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On the Complexity of Logistic Regression Models.

Nicola Bulso1, Matteo Marsili2, Yasser Roudi3

  • 1Kavli Institute for Systems Neuroscience and Centre for Neural Computation, Norwegian University of Science and Technology (NTNU), Trondheim, Norway nicola.bulso@ntnu.no.

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Summary

Logistic regression model complexity depends on input distribution, not just parameters. Input correlations reduce complexity, leading to a novel, robust model selection criterion outperforming standard methods.

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Area of Science:

  • Machine Learning
  • Statistical Modeling
  • Information Theory

Background:

  • Standard complexity measures for logistic regression models do not fully account for input data distributions.
  • Input correlations can significantly influence model complexity in ways not captured by traditional analyses.
  • Understanding model complexity is crucial for effective model selection and generalization.

Purpose of the Study:

  • To investigate the complexity of logistic regression models, considering the role of input distributions.
  • To develop a novel model selection criterion that incorporates input data entropy.
  • To evaluate the performance of the proposed criterion against established methods.

Main Methods:

  • Analytical derivation of complexity bounds for logistic models with binary inputs.
  • Investigation of parameter support effects on model complexity.
  • Development and Bayesian model selection framework implementation of a new criterion based on input entropy.
  • Numerical testing comparing the new criterion with AIC, BIC, and regularization methods.

Main Results:

  • Input correlations effectively reduce logistic model complexity.
  • Finite parameter support decreases complexity, dependent on domain size.
  • The proposed entropy-based criterion demonstrates superior reconstruction accuracy across varying sparsity, data size, and correlation levels.
  • Categorical input alphabet size has a minimal impact on complexity compared to parameter space dimension.

Conclusions:

  • Model complexity is intrinsically linked to input data characteristics, particularly correlations and distributions.
  • The novel entropy-aware model selection criterion offers robust performance, outperforming traditional methods.
  • Findings provide a more nuanced understanding of logistic regression complexity and model selection strategies.