Minimizing Stochastic Complexity with Ridge Regression

Area of Science:

• Statistical Learning Theory
• Machine Learning
• Information Theory

Background:

• Stochastic complexity (SC) is a principled information-theoretic approach for model selection.
• Traditional SC does not fully capture the impact of regularization on model complexity.
• Ridge regression is a widely used regularization technique to prevent overfitting.

Purpose of the Study:

• To derive a penalty strength criterion for ridge regression based on stochastic complexity.
• To adapt the stochastic complexity framework to account for regularization's effect on model complexity.
• To provide an analytical expression for the complexity penalty in ridge regression.

Main Methods:

• Modified the standard stochastic complexity coding scheme using a weighted ensemble of regularized model fits.
• Interpreted regularization as a mechanism for constraining model flexibility and reducing complexity.
• Derived an analytical complexity penalty term: the log determinant of the residual operator.

Main Results:

• Successfully derived a penalty strength criterion for ridge regression using the modified stochastic complexity.
• The derived penalty term offers an analytical solution, simplifying practical application.
• Demonstrated the effectiveness of the complexity penalty in a reservoir computing task and benchmark datasets.

Conclusions:

• The proposed method provides a principled way to determine ridge regression penalty strength via stochastic complexity.
• Regularization's role in complexity reduction is explicitly modeled, enhancing theoretical understanding.
• The analytical penalty term facilitates improved model selection and performance in regression tasks.