In this note, we study the relationship between the variational gap and the variance of the (log) likelihood ratio. We show that the gap can be upper bounded by some form of dispersion measure of the likelihood ratio, which suggests the bias of variational inference can be reduced by making the distribution of the likelihood ratio more concentrated, such as via averaging and variance reduction.
Recent variational inference methods use stochastic gradient estimators whose variance is not well understood. Theoretical guarantees for these estimators are important to understand when these methods will or will not work. This paper gives bounds for the common "reparameterization" estimators when the target is smooth and the variational family is a location-scale distribution. These bounds are unimprovable and thus provide the best possible guarantees under the stated assumptions.
We propose a simple and general variant of the standard reparameterized gradient estimator for the variational evidence lower bound. Specifically, we remove a part of the total derivative with respect to the variational parameters that corresponds to the score function. Removing this term produces an unbiased gradient estimator whose variance approaches zero as the approximate posterior approaches the exact posterior. We analyze the behavior of this gradient estimator theoretically and empirically, and generalize it to more complex variational distributions such as mixtures and importance-weig…
Stochastic optimization techniques are standard in variational inference algorithms. These methods estimate gradients by approximating expectations with independent Monte Carlo samples. In this paper, we explore a technique that uses correlated, but more representative , samples to reduce estimator variance. Specifically, we show how to generate antithetic samples that match sample moments with the true moments of an underlying importance distribution. Combining a differentiable antithetic sampler with modern stochastic variational inference, we showcase the effectiveness of this approach for …
Heidelberg University（海德堡大学）；Özyeğin University（厄兹耶因大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）
We propose a new Bayesian Neural Net formulation that affords variational inference for which the evidence lower bound is analytically tractable subject to a tight approximation. We achieve this tractability by (i) decomposing ReLU nonlinearities into the product of an identity and a Heaviside step function, (ii) introducing a separate path that decomposes the neural net expectation from its variance. We demonstrate formally that introducing separate latent binary variables to the activations allows representing the neural network likelihood as a chain of linear operations. Performing variatio…
College of Information and Computer Science（信息与计算机科学学院）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
Flexible variational distributions improve variational inference but are harder to optimize. In this work we present a control variate that is applicable for any reparameterizable distribution with known mean and covariance matrix, e.g. Gaussians with any covariance structure. The control variate is based on a quadratic approximation of the model, and its parameters are set using a double-descent scheme by minimizing the gradient estimator's variance. We empirically show that this control variate leads to large improvements in gradient variance and optimization convergence for inference with n…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Carnegie Mellon University（卡内基梅隆大学）
We propose the use of U-statistics to reduce variance for gradient estimation in importance-weighted variational inference. The key observation is that, given a base gradient estimator that requires $m > 1$ samples and a total of $n > m$ samples to be used for estimation, lower variance is achieved by averaging the base estimator on overlapping batches of size $m$ than disjoint batches, as currently done. We use classical U-statistic theory to analyze the variance reduction, and propose novel approximations with theoretical guarantees to ensure computational efficiency. We find empirically tha…
Princeton Neuroscience Institute, Princeton University（普林斯顿大学神经科学研究所）；Google AI Berlin（谷歌柏林人工智能研究院）；Department of Engineering, University of Cambridge（剑桥大学工程系）；Microsoft Research, Cambridge（微软剑桥研究院）
Bayesian neural networks (BNNs) hold great promise as a flexible and principled solution to deal with uncertainty when learning from finite data. Among approaches to realize probabilistic inference in deep neural networks, variational Bayes (VB) is theoretically grounded, generally applicable, and computationally efficient. With wide recognition of potential advantages, why is it that variational Bayes has seen very limited practical use for BNNs in real applications? We argue that variational inference in neural networks is fragile: successful implementations require careful initialization an…
Freie Universität Berlin（柏林自由大学）；BTU Cottbus-Senftenberg（科特布斯-森夫滕贝格工业大学）；dida Datenschmiede GmbH（迪达数据工坊有限公司）；University of Warwick（华威大学）；Universität Potsdam（波茨坦大学）；DeepMind（深度思维）；The Alan Turing Institute（艾伦·图灵研究所）
We analyse the properties of an unbiased gradient estimator of the ELBO for variational inference, based on the score function method with leave-one-out control variates. We show that this gradient estimator can be obtained using a new loss, defined as the variance of the log-ratio between the exact posterior and the variational approximation, which we call the $\textit{log-variance loss}$. Under certain conditions, the gradient of the log-variance loss equals the gradient of the (negative) ELBO. We show theoretically that this gradient estimator, which we call $\textit{VarGrad}$ due to its co…
We propose a variance reduction framework for variational inference using the Multilevel Monte Carlo (MLMC) method. Our framework is built on reparameterized gradient estimators and "recycles" parameters obtained from past update history in optimization. In addition, our framework provides a new optimization algorithm based on stochastic gradient descent (SGD) that adaptively estimates the sample size used for gradient estimation according to the ratio of the gradient variance. We theoretically show that, with our method, the variance of the gradient estimator decreases as optimization proceed…
Black box variational inference (BBVI) with reparameterization gradients triggered the exploration of divergence measures other than the Kullback-Leibler (KL) divergence, such as alpha divergences. In this paper, we view BBVI with generalized divergences as a form of estimating the marginal likelihood via biased importance sampling. The choice of divergence determines a bias-variance trade-off between the tightness of a bound on the marginal likelihood (low bias) and the variance of its gradient estimators. Drawing on variational perturbation theory of statistical physics, we use these insight…
The Black Box Variational Inference (Ranganath et al. (2014)) algorithm provides a universal method for Variational Inference, but taking advantage of special properties of the approximation family or of the target can improve the convergence speed significantly. For example, if the approximation family is a transformation family, such as a Gaussian, then switching to the reparameterization gradient (Kingma and Welling (2014)) often yields a major reduction in gradient variance. Ultimately, reducing the variance can reduce the computational cost and yield better approximations. We present a ne…
College of Information and Computer Science（信息与计算机科学学院）；University of Massachusetts（马萨诸塞大学）；Amherst, MA 01003（阿默斯特，马萨诸塞州 01003）
Variational inference is increasingly being addressed with stochastic optimization. In this setting, the gradient's variance plays a crucial role in the optimization procedure, since high variance gradients lead to poor convergence. A popular approach used to reduce gradient's variance involves the use of control variates. Despite the good results obtained, control variates developed for variational inference are typically looked at in isolation. In this paper we clarify the large number of control variates that are available by giving a systematic view of how they are derived. We also present…
We present a new algorithm for stochastic variational inference that targets at models with non-differentiable densities. One of the key challenges in stochastic variational inference is to come up with a low-variance estimator of the gradient of a variational objective. We tackle the challenge by generalizing the reparameterization trick, one of the most effective techniques for addressing the variance issue for differentiable models, so that the trick works for non-differentiable models as well. Our algorithm splits the space of latent variables into regions where the density of the variable…
University of New South Wales（新南威尔士大学）；UNSW Business School（新南威尔士大学商学院）
The reparameterization trick is widely used in variational inference as it yields more accurate estimates of the gradient of the variational objective than alternative approaches such as the score function method. Although there is overwhelming empirical evidence in the literature showing its success, there is relatively little research exploring why the reparameterization trick is so effective. We explore this under the idealized assumptions that the variational approximation is a mean-field Gaussian density and that the log of the joint density of the model parameters and the data is a quadr…
We introduce overdispersed black-box variational inference, a method to reduce the variance of the Monte Carlo estimator of the gradient in black-box variational inference. Instead of taking samples from the variational distribution, we use importance sampling to take samples from an overdispersed distribution in the same exponential family as the variational approximation. Our approach is general since it can be readily applied to any exponential family distribution, which is the typical choice for the variational approximation. We run experiments on two non-conjugate probabilistic models to …
Approximate Bayesian Computation (ABC) is a framework for performing likelihood-free posterior inference for simulation models. Stochastic Variational inference (SVI) is an appealing alternative to the inefficient sampling approaches commonly used in ABC. However, SVI is highly sensitive to the variance of the gradient estimators, and this problem is exacerbated by approximating the likelihood. We draw upon recent advances in variance reduction for SV and likelihood-free inference using deterministic simulations to produce low variance gradient estimators of the variational lower-bound. By the…
In this paper we address the problem of enhancing speech signals in noisy mixtures using a source separation approach. We explore the use of neural networks as an alternative to a popular speech variance model based on supervised non-negative matrix factorization (NMF). More precisely, we use a variational autoencoder as a speaker-independent supervised generative speech model, highlighting the conceptual similarities that this approach shares with its NMF-based counterpart. In order to be free of generalization issues regarding the noisy recording environments, we follow the approach of havin…
Many machine learning problems involve Monte Carlo gradient estimators. As a prominent example, we focus on Monte Carlo variational inference (MCVI) in this paper. The performance of MCVI crucially depends on the variance of its stochastic gradients. We propose variance reduction by means of Quasi-Monte Carlo (QMC) sampling. QMC replaces N i.i.d. samples from a uniform probability distribution by a deterministic sequence of samples of length N. This sequence covers the underlying random variable space more evenly than i.i.d. draws, reducing the variance of the gradient estimator. With our nove…
Highly expressive directed latent variable models, such as sigmoid belief networks, are difficult to train on large datasets because exact inference in them is intractable and none of the approximate inference methods that have been applied to them scale well. We propose a fast non-iterative approximate inference method that uses a feedforward network to implement efficient exact sampling from the variational posterior. The model and this inference network are trained jointly by maximizing a variational lower bound on the log-likelihood. Although the naive estimator of the inference model grad…
