This paper presents a novel algorithm that leverages Stochastic Gradient Descent strategies in conjunction with Random Features to augment the scalability of Conic Particle Gradient Descent (CPGD) specifically tailored for solving sparse optimisation problems on measures. By formulating the CPGD steps within a variational framework, we provide rigorous mathematical proofs demonstrating the following key findings: (i) The total variation norms of the solution measures along the descent trajectory remain bounded, ensuring stability and preventing undesirable divergence; (ii) We establish a global convergence guarantee with a convergence rate of $\mathcal{O}(\log(K)/\sqrt{K})$ over $K$ iterations, showcasing the efficiency and effectiveness of our algorithm; (iii) Additionally, we analyze and establish local control over the first-order condition discrepancy, contributing to a deeper understanding of the algorithm's behavior and reliability in practical applications.

翻译：本文提出了一种新型算法，该算法结合随机梯度下降策略与随机特征，旨在增强专用于求解测度稀疏优化问题的锥形粒子梯度下降（CPGD）的可扩展性。通过在变分框架内对CPGD步骤进行公式化，我们提供了严格的数学证明，证实了以下关键结论：（i）沿下降轨迹的解测度的总变差范数保持有界，确保了稳定性并防止了不良发散；（ii）我们建立了全局收敛性保证，在$K$次迭代中收敛速度为$\mathcal{O}(\log(K)/\sqrt{K})$，展示了我们算法的效率与有效性；（iii）此外，我们分析并建立了一阶条件偏差的局部控制，从而更深入地理解算法在实际应用中的行为与可靠性。