Self-concordant analysis for logistic regression (0910.4627v1)

Abstract: Most of the non-asymptotic theoretical work in regression is carried out for the square loss, where estimators can be obtained through closed-form expressions. In this paper, we use and extend tools from the convex optimization literature, namely self-concordant functions, to provide simple extensions of theoretical results for the square loss to the logistic loss. We apply the extension techniques to logistic regression with regularization by the $\ell_2$-norm and regularization by the $\ell_1$-norm, showing that new results for binary classification through logistic regression can be easily derived from corresponding results for least-squares regression.

• The paper introduces a novel framework that adapts self-concordant functions to logistic loss, establishing global Taylor expansion bounds for logistic regression.
• It leverages techniques from quadratic loss regression to derive non-asymptotic error bounds and performance guarantees for both ℓ2 and ℓ1 regularizations.
• The framework opens avenues for broader applications in exponential family distributions and generalized linear models, ensuring robust analysis across varied statistical settings.

Self-concordant Analysis for Logistic Regression

The paper "Self-concordant analysis for logistic regression," authored by Francis Bach, seeks to extend non-asymptotic theoretical results from the domain of least-squares regression, typically analyzed using square loss functions, to logistic regression, which involves logistic loss functions. The focus of this work lies in leveraging self-concordant function concepts from convex optimization to simplify the analysis and comparison of statistical results, specifically for logistic regression tasks.

In the context of regression analysis, closed-form solutions significantly simplify theoretical investigations, notably when the regression estimator minimizes a square loss. This closed-form solution offers classical, established methods for deriving both asymptotic and non-asymptotic results, which are prominently featured in probability theory. However, this simplicity diminishes when moving beyond quadratic loss functions, such as those found in logistic regression, which typically do not yield closed-form solutions.

The paper introduces a novel analytical framework by modifying the concept of self-concordance. Traditional self-concordant functions are well-known in optimization for improving the understanding of Newton’s method, with their applicability primarily tied to convex functions provided with second derivatives controlled by their third derivatives. As logistic loss functions do not naturally meet this classic self-concordance, the paper extends these notions with alternative control of third derivatives.

This approach facilitates two primary outcomes: globally bounding Taylor expansions for logistic loss functions and examining the behavior of Newton’s method within this adjusted context. The theoretical derivations and extended results pertain notably to logistic regression models that incorporate both $\ell_2$- and $\ell_1$-norm regularizations, offering straightforward means to adapt known results from least-squares regression.

The paper's exploration of $\ell_2$-norm regularization exemplifies this adaptability by providing a performance analysis that incorporates minimal assumptions, which is particularly useful for misspecified models. The theoretical bounds are structured around an oracle inequality that characterizes the generalization performance under both model consistency and well-specification assumptions. On the other hand, in $\ell_1$-norm regularization, the focus shifts to issues of model selection consistency and efficiency, providing clarity on conditions that align closely with those in the least-squares context.

A significant implication of the paper is its potential application across broader statistical estimation problems beyond logistic regression, encompassing other exponential family distributions and generalized linear models. These insights hint at practical applications in fields where logistic regression models are prevalent, enabling analysts to derive consistent estimates without cumbersome additional assumptions.

In summary, this research makes a noteworthy contribution by presenting modified theoretical tools—rooted in the convex optimization domain—and adapting them to manage logistic regression analyses transparently and effectively. Future prospects include further exploration into expanding these methods to other non-linear loss functions and high-dimensional statistical estimation frameworks. This paper thus potentially unfolds new avenues in robustly analyzing logistic and similar non-linear models across diverse applied settings within artificial intelligence.