# Adaptive Restart of the Optimized Gradient Method for Convex Optimization

arXiv: https://arxiv.org/abs/1703.04641 · HTML: https://mlsanity.com/paper/1703.04641/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Mar 2017.

## Built on by (most cited first, 6 of 6)

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- [DualFL: A Duality-based Federated Learning Algorithm with Communication Acceleration in the General Convex Regime](https://mlsanity.com/paper/2305.10294/) (2023)
- [Fast Empirical Scenarios](https://mlsanity.com/paper/2307.03927/) (2023)
- [Ensemble-based Hybrid Optimization of Bayesian Neural Networks and Traditional Machine Learning Algorithms](https://mlsanity.com/paper/2310.05456/) (2023)
- [Outer-Momentum Restarting in High-Dimensional Two-Phase Optimization](https://mlsanity.com/paper/2605.28585/) (2026)

## Abstract (arXiv)

First-order methods with momentum such as Nesterov's fast gradient method are very useful for convex optimization problems, but can exhibit undesirable oscillations yielding slow convergence rates for some applications. An adaptive restarting scheme can improve the convergence rate of the fast gradient method, when the parameter of a strongly convex cost function is unknown or when the iterates of the algorithm enter a locally strongly convex region. Recently, we introduced the optimized gradient method, a first-order algorithm that has an inexpensive per-iteration computational cost similar to that of the fast gradient method, yet has a worst-case cost function rate that is twice faster than that of the fast gradient method and that is optimal for large-dimensional smooth convex problems. Building upon the success of accelerating the fast gradient method using adaptive restart, this paper investigates similar heuristic acceleration of the optimized gradient method. We first derive a new first-order method that resembles the optimized gradient method for strongly convex quadratic problems with known function parameters, yielding a linear convergence rate that is faster than that of the analogous version of the fast gradient method. We then provide a heuristic analysis and numerical experiments that illustrate that adaptive restart can accelerate the convergence of the optimized gradient method. Numerical results also illustrate that adaptive restart is helpful for a proximal version of the optimized gradient method for nonsmooth composite convex functions.