# Generalizing the optimized gradient method for smooth convex minimization

arXiv: https://arxiv.org/abs/1607.06764 · HTML: https://mlsanity.com/paper/1607.06764/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2016.

## Built on by (most cited first, 5 of 5)

- [Optimal Gradient Sliding and its Application to Distributed Optimization Under Similarity](https://mlsanity.com/paper/2205.15136/) (2022)
- [Stabilized Proximal-Point Methods for Federated Optimization](https://mlsanity.com/paper/2407.07084/) (2024)
- [Potential Function-based Framework for Making the Gradients Small in Convex and Min-Max Optimization](https://mlsanity.com/paper/2101.12101/) (2021)
- [Practical Schemes for Finding Near-Stationary Points of Convex Finite-Sums](https://mlsanity.com/paper/2105.12062/) (2021)
- [Tradeoffs between convergence rate and noise amplification for momentum-based accelerated optimization algorithms](https://mlsanity.com/paper/2209.11920/) (2022)

## Abstract (arXiv)

This paper generalizes the optimized gradient method (OGM) that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization. Specifically, this paper studies a generalized formulation of OGM and analyzes its worst-case rates in terms of both the function value and the norm of the function gradient. This paper also develops a new algorithm called OGM-OG that is in the generalized family of OGM and that has the best known analytical worst-case bound with rate O(1/N1.5) on the decrease of the gradient norm among fixed-step first-order methods. This paper also proves that Nesterov's fast gradient method has an O(1/N1.5) worst-case gradient norm rate but with constant larger than OGM-OG. The proof is based on the worst-case analysis called Performance Estimation Problem.