# Accelerated gradient sliding for structured convex optimization

arXiv: https://arxiv.org/abs/1609.04905 · HTML: https://mlsanity.com/paper/1609.04905/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2016.

## Built on by (most cited first, 7 of 7)

- [Stochastic Distributed Optimization under Average Second-order Similarity: Algorithms and Analysis](https://mlsanity.com/paper/2304.07504/) (2023)
- [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)
- [A Whole New Ball Game: A Primal Accelerated Method for Matrix Games and Minimizing the Maximum of Smooth Functions](https://mlsanity.com/paper/2311.10886/) (2023)
- [Auto-conditioned primal-dual hybrid gradient method and alternating direction method of multipliers](https://mlsanity.com/paper/2410.01979/) (2024)
- [Accelerated Methods with Complexity Separation Under Data Similarity for Federated Learning Problems](https://mlsanity.com/paper/2601.08614/) (2026)
- [Decentralized Optimization with Cross-Coupled Mixed Affine Constraints](https://mlsanity.com/paper/2609.33021/) (2026)

## Abstract (arXiv)

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show that the AGS method can skip the gradient computation for one of these smooth components without slowing down the overall optimal rate of convergence. This result is much sharper than the classic black-box CP complexity results especially when the difference between the two Lipschitz constants associated with these components is large. We then consider an important class of bilinear saddle point problem whose objective function is given by the summation of a smooth component and a nonsmooth one with a bilinear saddle point structure. Using the aforementioned AGS method for smooth composite optimization and Nesterov's smoothing technique, we show that one only needs cal O(1/√ε) gradient computations for the smooth component while still preserving the optimal cal O(1/ε) overall iteration complexity for solving these saddle point problems. We demonstrate that even more significant savings on gradient computations can be obtained for strongly convex smooth and bilinear saddle point problems.