Stability of decision trees and logistic regression

TL;DR

This paper investigates the stability of decision trees and logistic regression, deriving new stability notions, analyzing their dependence on model parameters, and proposing a framework for empirical stability measurement.

Contribution

It introduces new stability concepts for decision trees and logistic regression, provides theoretical bounds, and develops a novel empirical stability measurement framework.

Findings

Stability of decision trees depends on the number of leaves.

Logistic regression stability depends on the Hessian's smallest eigenvalue.

Logistic regression is generally not a stable learning algorithm.

Abstract

Decision trees and logistic regression are one of the most popular and well-known machine learning algorithms, frequently used to solve a variety of real-world problems. Stability of learning algorithms is a powerful tool to analyze their performance and sensitivity and subsequently allow researchers to draw reliable conclusions. The stability of these two algorithms has remained obscure. To that end, in this paper, we derive two stability notions for decision trees and logistic regression: hypothesis and pointwise hypothesis stability. Additionally, we derive these notions for L2-regularized logistic regression and confirm existing findings that it is uniformly stable. We show that the stability of decision trees depends on the number of leaves in the tree, i.e., its depth, while for logistic regression, it depends on the smallest eigenvalue of the Hessian matrix of the cross-entropy loss. We show that logistic regression is not a stable learning algorithm. We construct the upper bounds on the generalization error of all three algorithms. Moreover, we present a novel stability measuring framework that allows one to measure the aforementioned notions of stability. The measures are equivalent to estimates of expected loss differences at an input example and then leverage bootstrap sampling to yield statistically reliable estimates. Finally, we apply this framework to the three algorithms analyzed in this paper to confirm our theoretical findings and, in addition, we discuss the possibilities of developing new training techniques to optimize the stability of logistic regression, and hence decrease its generalization error.