# Sparse Logistic Regression Learns All Discrete Pairwise Graphical Models

arXiv: https://arxiv.org/abs/1810.11905 · HTML: https://mlsanity.com/paper/1810.11905/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2018.

## Built on by (most cited first, 16 of 16)

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- [A Unified Approach to Learning Ising Models: Beyond Independence and Bounded Width](https://mlsanity.com/paper/2311.09197/) (2023)
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- [Learning the Sherrington-Kirkpatrick Model Even at Low Temperature](https://mlsanity.com/paper/2411.11174/) (2024)
- [Ising Model Selection Using ℓ1-Regularized Linear Regression: A Statistical Mechanics Analysis](https://mlsanity.com/paper/2102.03988/) (2021)
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- [Learning Gaussian DAG Models without Condition Number Bounds](https://mlsanity.com/paper/2511.06164/) (2025)
- [Estimating Ising Models in Total Variation Distance](https://mlsanity.com/paper/2511.21008/) (2025)
- [BClean: A Bayesian Data Cleaning System](https://mlsanity.com/paper/2311.06517/) (2023)
- [Discrete distributions are learnable from metastable samples](https://mlsanity.com/paper/2410.13800/) (2024)
- [Efficient Hamiltonian, structure and trace distance learning of Gaussian states](https://mlsanity.com/paper/2411.03163/) (2024)
- [Learning mathsfAC0 under Locally Sampleable Graphical Models](https://mlsanity.com/paper/2607.08303/) (2026)
- [Recoverability Landscape of Tree Structured Markov Random Fields under Symmetric Noise](https://mlsanity.com/paper/2102.08554/) (2021)
- [Finite Sample Bounds for Learning with Score Matching](https://mlsanity.com/paper/2605.14168/) (2026)
- [Efficient Learning of Truncated Boolean Product Distributions: Influence to the Rescue](https://mlsanity.com/paper/2607.22889/) (2026)

## Abstract (arXiv)

We characterize the effectiveness of a classical algorithm for recovering the Markov graph of a general discrete pairwise graphical model from i.i.d. samples. The algorithm is (appropriately regularized) maximum conditional log-likelihood, which involves solving a convex program for each node; for Ising models this is ℓ1-constrained logistic regression, while for more general alphabets an ℓ2,1 group-norm constraint needs to be used. We show that this algorithm can recover any arbitrary discrete pairwise graphical model, and also characterize its sample complexity as a function of model width, alphabet size, edge parameter accuracy, and the number of variables. We show that along every one of these axes, it matches or improves on all existing results and algorithms for this problem. Our analysis applies a sharp generalization error bound for logistic regression when the weight vector has an ℓ1 constraint (or ℓ2,1 constraint) and the sample vector has an ℓ∞ constraint (or ℓ2, ∞ constraint). We also show that the proposed convex programs can be efficiently solved in tildeO(n2) running time (where n is the number of variables) under the same statistical guarantees. We provide experimental results to support our analysis.