Volumes of logistic regression models with applications to model selection
This work provides a theoretical foundation for model selection in logistic regression, offering a complexity measure that spontaneously encourages sparsity, which is incremental but novel in its geometric approach.
The authors tackled the problem of quantifying model complexity in logistic regression by proving that Fisher information volumes are bounded between π^q and (n choose q)π^q, showing these volumes are always finite and can serve as a complexity measure. They demonstrated that this volume is discontinuous at sparse design matrices, leading to a model-selection criterion that naturally prefers sparse models without explicit regularization.
Logistic regression models with $n$ observations and $q$ linearly-independent covariates are shown to have Fisher information volumes which are bounded below by $π^q$ and above by ${n \choose q} π^q$. This is proved with a novel generalization of the classical theorems of Pythagoras and de Gua, which is of independent interest. The finding that the volume is always finite is new, and it implies that the volume can be directly interpreted as a measure of model complexity. The volume is shown to be a continuous function of the design matrix $X$ at generic $X$, but to be discontinuous in general. This means that models with sparse design matrices can be significantly less complex than nearby models, so the resulting model-selection criterion prefers sparse models. This is analogous to the way that $\ell^1$-regularisation tends to prefer sparse model fits, though in our case this behaviour arises spontaneously from general principles. Lastly, an unusual topological duality is shown to exist between the ideal boundaries of the natural and expectation parameter spaces of logistic regression models.