# Error bound conditions and convergence of optimization methods on smooth and proximally smooth manifolds

arXiv: https://arxiv.org/abs/1912.04660 · HTML: https://mlsanity.com/paper/1912.04660/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2019.

## Built on by (most cited first, 3 of 3)

- [Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Data](https://mlsanity.com/paper/2406.08465/) (2024)
- [Decentralized Weakly Convex Optimization Over the Stiefel Manifold](https://mlsanity.com/paper/2303.17779/) (2023)
- [Quantitative Convergence Analysis of Projected Stochastic Gradient Descent for Non-Convex Losses via the Goldstein Subdifferential](https://mlsanity.com/paper/2510.02735/) (2025)

## Abstract (arXiv)

We analyse the convergence of the gradient projection algorithm, which is finalized with the Newton method, to a stationary point for the problem of nonconvex constrained optimization minx ∈ S f(x) with a proximally smooth set S = x ∈ Rn : g(x) = 0 , ; g : Rn rightarrow Rm and a smooth function f. We propose new Error bound (EB) conditions for the gradient projection method which lead to the convergence domain of the Newton method. We prove that these EB conditions are typical for a wide class of optimization problems. It is possible to reach high convergence rate of the algorithm by switching to the Newton method.