Evaluating expectations on an Ising model (or Boltzmann machine) is essential for various applications, including statistical machine learning. However, in general, the evaluation is computationally difficult because it involves intractable multiple summations or integrations; therefore, it requires approximation. Monte Carlo integration (MCI) is a well-known approximation method; a more effective MCI-like approximation method was proposed recently, called spatial Monte Carlo integration (SMCI). However, the estimations obtained using SMCI (and MCI) exhibit a low accuracy in Ising models under a low temperature owing to degradation of the sampling quality. Annealed importance sampling (AIS) is a type of importance sampling based on Markov chain Monte Carlo methods that can suppress performance degradation in low-temperature regions with the force of importance weights. In this study, a new method is proposed to evaluate the expectations on Ising models combining AIS and SMCI. The proposed method performs efficiently in both high- and low-temperature regions, which is demonstrated theoretically and numerically.
The University of Chicago（芝加哥大学）；The Grossman Institute for Neuroscience, Quantitative Biology and Human Behavior（格罗斯曼神经科学、定量生物学与人类行为研究所）
Variational inference (VI) and Markov chain Monte Carlo (MCMC) are two main approximate approaches for learning deep generative models by maximizing marginal likelihood. In this paper, we propose using annealed importance sampling for learning deep generative models. Our proposed approach bridges VI with MCMC. It generalizes VI methods such as variational auto-encoders and importance weighted auto-encoders (IWAE) and the MCMC method proposed in (Hoffman, 2017). It also provides insights into why running multiple short MCMC chains can help learning deep generative models. Through experiments, w…
The evaluation of the free energy of a stochastic model is considered a significant issue in various fields of physics and machine learning. However, the exact free energy evaluation is computationally infeasible because the free energy expression includes an intractable partition function. Annealed importance sampling (AIS) is a type of importance sampling based on the Markov chain Monte Carlo method that is similar to a simulated annealing and can effectively approximate the free energy. This study proposes an AIS-based approach, which is referred to as marginalized AIS (mAIS). The statistic…
Stellenbosch University（斯泰伦博斯大学）；National Institute for Theoretical Physics（国家理论物理研究所）
We consider estimating the marginal likelihood in settings with independent and identically distributed (i.i.d.) data. We propose estimating the predictive distributions in a sequential factorization of the marginal likelihood in such settings by using stochastic gradient Markov Chain Monte Carlo techniques. This approach is far more efficient than traditional marginal likelihood estimation techniques such as nested sampling and annealed importance sampling due to its use of mini-batches to approximate the likelihood. Stability of the estimates is provided by an adaptive annealing schedule. Th…
ETH Zurich（苏黎世联邦理工学院）；Swiss Data Science Center（瑞士数据科学中心）
Annealed Importance Sampling (AIS) is a popular algorithm used to estimates the intractable marginal likelihood of deep generative models. Although AIS is guaranteed to provide unbiased estimate for any set of hyperparameters, the common implementations rely on simple heuristics such as the geometric average bridging distributions between initial and the target distribution which affect the estimation performance when the computation budget is limited. In order to reduce the number of sampling iterations, we present a parameteric AIS process with flexible intermediary distributions defined by …
More than twenty years after its introduction, Annealed Importance Sampling (AIS) remains one of the most effective methods for marginal likelihood estimation. It relies on a sequence of distributions interpolating between a tractable initial distribution and the target distribution of interest which we simulate from approximately using a non-homogeneous Markov chain. To obtain an importance sampling estimate of the marginal likelihood, AIS introduces an extended target distribution to reweight the Markov chain proposal. While much effort has been devoted to improving the proposal distribution…
This note introduces a method for sampling Ising models with mixed boundary conditions. As an application of annealed importance sampling and the Swendsen-Wang algorithm, the method adopts a sequence of intermediate distributions that keeps the temperature fixed but turns on the boundary condition gradually. The numerical results show that the variance of the sample weights is relatively small.
