# Logarithmic Regret for parameter-free Online Logistic Regression

## TL;DR

This paper introduces a new second-order online logistic regression algorithm with logarithmic regret bounds in adversarial settings, advancing the theoretical understanding of parameter-free methods.

## Contribution

The paper proposes the Semi-Online Step (SOS) algorithm, achieving the first logarithmic regret bound for parameter-free online logistic regression in adversarial environments.

## Key findings

- SOS achieves O(log n) regret in adversarial settings.
- EKF attains O(log n) regret in constant dynamics and well-specified models.
- First logarithmic regret bounds for parameter-free online logistic regression.

## Abstract

We consider online optimization procedures in the context of logistic regression, focusing on the Extended Kalman Filter (EKF). We introduce a second-order algorithm close to the EKF, named Semi-Online Step (SOS), for which we prove a O(log(n)) regret in the adversarial setting, paving the way to similar results for the EKF. This regret bound on SOS is the first for such parameter-free algorithm in the adversarial logistic regression. We prove for the EKF in constant dynamics a O(log(n)) regret in expectation and in the well-specified logistic regression model.