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# Sampling from Non-Log-Concave Distributions via Variance-Reduced Gradient Langevin Dynamics

; Proceedings of Machine Learning Research, PMLR 89:2936-2945, 2019.

#### Abstract

We study stochastic variance reduction-based Langevin dynamic algorithms, SVRG-LD and SAGA-LD \citep{dubey2016variance}, for sampling from non-log-concave distributions. Under certain assumptions on the log density function, we establish the convergence guarantees of SVRG-LD and SAGA-LD in $2$-Wasserstein distance. More specifically, we show that both SVRG-LD and SAGA-LD require $ \tilde O\big(n+n^{3/4}/\epsilon^2 + n^{1/2}/\epsilon^4\big)\cdot \exp\big(\tilde O(d+\gamma)\big)$ stochastic gradient evaluations to achieve $\epsilon$-accuracy in $2$-Wasserstein distance, which outperforms the $ \tilde O\big(n/\epsilon^4\big)\cdot \exp\big(\tilde O(d+\gamma)\big)$ gradient complexity achieved by Langevin Monte Carlo Method \citep{raginsky2017non}. Experiments on synthetic data and real data back up our theory.