We introduce an extension to annealed importance sampling that uses Hamiltonian dynamics to rapidly estimate normalization constants. We demonstrate this method by computing log likelihoods in directed and undirected probabilistic image models. We compare the performance of linear generative models with both Gaussian and Laplace priors, product of experts models with Laplace and Student's t experts, the mc-RBM, and a bilinear generative model. We provide code to compare additional models.
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…
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…
Max Planck Institute for Intelligent Systems Tübingen（马克斯·普朗克智能系统研究所图宾根）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Technical University of Munich（慕尼黑技术大学）；Isomorphic Labs（同构实验室）；University of Cambridge（剑桥大学）；University of Bochum（博德姆大学）
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…
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…
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…
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…
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…
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）
Fields Institute（菲尔兹研究所）；University of Toronto（多伦多大学）
Markov random fields (MRFs) are difficult to evaluate as generative models because computing the test log-probabilities requires the intractable partition function. Annealed importance sampling (AIS) is widely used to estimate MRF partition functions, and often yields quite accurate results. However, AIS is prone to overestimate the log-likelihood with little indication that anything is wrong. We present the Reverse AIS Estimator (RAISE), a stochastic lower bound on the log-likelihood of an approximation to the original MRF model. RAISE requires only the same MCMC transition operators as stand…
The ratio between the probability that two distributions $R$ and $P$ give to points $x$ are known as importance weights or propensity scores and play a fundamental role in many different fields, most notably, statistics and machine learning. Among its applications, importance weights are central to domain adaptation, anomaly detection, and estimations of various divergences such as the KL divergence. We consider the common setting where $R$ and $P$ are only given through samples from each distribution. The vast literature on estimating importance weights is either heuristic, or makes strong as…
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…
University of Oxford（牛津大学）；Northeastern University（东北大学）
We study adaptive importance sampling (AIS) as an online learning problem and argue for the importance of the trade-off between exploration and exploitation in this adaptation. Borrowing ideas from the bandits literature, we propose Daisee, a partition-based AIS algorithm. We further introduce a notion of regret for AIS and show that Daisee has $\mathcal{O}(\sqrt{T}(\log T)^{\frac{3}{4}})$ cumulative pseudo-regret, where $T$ is the number of iterations. We then extend Daisee to adaptively learn a hierarchical partitioning of the sample space for more efficient sampling and confirm the performa…
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…
DISA: Offline Importance Sampling for Distribution-Matching LLM-RL
Shanghai Jiao Tong University（上海交通大学）；Qwen Team, Alibaba Group（阿里集团Qwen团队）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；The University of Science and Technology of China（中国科学技术大学）；Nanjing University（南京大学）；The University of Hong Kong（香港大学）
We consider MAP estimators for structured prediction with exponential family models. In particular, we concentrate on the case that efficient algorithms for uniform sampling from the output space exist. We show that under this assumption (i) exact computation of the partition function remains a hard problem, and (ii) the partition function and the gradient of the log partition function can be approximated efficiently. Our main result is an approximation scheme for the partition function based on Markov Chain Monte Carlo theory. We also show that the efficient uniform sampling assumption holds …
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…
Log-linear models are arguably the most successful class of graphical models for large-scale applications because of their simplicity and tractability. Learning and inference with these models require calculating the partition function, which is a major bottleneck and intractable for large state spaces. Importance Sampling (IS) and MCMC-based approaches are lucrative. However, the condition of having a "good" proposal distribution is often not satisfied in practice. In this paper, we add a new dimension to efficient estimation via sampling. We propose a new sampling scheme and an unbiased esti…
