Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models (1605.07252v3)

Abstract: We consider the problem of learning the underlying graph of an unknown Ising model on p spins from a collection of i.i.d. samples generated from the model. We suggest a new estimator that is computationally efficient and requires a number of samples that is near-optimal with respect to previously established information-theoretic lower-bound. Our statistical estimator has a physical interpretation in terms of "interaction screening". The estimator is consistent and is efficiently implemented using convex optimization. We prove that with appropriate regularization, the estimator recovers the underlying graph using a number of samples that is logarithmic in the system size p and exponential in the maximum coupling-intensity and maximum node-degree.

• The paper presents the Interaction Screening Objective (ISO) and the RISE estimator to efficiently recover sparse Ising models.
• It leverages ℓ1-regularization and convex optimization techniques to achieve robust error bounds and near-optimal sample complexity.
• Experimental validations on grid and spin glass models demonstrate that sample requirements scale logarithmically with system size and exponentially with coupling intensity.

Overview of Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models

This paper addresses a critical problem within graphical model (GM) theory: learning the underlying graphical structure of Ising models from independent and identically distributed (i.i.d.) samples. The authors propose the Interaction Screening estimator as a computationally efficient and sample-optimal method for reconstructing Ising models on $p$ spins, enhancing the fidelity and reducing the sample complexity needed relative to previous approaches. Their approach leverages convex optimization and $\ell_{1}$-regularization to perform interaction screening and demonstrate both theoretical robustness and empirical efficacy across varied test cases.

Key Contributions

The paper makes several contributions to the field of computational statistical mechanics and machine learning:

1. Interaction Screening Objective (ISO): The development of a convex objective function which is sensitive to spin interactions, coined the Interaction Screening Objective (ISO), is central to this method. The ISO empirically adapts to minimize at true parameter values, screening out the spin-related interactions in the Ising model—effectively isolating each spin from its neighbors in the sample dataset context.
2. Regularized Interaction Screening Estimator (RISE): The authors introduce an estimator that infers spin couplings with sparse regularization via $\ell_{1}$ norm penalties, minimizing statistical noise and bias in low sample regimes, which guarantees near-optimal number requirements and computational efficiency.
3. Structure-RISE: This step involves thresholding recovered couplings, allowing precise reconstruction of the GM structure and providing strong guarantees for exact recovery.

Theoretical Results

Through the presented theorems, the authors provide bounds on the sample complexity necessary to achieve both accurate coupling parameter estimations and exact graph structure recovery. Specifically:

• Sample Complexity: The number of samples required scales logarithmically with the number of spins $p$ yet exponentially with coupling intensity and node degree, aligning well with established lower bounds based on information theoretic principles.
• Error Guarantees: The penalty parameter $\lambda$ utilized in RISE ensures robust $\ell_{1}$-regularization when tailored adequately, achieving bounds on the estimation error which confirm theoretical efficiency.

Experimental Validation

Empirical results validate the theoretical claims regarding sample complexity, demonstrating the logarithmic relationship between sample requirements and system size and exponential relationship with coupling intensity during tests conducted on Ising models on grid structures. These simulations further highlight the efficiency gains over prior methods.

• Grid Models: 2D grid configurations with uniform coupling intensities provided a basis for assessing scalability with system size, supporting theoretical expectations regarding logarithmic scaling.
• Coupling Intensity: Spin glass models with variable coupling intensities confirmed exponential scaling in sample requirements, demonstrating the versatility of RISE in handling diverse interaction scenarios.

Implications and Future Directions

The authors elucidate pathways to extend the screening method beyond pairwise binary interactions, suggesting adaptations for broader classes of GMs, including those with higher-order or mixed interactions. Such extensions could enhance model applicability across fields such as gene expression analysis, neuroscience, image processing, and beyond. The groundwork laid by the Interaction Screening estimator could foster advancements in statistical inference for high-dimensional systems, shaping future research trajectories in statistical physics and machine learning.

The rigorous framework and promising results presented make Interaction Screening an important methodological advance, setting benchmarks for efficiency and precision in graphical model learning tasks. However, given the paper's computational focus, further exploration into interaction screening's adaptability and scalability in real-world datasets remains a fruitful avenue for subsequent research.