# Oracle Complexity of Second-Order Methods for Smooth Convex Optimization

## TL;DR

This paper establishes tight bounds on the number of iterations second-order methods need to optimize smooth convex functions, clarifying their advantages and limitations compared to gradient-based methods.

## Contribution

It provides the first tight bounds on the oracle complexity of second-order methods for smooth convex optimization, including higher-order generalizations.

## Key findings

- Second-order methods can match or outperform gradient methods under certain conditions.
- Tight bounds reveal when second-order methods are advantageous or limited.
- Results extend to higher-order optimization techniques.

## Abstract

Second-order methods, which utilize gradients as well as Hessians to optimize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth convex functions, or equivalently, the worst-case number of iterations required to optimize such functions to a given accuracy. In particular, these bounds indicate when such methods can or cannot improve on gradient-based methods, whose oracle complexity is much better understood. We also provide generalizations of our results to higher-order methods.