# Sharper Bounds for Proximal Gradient Algorithms with Errors

arXiv: https://arxiv.org/abs/2203.02204 · HTML: https://mlsanity.com/paper/2203.02204/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 51 references that are in the map.
- First posted on arXiv 4 Mar 2022.

## Builds on (most cited first, 7 of 7)

- [First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems](https://mlsanity.com/paper/1706.06461/) (2017)
- [Convergence of Stochastic Proximal Gradient Algorithm](https://mlsanity.com/paper/1403.5074/) (2014)
- [Distributed Proximal Gradient Algorithm for Partially Asynchronous Computer Clusters](https://mlsanity.com/paper/1704.03540/) (2017)
- [Maximum Hands-Off Control: A Paradigm of Control Effort Minimization](https://mlsanity.com/paper/1408.3025/) (2014)
- [On perturbed proximal gradient algorithms](https://mlsanity.com/paper/1402.2365/) (2014)
- [Non-smooth Non-convex Bregman Minimization: Unification and new Algorithms](https://mlsanity.com/paper/1707.02278/) (2017)
- [The Sound of APALM Clapping: Faster Nonsmooth Nonconvex Optimization with Stochastic Asynchronous PALM](https://mlsanity.com/paper/1606.02338/) (2016)

## Built on by (most cited first, 1 of 1)

- [Accelerating nuclear-norm regularized low-rank matrix optimization through Burer-Monteiro decomposition](https://mlsanity.com/paper/2204.14067/) (2022)

## Abstract (arXiv)

We analyse the convergence of the proximal gradient algorithm for convex composite problems in the presence of gradient and proximal computational inaccuracies. We derive new tighter deterministic and probabilistic bounds that we use to verify a simulated (MPC) and a synthetic (LASSO) optimization problems solved on a reduced-precision machine in combination with an inaccurate proximal operator. We also show how the probabilistic bounds are more robust for algorithm verification and more accurate for application performance guarantees. Under some statistical assumptions, we also prove that some cumulative error terms follow a martingale property. And conforming to observations, e.g., in citeschmidt2011convergence, we also show how the acceleration of the algorithm amplifies the gradient and proximal computational errors.