# Fast Spawn&Prune (FS&P): Global convergence of stochastic conic particle gradient descent via birth/death process

arXiv: https://arxiv.org/abs/2605.19784 · HTML: https://mlsanity.com/paper/2605.19784/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 17 references that are in the map.
- First posted on arXiv 19 May 2026.

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

- [Riemannian Gradient Descent for Gaussian Mixture Models with unknown diagonal covariances](https://mlsanity.com/paper/2609.30220/) (2026)

## Abstract (arXiv)

We investigate the global optimization of the objective function arising in continuous sparse regression, specifically the Beurling LASSO (BLASSO), over the space of measures. While Conic Particle Gradient Descent (CPGD) methods are computationally efficient, they may become trapped in local minima due to the non-convexity of the parameterization. To overcome this limitation, we introduce Fast Spawn&Prune (FS&P), a stochastic algorithm that extends FastPart introduced in De Castro et al. (2025) and combines CPGD with a birth-death process. The birth mechanism ensures asymptotic global exploration by introducing particles in regions where first-order optimality conditions are violated, while the death process preserves computational efficiency by pruning non-informative particles. We provide the first theoretical guarantee of global convergence for this class of discrete-time stochastic algorithms, without requiring exponentially large initializations. Furthermore, we derive explicit convergence rates for the excess risk, which scale as Obig(left(log K / Kright)^frac12(2+d)big), where K denotes the number of iterations and d the dimension of the domain, thereby quantifying the trade-off between global exploration and local refinement. Moreover, the sample complexity is Obig(N^-frac14(2+d)big) (up to logarithmic factors). We also propose a horizon-free variant that does not require prior knowledge of the iteration budget.