Universitat Politècnica de Catalunya - BarcelonaTech（加泰罗尼亚理工大学）
Probabilistic models based on Restricted Boltzmann Machines (RBMs) imply the evaluation of normalized Boltzmann factors, which in turn require from the evaluation of the partition function Z. The exact evaluation of Z, though, becomes a forbiddingly expensive task as the system size increases. This even worsens when one considers most usual learning algorithms for RBMs, where the exact evaluation of the gradient of the log-likelihood of the empirical distribution of the data includes the computation of Z at each iteration. The Annealed Importance Sampling (AIS) method provides a tool to stocha…
University of Cambridge（剑桥大学）；InstaDeep（InstaDeep公司）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Normalizing flows are tractable density models that can approximate complicated target distributions, e.g. Boltzmann distributions of physical systems. However, current methods for training flows either suffer from mode-seeking behavior, use samples from the target generated beforehand by expensive MCMC methods, or use stochastic losses that have high variance. To avoid these problems, we augment flows with annealed importance sampling (AIS) and minimize the mass-covering $α$-divergence with $α=2$, which minimizes importance weight variance. Our method, Flow AIS Bootstrap (FAB), uses AIS to ge…
Variational inference is a powerful paradigm for approximate Bayesian inference with a number of appealing properties, including support for model learning and data subsampling. By contrast MCMC methods like Hamiltonian Monte Carlo do not share these properties but remain attractive since, contrary to parametric methods, MCMC is asymptotically unbiased. For these reasons researchers have sought to combine the strengths of both classes of algorithms, with recent approaches coming closer to realizing this vision in practice. However, supporting data subsampling in these hybrid methods can be a c…
University of Toronto（多伦多大学）；Vector Institute（矢量研究所）；Stanford University（斯坦福大学）
Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation, but are not fully differentiable due to the use of Metropolis-Hastings correction steps. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective using gradient-based methods. To this end, we propose Differentiable AIS (DAIS), a variant of AIS which ensures differentiability by abandoning the Metropolis-Hastings corrections. As a further advantage, DAIS allows for mini-batch gradients. We provide a detailed…
University of Montreal（蒙特利尔大学）；MILA（蒙特利尔学习算法研究所）；Element AI
Despite the advances in the representational capacity of approximate distributions for variational inference, the optimization process can still limit the density that is ultimately learned. We demonstrate the drawbacks of biasing the true posterior to be unimodal, and introduce Annealed Variational Objectives (AVO) into the training of hierarchical variational methods. Inspired by Annealed Importance Sampling, the proposed method facilitates learning by incorporating energy tempering into the optimization objective. In our experiments, we demonstrate our method's robustness to deterministic w…
Annealed Importance Sampling (AIS) and its Sequential Monte Carlo (SMC) extensions are state-of-the-art methods for estimating normalizing constants of probability distributions. We propose here a novel Monte Carlo algorithm, Annealed Flow Transport (AFT), that builds upon AIS and SMC and combines them with normalizing flows (NFs) for improved performance. This method transports a set of particles using not only importance sampling (IS), Markov chain Monte Carlo (MCMC) and resampling steps - as in SMC, but also relies on NFs which are learned sequentially to push particles towards the successi…
Many high dimensional optimization problems can be reformulated into a problem of finding theoptimal state path under an equivalent state space model setting. In this article, we present a general emulation strategy for developing a state space model whose likelihood function (or posterior distribution) shares the same general landscape as the objective function of the original optimization problem. Then the solution of the optimization problem is the same as the optimal state path that maximizes the likelihood function or the posterior distribution under the emulated system. To find such an o…
Improving Mutual Information Estimation with Annealed and Energy-Based Bounds
Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）；Vector Institute（向量研究所）；University of Toronto（多伦多大学）；MIT EECS / IMES / CSAIL（麻省理工学院电气工程与计算机科学系 / 医学工程与科学研究所 / 计算机科学与人工智能实验室）
University of Torontoland（多伦多兰大学）；University of Edenborrow（伊甸伯勒大学）；HSE University（高等经济学院）；Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；University of Oxford（牛津大学）；Centre de recherche Lagrange en mathematiques et calcul（拉格朗日数学与计算研究中心）
Variational auto-encoders (VAE) are popular deep latent variable models which are trained by maximizing an Evidence Lower Bound (ELBO). To obtain tighter ELBO and hence better variational approximations, it has been proposed to use importance sampling to get a lower variance estimate of the evidence. However, importance sampling is known to perform poorly in high dimensions. While it has been suggested many times in the literature to use more sophisticated algorithms such as Annealed Importance Sampling (AIS) and its Sequential Importance Sampling (SIS) extensions, the potential benefits broug…
Normalizing flows are flexible, parameterized distributions that can be used to approximate expectations from intractable distributions via importance sampling. However, current flow-based approaches are limited on challenging targets where they either suffer from mode seeking behaviour or high variance in the training loss, or rely on samples from the target distribution, which may not be available. To address these challenges, we combine flows with annealed importance sampling (AIS), while using the $α$-divergence as our objective, in a novel training procedure, FAB (Flow AIS Bootstrap). The…
