# Broximal Gradient Descent: A Projection-Free Sister of Projected Gradient Descent

arXiv: https://arxiv.org/abs/2605.08850 · HTML: https://mlsanity.com/paper/2605.08850/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 85 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 20 of 20)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Safe RLHF: Safe Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2310.12773/) (2023)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)](https://mlsanity.com/paper/2505.13416/) (2025)
- [Projection onto the probability simplex: An efficient algorithm with a simple proof, and an application](https://mlsanity.com/paper/1309.1541/) (2013)
- [Convergence Rate of Frank-Wolfe for Non-Convex Objectives](https://mlsanity.com/paper/1607.00345/) (2016)
- [Distance-Based Regularisation of Deep Networks for Fine-Tuning](https://mlsanity.com/paper/2002.08253/) (2020)
- [The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle](https://mlsanity.com/paper/1309.5550/) (2013)
- [Deep Neural Network Training with Frank-Wolfe](https://mlsanity.com/paper/2010.07243/) (2020)
- [Conditional Gradient Methods](https://mlsanity.com/paper/2211.14103/) (2022)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [The Iterates of the Frank-Wolfe Algorithm May Not Converge](https://mlsanity.com/paper/2202.08711/) (2022)
- [The Ball-Proximal (="Broximal") Point Method: a New Algorithm, Convergence Theory, and Applications](https://mlsanity.com/paper/2502.02002/) (2025)
- [Lower Bounds for Frank-Wolfe on Strongly Convex Sets](https://mlsanity.com/paper/2602.04378/) (2026)
- [Lower Bounds for Linear Minimization Oracle Methods Optimizing over Strongly Convex Sets](https://mlsanity.com/paper/2602.22608/) (2026)
- [Solving the Optimal Experiment Design Problem with Mixed-Integer Convex Methods](https://mlsanity.com/paper/2312.11200/) (2023)
- [Don't Be Greedy, Just Relax! Pruning LLMs via Frank-Wolfe](https://mlsanity.com/paper/2510.13713/) (2025)
- [Frank-Wolfe Algorithms for (L0, L1)-smooth functions](https://mlsanity.com/paper/2510.16468/) (2025)
- [Convergence Analysis of the Frank-Wolfe Algorithm and Its Generalization in Banach Spaces](https://mlsanity.com/paper/1710.07367/) (2017)

## Abstract (arXiv)

We propose Broximal Gradient Descent (BroxGD), a projection-free sister method to projected gradient descent for constrained optimization. Its forward--backward construction replaces the proximal backward operation by the broximal operation of Gruntkowska et al. (2025). Instead of projecting, each step minimizes a linear function over the intersection of the constraint set X and a ball mathbbB(xk,tk) centered at the current iterate xk, of suitable radius tk>0: [ xk+1∈argmin_z∈XcapmathbbB(xk,tk)langlenabla f(xk),zrangle. ] We develop a comprehensive convergence theory spanning a wide range of optimization regimes and radius rules. We expect BroxGD to find many applications and inspire numerous extensions, much like projected gradient descent. Our contribution is theoretical; potential applications and toy experiments illustrate the method and suggest directions for future work.