
Langevin Monte Carlo: random coordinate descent and variance reduction
Sampling from a logconcave distribution function on ℝ^d (with d≫ 1) is ...
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Variance Reduction with Sparse Gradients
Variance reduction methods such as SVRG and SpiderBoost use a mixture of...
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Random Coordinate Underdamped Langevin Monte Carlo
The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Mo...
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Random Coordinate Langevin Monte Carlo
Langevin Monte Carlo (LMC) is a popular Markov chain Monte Carlo samplin...
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Computational cost for determining an approximate global minimum using the selection and crossover algorithm
This work examines the expected computational cost to determine an appro...
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Reducing Reparameterization Gradient Variance
Optimization with noisy gradients has become ubiquitous in statistics an...
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Langevin Monte Carlo without Smoothness
Langevin Monte Carlo (LMC) is an iterative algorithm used to generate sa...
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Variance reduction for Langevin Monte Carlo in high dimensional sampling problems
Sampling from a logconcave 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 finitedifferencing approximations as surrogates, and the method is expensive in highdimensions. 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 counterexample 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 RCADOLMC and RCADULMC 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.
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