  For many applications in signal processing and machine learning, we are
tasked with minimizing a large sum of convex functions subject to a large
number of convex constraints. In this paper, we devise a new random projection
method (RPM) to efficiently solve this problem. Compared with existing RPMs,
our proposed algorithm features two useful algorithmic ideas. First, at each
iteration, instead of projecting onto the subset defined by one of the
constraints, our algorithm only requires projecting onto a half-space
approximation of the subset, which significantly reduces the computational cost
as it admits a closed-form formula. Second, to exploit the structure that the
objective is a sum, variance reduction is incorporated into our algorithm to
further improve the performance. As theoretical contributions, under a novel
error bound condition and other standard assumptions, we prove that the
proposed RPM converges to an optimal solution and that both optimality and
feasibility gaps vanish at a sublinear rate. In particular, via a new analysis
framework, we show that our RPM attains a faster convergence rate in optimality
gap than existing RPMs when the objective function has a Lipschitz continuous
gradient, capitalizing the benefit of the variance reduction. We also provide
sufficient conditions for the error bound condition to hold. Experiments on a
beamforming problem and a robust classification problem are also presented to
demonstrate the superiority of our RPM over existing ones.
