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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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Sparse Poisson regression via mixed-integer optimization.

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We introduce a mixed-integer optimization approach for sparse Poisson regression, enhancing variable selection. This method improves prediction accuracy compared to existing techniques, especially in low-noise environments.

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Area of Science:

  • Statistics
  • Optimization
  • Machine Learning

Background:

  • Sparse regression methods aim to identify relevant variables from large datasets.
  • Mixed-integer optimization (MIO) offers a robust framework for subset selection in regression.
  • Advances in algorithms and hardware have revitalized MIO for statistical modeling.

Purpose of the Study:

  • To develop a mixed-integer quadratic optimization (MIQO) formulation for sparse Poisson regression.
  • To optimize a weighted combination of log-likelihood and L2-regularization.
  • To propose efficient methods for piecewise-linear approximation in the optimization process.

Main Methods:

  • Formulating sparse Poisson regression as a mixed-integer quadratic optimization (MIQO) problem.
  • Employing piecewise-linear approximation for the log-likelihood function.
  • Developing novel tangent line selection strategies for approximation accuracy.

Main Results:

  • The proposed MIQO formulation is solvable to optimality using standard software.
  • The tangent line selection methods outperform conventional greedy algorithms in maximizing log-likelihood.
  • The MIQO approach demonstrates superior out-of-sample prediction performance over stepwise selection and L1-regularization.

Conclusions:

  • The MIQO approach provides an effective and accurate method for sparse Poisson regression.
  • This technique offers significant advantages in variable selection and predictive performance.
  • The method is particularly beneficial in scenarios with limited noise.