Condition Number Analysis of Logistic Regression, and its Implications for Standard First-Order Solution Methods
This work addresses a theoretical gap for practitioners using logistic regression in binary classification, particularly in large-scale settings, by providing insights into algorithm behavior, though it is incremental as it builds on prior analysis.
The paper tackles the problem of understanding how the separability or non-separability of data affects the convergence of first-order methods in logistic regression, showing that condition numbers measuring these properties inform the convergence guarantees for steepest descent and stochastic gradient descent, including linear convergence without strong convexity and approximate-maximum-margin solutions in separable cases.
Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of non-separability or separability of a given dataset in the setting of binary classification, and we study how these condition numbers relate to and inform the properties and the convergence guarantees of first-order methods. When the training data is non-separable, we show that the degree of non-separability naturally enters the analysis and informs the properties and convergence guarantees of two standard first-order methods: steepest descent (for any given norm) and stochastic gradient descent. Expanding on the work of Bach, we also show how the degree of non-separability enters into the analysis of linear convergence of steepest descent (without needing strong convexity), as well as the adaptive convergence of stochastic gradient descent. When the training data is separable, first-order methods rather curiously have good empirical success, which is not well understood in theory. In the case of separable data, we demonstrate how the degree of separability enters into the analysis of $\ell_2$ steepest descent and stochastic gradient descent for delivering approximate-maximum-margin solutions with associated computational guarantees as well. This suggests that first-order methods can lead to statistically meaningful solutions in the separable case, even though the ML solution does not exist.