  Many Markov Chain Monte Carlo (MCMC) methods leverage gradient information of
the potential function of target distribution to explore sample space
efficiently. However, computing gradients can often be computationally
expensive for large scale applications, such as those in contemporary machine
learning. Stochastic Gradient (SG-)MCMC methods approximate gradients by
stochastic ones, commonly via uniformly subsampled data points, and achieve
improved computational efficiency, however at the price of introducing sampling
error. We propose a non-uniform subsampling scheme to improve the sampling
accuracy. The proposed exponentially weighted stochastic gradient (EWSG) is
designed so that a non-uniform-SG-MCMC method mimics the statistical behavior
of a batch-gradient-MCMC method, and hence the inaccuracy due to SG
approximation is reduced. EWSG differs from classical variance reduction (VR)
techniques as it focuses on the entire distribution instead of just the
variance; nevertheless, its reduced local variance is also proved. EWSG can
also be viewed as an extension of the importance sampling idea, successful for
stochastic-gradient-based optimizations, to sampling tasks. In our practical
implementation of EWSG, the non-uniform subsampling is performed efficiently
via a Metropolis-Hastings chain on the data index, which is coupled to the MCMC
algorithm. Numerical experiments are provided, not only to demonstrate EWSG's
effectiveness, but also to guide hyperparameter choices, and validate our
\emph{non-asymptotic global error bound} despite of approximations in the
implementation. Notably, while statistical accuracy is improved, convergence
speed can be comparable to the uniform version, which renders EWSG a practical
alternative to VR (but EWSG and VR can be combined too).
