2020/06/10 by Zhiyan Ding, Qin Li, Ding, Zhiyan +1
Computer Science · Mathematics · #Algorithm #Applied mathematics #Artificial intelligence #Artificial neural network #Computation #Computer science #Control variates #Convergence (economics) #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Hybrid Monte Carlo #Importance sampling #Langevin dynamics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Markov chain Monte Carlo #Mathematical optimization #Mathematics #Monte Carlo method #Randomness #Rate of convergence #Statistical Methods and Inference #Statistics #Statistics Theory (math.ST) #Stochastic gradient descent #Variance reduction #cs.LG #math.ST #stat.ML #stat.TH
paper · pdf · doi:10.48550/arxiv.2006.06068
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2020/06/10 · arxiv created 2020/10/22 · arxiv updated 2020/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions. A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbation stochastic approximation (SPSA). We show by a counter-example that blindly applying RCD does not achieve the goal in the most general setting. The high variance induced by the randomness means a larger number of iterations are needed, and this balances out the saving in each iteration. We then introduce a new variance reduction approach, termed Randomized Coordinates Averaging Descent (RCAD), and incorporate it with both overdamped and underdamped LMC. The methods are termed RCAD-O-LMC and RCAD-U-LMC respectively. The methods still sit in the random gradient approximation framework, and thus the computational cost in each iteration is low. However, by employing RCAD, the variance is reduced, so the methods converge within the same number of iterations as the classical overdamped and underdamped LMC. This leads to a computational saving overall.