Langevin Monte Carlo: random coordinate descent and variance reduction

Sampling from a log-concave distribution function is a popular problem that has wide applications in Bayesian statistics and machine learning. Among many popular sampling methods, the Langevin Monte Carlo (LMC) method stands out for its fast convergence rate. One key drawback, however, is that it requires the computation of the full-gradient in each iteration. For a problem on $\mathbb{R}^d$, this means $d$ finite differencing type approximations per iteration, and if $d\gg 1$, the computational cost is high. Such challenge is also seen in optimization when gradient descent based methods are used. Some random directional gradient approximations were developed in recent years to partially overcome this difficulty. One example is the random coordinate descent (RCD) method, where the full gradient is replaced by some random directional derivatives in each iteration. We investigate the application of RCD in sampling. There are two sides of the theory: