# Some Worst-Case Datasets of Deterministic First-Order Methods for Solving Binary Logistic Regression

## TL;DR

This paper constructs worst-case datasets demonstrating that deterministic first-order methods require at least on the order of 1/√ε oracle inquiries to solve large-scale binary logistic regression problems to a given accuracy, highlighting limitations in their efficiency.

## Contribution

It introduces new worst-case datasets for binary logistic regression, showing these problems are among the hardest for deterministic first-order methods and establishing lower bounds on their iteration complexity.

## Key findings

- Deterministic first-order methods need at least O(1/√ε) oracle calls.
- The datasets are new worst-case instances for smooth convex optimization.
- Binary logistic regression can be as hard as the worst-case smooth convex problems.

## Abstract

We present in this paper some worst-case datasets of deterministic first-order methods for solving large-scale binary logistic regression problems. Under the assumption that the number of algorithm iterations is much smaller than the problem dimension, with our worst-case datasets it requires at least $\mathcal{O}(1/\sqrt{\varepsilon})$ first-order oracle inquiries to compute an $\varepsilon$-approximate solution. From traditional iteration complexity analysis point of view, the binary logistic regression loss functions with our worst-case datasets are new worst-case function instances among the class of smooth convex optimization problems.