjee¹,

Niu²,

Halder³

et al. 2021

Preprint

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Logistic regression is one of the most fundamental methods for modeling the probability of a binary outcome based on a collection of covariates. However, the classical formulation of logistic regression relies on the independent sampling assumption, which is often violated when the outcomes interact through an underlying network structure, such as over a temporal/spatial domain or on a social network. This necessitates the development of models that can simultaneously handle both the network 'peer-effect' (arising from neighborhood interactions) and the effect of (possibly) high-dimensional covariates. In this paper, we develop a framework for incorporating such dependencies in a high-dimensional logistic regression model by introducing a quadratic interaction term, as in the Ising model, designed to capture the pairwise interactions from the underlying network. The resulting model can also be viewed as an Ising model, where the node-dependent external fields linearly encode the high-dimensional covariates. We propose a penalized maximum pseudo-likelihood method for estimating the network peer-effect and the effect of the covariates (the regression coefficients), which, in addition to handling the high-dimensionality of the parameters, conveniently avoids the computational intractability of the maximum likelihood approach. Consequently, our method is computational efficient and, under various standard regularity conditions, we show that the corresponding estimate attains the classical high-dimensional rate of consistency. In particular, our results imply that even under network dependence it is possible to consistently estimate the model parameters at the same rate as in classical (independent) logistic regression, when the true parameter is sparse and the underlying network is not too dense. As a consequence of the general results, we derive the rates of consistency of our proposed estimator for various natural graph ensembles, such as bounded degree graphs, sparse Erdős-Rényi random graphs, and stochastic block models.

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Section: 1mentioning

confidence: 83%

Section: Proof Of Theoremmentioning

confidence: 99%

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High Dimensional Logistic Regression Under Network Dependence

Mukherjee¹,

Niu²,

Halder³

et al. 2021

Preprint

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“…The random vector X is Markov on a tree G = (V, E) and, in this paper, we focus on models in which each random variable has zero mean and the interactions between the nodes that are connected by an edge are identical. This is also known as a homogeneous Ising model with zero external field [DDK19,TYT21]. The probability mass function can be expressed as…”

Section: System Modelmentioning

confidence: 99%

Active-LATHE: An Active Learning Algorithm for Boosting the Error Exponent for Learning Homogeneous Ising Trees

Zhang¹,

Tandon²,

Tan³

2021

Preprint

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The Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) has been a mainstay for the learning of tree-structured graphical models from i.i.d. sampled data vectors. Its theoretical properties have been well-studied and are well-understood. In this paper, we focus on the class of trees that are arguably even more fundamental, namely homogeneous trees in which each pair of nodes that forms an edge has the same correlation ρ. We ask whether we are able to further reduce the error probability of learning the structure of the homogeneous tree model when active learning or active sampling of nodes or variables is allowed. Our figure of merit is the error exponent, which quantifies the exponential rate of decay of the error probability with an increasing number of data samples. At first sight, an improvement in the error exponent seems impossible, as all the edges are statistically identical. We design and analyze an algorithm Active Learning Algorithm for Trees with Homogeneous Edges (Active-LATHE), which surprisingly boosts the error exponent by at least 40% when ρ is at least 0.8. For all other values of ρ, we also observe commensurate, but more modest, improvements in the error exponent. Our analysis hinges on judiciously exploiting the minute but detectable statistical variation of the samples to allocate more data to parts of the graph in which we are less confident of being correct.

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“…For the ferromagnetic (attractive) Ising model, Daskalakis et al (2018) presented a polynomial time algorithm for identity testing. We prove hardness results in the antiferromagnetic (repulsive) setting in the same regime of parameters where structure learning is known to require a super-polynomial number of samples.…”

mentioning

confidence: 99%