Annealed importance sampling (AIS) is a common algorithm to estimate partition functions of useful stochastic models. One important problem for obtaining accurate AIS estimates is the selection of an annealing schedule. Conventionally, an annealing schedule is often determined heuristically or is simply set as a linearly increasing sequence. In this paper, we propose an algorithm for the optimal schedule by deriving a functional that dominates the AIS estimation error and by numerically minimizing this functional. We experimentally demonstrate that the proposed algorithm mostly outperforms con…
We propose to use deep neural networks for generating samples in Monte Carlo integration. Our work is based on non-linear independent components estimation (NICE), which we extend in numerous ways to improve performance and enable its application to integration problems. First, we introduce piecewise-polynomial coupling transforms that greatly increase the modeling power of individual coupling layers. Second, we propose to preprocess the inputs of neural networks using one-blob encoding, which stimulates localization of computation and improves inference. Third, we derive a gradient-descent-ba…
Many applications in computational sciences and statistical inference require the computation of expectations with respect to complex high-dimensional distributions with unknown normalization constants, as well as the estimation of these constants. Here we develop a method to perform these calculations based on generating samples from a simple base distribution, transporting them by the flow generated by a velocity field, and performing averages along these flowlines. This non-equilibrium importance sampling (NEIS) strategy is straightforward to implement and can be used for calculations with …
LEAPS: A discrete neural sampler via locally equivariant networks
Massachusetts Institute of Technology（麻省理工学院）；Society of Fellow, Harvard University（哈佛大学 fellows 会社）；Institute for Artificial Intelligence and Fundamental Interactions（人工智能与基本相互作用研究所）
We present a computational strategy for the evaluation of multidimensional integrals on hyper-rectangles based on Markovian stochastic exploration of the integration domain while the integrand is being morphed by starting from an initial appropriate profile. Thanks to an abstract reformulation of Jarzynski's equality applied in stochastic thermodynamics to evaluate the free-energy profiles along selected reaction coordinates via non-equilibrium transformations, it is possible to cast the original integral into the exponential average of the distribution of the pseudo-work (that we may term "co…
We propose a novel sampling framework for inference in probabilistic models: an active learning approach that converges more quickly (in wall-clock time) than Markov chain Monte Carlo (MCMC) benchmarks. The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unknown (hyper-)parameters (for example to compute the marginal likelihood or a partition function). MCMC has provided approaches to numerical integration that deliver state-of-the-art inference, but can suffer from sample inefficiency and poor convergence diagnostics. Bayesian quad…
Importance sampling (IS) and numerical integration methods are usually employed for approximating moments of complicated target distributions. In its basic procedure, the IS methodology randomly draws samples from a proposal distribution and weights them accordingly, accounting for the mismatch between the target and proposal. In this work, we present a general framework of numerical integration techniques inspired by the IS methodology. The framework can also be seen as an incorporation of deterministic rules into IS methods, reducing the error of the estimators by several orders of magnitude…
Ratios of normalizing constants for two distributions are needed in both Bayesian statistics, where they are used to compare models, and in statistical physics, where they correspond to differences in free energy. Two approaches have long been used to estimate ratios of normalizing constants. The `simple importance sampling' (SIS) or `free energy perturbation' method uses a sample drawn from just one of the two distributions. The `bridge sampling' or `acceptance ratio' estimate can be viewed as the ratio of two SIS estimates involving a bridge distribution. For both methods, difficult problems…
The variance reduction established by importance sampling strongly depends on the choice of the importance sampling distribution. A good choice is often hard to achieve especially for high-dimensional integration problems. Nonparametric estimation of the optimal importance sampling distribution (known as nonparametric importance sampling) is a reasonable alternative to parametric approaches.In this article nonparametric variants of both the self-normalized and the unnormalized importance sampling estimator are proposed and investigated. A common critique on nonparametric importance sampling is…
University of Toronto（多伦多大学）；University of Cambridge（剑桥大学）；Twitter（推特）；Harvard University（哈佛大学）
Computing the marginal likelihood (ML) of a model requires marginalizing out all of the parameters and latent variables, a difficult high-dimensional summation or integration problem. To make matters worse, it is often hard to measure the accuracy of one's ML estimates. We present bidirectional Monte Carlo, a technique for obtaining accurate log-ML estimates on data simulated from a model. This method obtains stochastic lower bounds on the log-ML using annealed importance sampling or sequential Monte Carlo, and obtains stochastic upper bounds by running these same algorithms in reverse startin…
Peking University（北京大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Lab（劳伦斯伯克利国家实验室）
