  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.
