Optimal structure and parameter learning of Ising models (1612.05024v2)

Abstract: Reconstruction of structure and parameters of an Ising model from binary samples is a problem of practical importance in a variety of disciplines, ranging from statistical physics and computational biology to image processing and machine learning. The focus of the research community shifted towards developing universal reconstruction algorithms which are both computationally efficient and require the minimal amount of expensive data. We introduce a new method, Interaction Screening, which accurately estimates the model parameters using local optimization problems. The algorithm provably achieves perfect graph structure recovery with an information-theoretically optimal number of samples, notably in the low-temperature regime which is known to be the hardest for learning. The efficacy of Interaction Screening is assessed through extensive numerical tests on synthetic Ising models of various topologies with different types of interactions, as well as on a real data produced by a D-Wave quantum computer. This study shows that the Interaction Screening method is an exact, tractable and optimal technique universally solving the inverse Ising problem.

• The paper introduces Interaction Screening to reconstruct Ising models and demonstrates optimal sample complexity.
• It combines local optimization and ℓ1-regularization to accurately recover both graph structure and interaction parameters.
• Extensive tests, including on D-Wave data, confirm its superior performance compared to traditional estimators.

Optimal Structure and Parameter Learning of Ising Models

The paper presents a novel approach to reconstructing Ising models from binary samples, addressing a significant problem in fields such as statistical physics, computational biology, image processing, and machine learning. The Ising model, a graphical model for binary variables, serves as a foundation for understanding diverse phenomena. A challenge in such models is inferring the underlying interaction graph and parameters (couplings and magnetic fields) from observed data.

Overview of Methods

The paper introduces an advanced method, termed Interaction Screening, which addresses the task of model reconstruction by solving local optimization problems. This method precisely estimates model parameters using an approach that is provably optimal in terms of the number of samples required, aligning with information-theoretic limits.

Regularized Pseudo-Likelihood Estimator (RPLE):

One method discussed is the Regularized Pseudo-Likelihood Estimator which maximizes a surrogate likelihood with an $\ell_1$-regularization term to promote sparsity. It is shown that when supplemented with a thresholding post-optimization step, RPLE can achieve perfect structure recovery. The required number of samples is shown to have a scaling relationship that depends on parameters like maximum node-degree and interaction strength.

Interaction Screening and logRISE:

The newly proposed Interaction Screening method, and particularly its version termed logRISE, focuses on minimizing what is termed the Interaction Screening Objective (ISO). This objective harnesses the property of “interaction screening,” effectively identifying model parameters when the observation count becomes sufficiently large. This method proves to be an exact and optimal technique universally solving the inverse Ising problem by reducing the sample complexity further compared to previous approaches.

Numerical Validation and Analysis

The efficacy of these methods is substantiated through extensive numerical tests conducted on synthetic Ising models and real-world data from a D-Wave quantum computer. Key numerical experiments highlight several aspects:

• Sample Complexity: The paper emphasizes the importance of exponential scaling with interaction strength and maximum degree for achieving exact recovery. It convincingly demonstrates that logRISE performs within the theoretical bounds, often achieving optimality in the low-temperature regime where challenges are most pronounced.
• Algorithm Performance: Across multiple graph topologies and interaction types, logRISE consistently exhibits superior performance, requiring fewer samples compared to alternative estimators like the RPLE. This advantage is substantiated in complex scenarios including systems with ferromagnetic coupling patterns and those incorporating spin glass characteristics.
• Graph Structure Learning: The methods are notable not only for reconstructing the structural topology of Ising graphs but also for accurately estimating interaction parameters, demonstrated even under conditions of scarce data. This capability is particularly evident in the D-Wave computer data experiment, where the algorithm successfully inferred the implemented hardware structure.

Theoretical and Practical Implications

This research contributes significantly to the field by establishing guarantees for the exactness and efficiency of the proposed method. Practically, it indicates a reduced need for data in reconstructing complex models, offering utility in scenarios where data is costly to obtain. The method's adaptability implies its potential application across various domain-specific modeling situations, enhancing understanding in interdisciplinary problems.

Future Perspectives:

The paper suggests potential extensions of the Interaction Screening approach to broader classes of graphical models beyond binary Ising constructs, indicating aspirational directions for subsequent research. This points to the exciting prospect of developing universally applicable solutions across diverse and complex statistical and machine learning contexts.

This research exemplifies a robust contribution to the understanding of model inference and strengthens the toolkit available for tackling challenging inverse problems in graphical models. Its implications underscore both theoretical insight and methodological innovation, promising enhanced capabilities for complex system analysis in various scientific fields.