# Dynamic Online Gradient Descent with Improved Query Complexity: A Theoretical Revisit

## TL;DR

This paper introduces a new theoretical analysis framework for online gradient descent in dynamic environments, achieving optimal regret with fewer gradient queries, especially benefiting ill-conditioned problems.

## Contribution

The paper presents a novel analysis framework that reduces gradient query complexity to O(1) for dynamic regret, improving over previous methods that depended on condition number.

## Key findings

- Achieves state-of-the-art dynamic regret without extra gradient queries
- Reduces gradient query complexity from O(κ) to O(1) for strongly convex and smooth functions
- Significantly benefits ill-conditioned problems with large condition numbers

## Abstract

We provide a new theoretical analysis framework to investigate online gradient descent in the dynamic environment. Comparing with the previous work, the new framework recovers the state-of-the-art dynamic regret, but does not require extra gradient queries for every iteration. Specifically, when functions are $\alpha$ strongly convex and $\beta$ smooth, to achieve the state-of-the-art dynamic regret, the previous work requires $O(\kappa)$ with $\kappa = \frac{\beta}{\alpha}$ queries of gradients at every iteration. But, our framework shows that the query complexity can be improved to be $O(1)$, which does not depend on $\kappa$. The improvement is significant for ill-conditioned problems because that their objective function usually has a large $\kappa$.