Monte Carlo Methods

Jun 13, 2019



Sliced-Wasserstein Flows: Nonparametric Generative Modeling via Optimal Transport and Diffusions By building upon the recent theory that established the connection between implicit generative modeling (IGM) and optimal transport, in this study, we propose a novel parameter-free algorithm for learning the underlying distributions of complicated datasets and sampling from them. The proposed algorithm is based on a functional optimization problem, which aims at finding a measure that is close to the data distribution as much as possible and also expressive enough for generative modeling purposes. We formulate the problem as a gradient flow in the space of probability measures. The connections between gradient flows and stochastic differential equations let us develop a computationally efficient algorithm for solving the optimization problem. We provide formal theoretical analysis where we prove finite-time error guarantees for the proposed algorithm. To the best of our knowledge, the proposed algorithm is the first nonparametric IGM algorithm with explicit theoretical guarantees. Our experimental results support our theory and show that our algorithm is able to successfully capture the structure of different types of data distributions. Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimization Recent studies on diffusion-based sampling methods have shown that Langevin Monte Carlo (LMC) algorithms can be beneficial for non-convex optimization, and rigorous theoretical guarantees have been proven for both asymptotic and finite-time regimes. Algorithmically, LMC-based algorithms resemble the well-known gradient descent (GD) algorithm, where the GD recursion is perturbed by an additive Gaussian noise whose variance has a particular form. Fractional Langevin Monte Carlo (FLMC) is a recently proposed extension of LMC, where the Gaussian noise is replaced by a heavy-tailed α-stable noise. As opposed to its Gaussian counterpart, these heavy-tailed perturbations can incur large jumps and it has been empirically demonstrated that the choice of α-stable noise can provide several advantages in modern machine learning problems, both in optimization and sampling contexts. However, as opposed to LMC, only asymptotic convergence properties of FLMC have been yet established. In this study, we analyze the non-asymptotic behavior of FLMC for non-convex optimization and prove finite-time bounds for its expected suboptimality. Our results show that the weak-error of FLMC increases faster than LMC, which suggests using smaller step-sizes in FLMC. We finally extend our results to the case where the exact gradients are replaced by stochastic gradients and show that similar results hold in this setting as well. Unifying Orthogonal Monte Carlo Methods Many machine learning methods making use of Monte Carlo sampling in vector spaces have been shown to be improved by conditioning samples to be mutually orthogonal. Exact orthogonal coupling of samples is computationally intensive, hence approximate methods have been of great interest. In this paper, we present a unifying perspective of many approximate methods by considering Givens transformations, propose new approximate methods based on this framework, and demonstrate the first statistical guarantees for families of approximate methods in kernel approximation. We provide extensive empirical evaluations with guidance for practitioners. Adaptive Monte Carlo Multiple Testing via Multi-Armed Bandits Monte Carlo (MC) permutation testing is considered the gold standard for statistical hypothesis testing, especially when standard parametric assumptions are not clear or likely to fail. However, in modern data science settings where a large number of hypothesis tests need to be performed simultaneously, it is rarely used due to its prohibitive computational cost. In genome-wide association studies, for example, the number of hypothesis tests m is around 10^6 while the number of MC samples n for each test could be greater than 10^8, totaling more than nm=10^14 samples. In this paper, we propose Adaptive MC Testing (AMT) to estimate MC p-values and control false discovery rate in multiple testing. The algorithm outputs the same result as the standard full MC approach with high probability while requiring only \tilde{O}(\sqrt{n}m) samples. This sample complexity is shown to be optimal. On a Parkinson GWAS dataset, the algorithm reduces the running time from 2 months for full MC to an hour. The AMT algorithm is derived based on the theory of multi-armed bandits. Metropolis-Hastings Generative Adversarial Networks We introduce the Metropolis-Hastings generative adversarial network (MH-GAN), which combines aspects of Markov chain Monte Carlo and GANs. The MH-GAN draws samples from the distribution implicitly defined by a GAN's discriminator-generator pair, as opposed to standard GANs which draw samples from the distribution defined by only the generator. It uses the discriminator from GAN training to build a wrapper around the generator for improved sampling. With a perfect discriminator, this wrapped generator samples from the true distribution on the data exactly even when the generator is imperfect. We demonstrate the benefits of the improved generator on multiple benchmark datasets, including CIFAR-10 and CelebA, using the DCGAN, WGAN, and progressive GAN. Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets Bayesian inference via standard Markov Chain Monte Carlo (MCMC) methods such as Metropolis--Hastings is too computationally intensive to handle large datasets, since the cost per step usually scales like O(n) in the number of data points n. We propose the \emph{Scalable Metropolis--Hastings} (SMH) kernel that only requires processing on average O(1) or even O(1/√n) data points per step. This scheme is based on a combination of factorized acceptance probabilities, procedures for fast simulation of Bernoulli processes, and control variate ideas. Contrary to many MCMC subsampling schemes such as fixed step-size Stochastic Gradient Langevin Dynamics, our approach is exact insofar as the invariant distribution is the true posterior and not an approximation to it. We characterise the performance of our algorithm theoretically, and give realistic and verifiable conditions under which it is geometrically ergodic. This theory is borne out by empirical results that demonstrate overall performance benefits over standard Metropolis-Hastings and various subsampling algorithms. Replica Conditional Sequential Monte Carlo We propose a Markov chain Monte Carlo (MCMC) scheme to perform state inference in non-linear non-Gaussian state-space models. Current state-of-the-art methods to address this problem rely on particle MCMC techniques and its variants, such as the iterated conditional Sequential Monte Carlo (cSMC) scheme, which uses a Sequential Monte Carlo (SMC) type proposal within MCMC. A deficiency of standard SMC proposals is that they only use observations up to time t to propose states at time t when an entire observation sequence is available. More sophisticated SMC based on lookahead techniques could be used but they can be difficult to put in practice. We propose here replica cSMC where we build SMC proposals for one replica using information from the entire observation sequence by conditioning on the states of the other replicas. This approach is easily parallelizable and we demonstrate its excellent empirical performance when compared to the standard iterated cSMC scheme at fixed computational complexity. A Polynomial Time MCMC Method for Sampling from Continuous Determinantal Point Processes We study the Gibbs sampling algorithm for discrete and continuous k-determinantal point processes. We show that in both cases, the spectral gap of the chain is bounded by a polynomial of k and it is independent of the size of the domain. As an immediate corollary, we obtain sublinear time algorithms for sampling from discrete k-DPPs given access to polynomially many processors. In the continuous setting, our result leads to the first class of rigorously analyzed efficient algorithms to generate random samples of continuous k-DPPs. We achieve this by showing that the Gibbs sampler for a large family of continuous k -DPPs can be simulated efficiently when the spectrum is not concentrated on the top keigenvalues. Adaptive Antithetic Sampling for Variance Reduction Variance reduction techniques are crucial in stochastic estimation and optimization problems. Antithetic sampling techniques reduce the variance of a Monte Carlo estimator by drawing correlated, rather than independent, samples. Designing the right correlation structure, however, is challenging and application specific, thus limiting the practical applicability of these methods. In this paper, we propose a general-purpose adaptive antithetic sampling framework. We leverage advances in generative models and stochastic computation graphs to define a flexible family of antithetic samplers. We provide gradient-based and gradient-free methods to train the samplers such that they reduce variance while ensuring that the underlying Monte Carlo estimator is provably unbiased. We demonstrate the effectiveness of our approach on Bayesian inference and generative model training tasks, where it reduces variance and improves task performance with little or no computational overhead. Accelerated Flow for Probability Distributions This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in (Wibisono2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton's equations for the optimal gradient flow. The Hamilton's equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton's equations as a system of N interacting particles. The continuous limit of the Nesterov's algorithm is shown to be a special case with N=1. The algorithm is illustrated with numerical examples and the performance is compared with the MCMC and Hamiltonian MCMC algorithms.



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