Normalizing constant (also called partition function, Bayesian evidence, or marginal likelihood) is one of the central goals of Bayesian inference, yet most of the existing methods are both expensive and inaccurate. Here we develop a new approach, starting from posterior samples obtained with a standard Markov Chain Monte Carlo (MCMC). We apply a novel Normalizing Flow (NF) approach to obtain an analytic density estimator from these samples, followed by Optimal Bridge Sampling (OBS) to obtain the normalizing constant. We compare our method which we call Gaussianized Bridge Sampling (GBS) to ex…
Markov chain Monte Carlo (MCMC) is one of the main workhorses of probabilistic inference, but it is notoriously hard to measure the quality of approximate posterior samples. This challenge is particularly salient in black box inference methods, which can hide details and obscure inference failures. In this work, we extend the recently introduced bidirectional Monte Carlo technique to evaluate MCMC-based posterior inference algorithms. By running annealed importance sampling (AIS) chains both from prior to posterior and vice versa on simulated data, we upper bound in expectation the symmetrized…
We propose a global optimization algorithm based on the Sequential Monte Carlo (SMC) sampling framework. In this framework, the objective function is normalized to be a probabilistic density function (pdf), based on which a sequence of annealed target pdfs is designed to asymptotically converge on the set of global optima. A sequential importance sampling (SIS) procedure is performed to simulate the resulting targets, and the maxima of the objective function is assessed from the yielded samples. The disturbing issue lies in the design of the importance sampling (IS) pdf, which crucially influe…
Kernel methods have revolutionized the fields of pattern recognition and machine learning. Their success, however, critically depends on the choice of kernel parameters. Using Gaussian process (GP) classification as a working example, this paper focuses on Bayesian inference of covariance (kernel) parameters using Markov chain Monte Carlo (MCMC) methods. The motivation is that, compared to standard optimization of kernel parameters, they have been systematically demonstrated to be superior in quantifying uncertainty in predictions. Recently, the Pseudo-Marginal MCMC approach has been proposed …
Partition functions of probability distributions are important quantities for model evaluation and comparisons. We present a new method to compute partition functions of complex and multimodal distributions. Such distributions are often sampled using simulated tempering, which augments the target space with an auxiliary inverse temperature variable. Our method exploits the multinomial probability law of the inverse temperatures, and provides estimates of the partition function in terms of a simple quotient of Rao-Blackwellized marginal inverse temperature probability estimates, which are updat…
Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC) method for performing approximate inference in complex probabilistic models of continuous variables. In common with many MCMC methods, however, the standard HMC approach performs poorly in distributions with multiple isolated modes. We present a method for augmenting the Hamiltonian system with an extra continuous temperature control variable which allows the dynamic to bridge between sampling a complex target distribution and a simpler unimodal base distribution. This augmentation both helps improve mixing in multimod…
In this paper, we propose an efficient simulation method based on adaptive importance sampling, which can automatically find the optimal proposal within the Gaussian family based on previous samples, to evaluate the probability of bit error rate (BER) or word error rate (WER). These two measures, which involve high-dimensional black-box integration and rare-event sampling, can characterize the performance of coded modulation. We further integrate the quasi-Monte Carlo method within our framework to improve the convergence speed. The proposed importance sampling algorithm is demonstrated to hav…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the curse of dimensionality. However, due to the curse of dimensionality, deterministic numerical methods are inefficient in high-dimensional settings. Consequentially, for many practical problems, one must resort to Monte Carlo estimation. In this paper, we introduce a novel Sequential Monte Carlo technique called Stratified Splitting. The method provides unbi…
Recently developed particle flow algorithms provide an alternative to importance sampling for drawing particles from a posterior distribution, and a number of particle filters based on this principle have been proposed. Samples are drawn from the prior and then moved according to some dynamics over an interval of pseudo-time such that their final values are distributed according to the desired posterior. In practice, implementing a particle flow sampler requires multiple layers of approximation, with the result that the final samples do not in general have the correct posterior distribution. I…
The efficient importance sampling (EIS) method is a general principle for the numerical evaluation of high-dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers. Despite a number of successful applications in high dimensions, it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases. We solve this problem by recognising that the EIS framework has an offline sequential Monte Carlo interpretation. The particle EIS method is based on no…
Interacting particle methods are increasingly used to sample from complex and high-dimensional distributions. These stochastic particle integration techniques can be interpreted as an universal acceptance-rejection sequential particle sampler equipped with adaptive and interacting recycling mechanisms. Practically, the particles evolve randomly around the space independently and to each particle is associated a positive potential function. Periodically, particles with high potentials duplicate at the expense of low potential particle which die. This natural genetic type selection scheme appear…
