# Adaptive Three Operator Splitting

arXiv: https://arxiv.org/abs/1804.02339 · HTML: https://mlsanity.com/paper/1804.02339/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Apr 2018.

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

- [Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient](https://mlsanity.com/paper/2301.04431/) (2023)
- [Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting](https://mlsanity.com/paper/2301.13395/) (2023)
- [A first-order primal-dual method with adaptivity to local smoothness](https://mlsanity.com/paper/2110.15148/) (2021)
- [Strong Screening Rules for Group-based SLOPE Models](https://mlsanity.com/paper/2405.15357/) (2024)
- [Stochastic Projective Splitting: Solving Saddle-Point Problems with Multiple Regularizers](https://mlsanity.com/paper/2106.13067/) (2021)
- [Operator Splitting for Learning to Predict Equilibria in Convex Games](https://mlsanity.com/paper/2106.00906/) (2021)
- [Dual Feature Reduction for the Sparse-group Lasso and its Adaptive Variant](https://mlsanity.com/paper/2405.17094/) (2024)

## Abstract (arXiv)

We propose and analyze an adaptive step-size variant of the Davis-Yin three operator splitting. This method can solve optimization problems composed by a sum of a smooth term for which we have access to its gradient and an arbitrary number of potentially non-smooth terms for which we have access to their proximal operator. The proposed method sets the step-size based on local information of the objective --hence allowing for larger step-sizes--, only requires two extra function evaluations per iteration and does not depend on any step-size hyperparameter besides an initial estimate. We provide an iteration complexity analysis that matches the best known results for the non-adaptive variant: sublinear convergence for general convex functions and linear convergence under strong convexity of the smooth term and smoothness of one of the proximal terms. Finally, an empirical comparison with related methods on 6 different problems illustrates the computational advantage of the proposed method.