Variational Bayes (VB) is a recent approximate method for Bayesian inference. It has the merit of being a fast and scalable alternative to Markov Chain Monte Carlo (MCMC) but its approximation error is often unknown. In this paper, we derive the approximation error of VB in terms of mean, mode, variance, predictive density and KL divergence for the linear Gaussian multi-equation regression. Our results indicate that VB approximates the posterior mean perfectly. Factors affecting the magnitude of underestimation in posterior variance and mode are revealed. Importantly, We demonstrate that VB es…
Recently, we and several other authors have written about the possibilities of using stochastic approximation techniques for fitting variational approximations to intractable Bayesian posterior distributions. Naive implementations of stochastic approximation suffer from high variance in this setting. Several authors have therefore suggested using control variates to reduce this variance, while we have taken a different but analogous approach to reducing the variance which we call stochastic linear regression. In this note we take the former perspective and derive the ideal set of control varia…
Variational autoencoder (VAE) is a very popular and well-investigated generative model in neural learning research. To leverage VAE in practical tasks dealing with a massive dataset of large dimensions, it is required to deal with the difficulty of building low variance evidence lower bounds (ELBO). Markov Chain Monte Carlo (MCMC) is an effective approach to tighten the ELBO for approximating the posterior distribution and Hamiltonian Variational Autoencoder (HVAE) is an effective MCMC inspired approach for constructing a low-variance ELBO that is amenable to the reparameterization trick. The …
University of Pennsylvania（宾夕法尼亚大学）；University of Rochester（罗切斯特大学）；Baylor College of Medicine（贝勒医学院）；Rice University（莱斯大学）
Sampling and Variational Inference (VI) are two large families of methods for approximate inference that have complementary strengths. Sampling methods excel at approximating arbitrary probability distributions, but can be inefficient. VI methods are efficient, but may misrepresent the true distribution. Here, we develop a general framework where approximations are stochastic mixtures of simple component distributions. Both sampling and VI can be seen as special cases: in sampling, each mixture component is a delta-function and is chosen stochastically, while in standard VI a single component …
The Tweedie Compound Poisson-Gamma model is routinely used for modeling non-negative continuous data with a discrete probability mass at zero. Mixed models with random effects account for the covariance structure related to the grouping hierarchy in the data. An important application of Tweedie mixed models is pricing the insurance policies, e.g. car insurance. However, the intractable likelihood function, the unknown variance function, and the hierarchical structure of mixed effects have presented considerable challenges for drawing inferences on Tweedie. In this study, we tackle the Bayesian…
Columbia University（哥伦比亚大学）；Princeton University（普林斯顿大学）
Variational inference (VI) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$. Closeness is usually measured via a divergence $D(q || p)$ from $q$ to $p$. While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose CHIVI, a black-box variational inference algorithm that minimizes $D_χ(p || q)$, the $χ$-divergence from $p$ to $q$. CHIVI minimizes an upper bound of the model evi…
We propose a new marginal data density estimator (MDDE) that uses the variational Bayes posterior density as a weighting density of the reciprocal importance sampling (RIS) MDDE. This computationally convenient estimator is based on variational Bayes posterior densities that are available for many models and requires simulated draws only from the posterior distribution. It provides accurate estimates with a moderate number of posterior draws, has a finite variance, and provides a minimum variance candidate for the class of RIS MDDEs. Its reciprocal is consistent, asymptotically normally distri…
In this article we consider importance sampling (IS) and sequential Monte Carlo (SMC) methods in the context of 1-dimensional random walks with absorbing barriers. In particular, we develop a very precise variance analysis for several IS and SMC procedures. We take advantage of some explicit spectral formulae available for these models to derive sharp and explicit estimates; this provides stability properties of the associated normalized Feynman-Kac semigroups. Our analysis allows one to compare the variance of SMC and IS techniques for these models. The work in this article, is one of the few…
Columbia University（哥伦比亚大学）；New York Genome Center（纽约基因组中心）
Brittle optimization has been observed to adversely impact model likelihoods for regression and VAEs when simultaneously fitting neural network mappings from a (random) variable onto the mean and variance of a dependent Gaussian variable. Previous works have bolstered optimization and improved likelihoods, but fail other basic posterior predictive checks (PPCs). Under the PPC framework, we propose critiques to test predictive mean and variance calibration and the predictive distribution's ability to generate sensible data. We find that our attractively simple solution, to treat heteroscedastic…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Linköping University（林雪平大学）
We propose stochastic variance reduced algorithms for solving convex-concave saddle point problems, monotone variational inequalities, and monotone inclusions. Our framework applies to extragradient, forward-backward-forward, and forward-reflected-backward methods both in Euclidean and Bregman setups. All proposed methods converge in the same setting as their deterministic counterparts and they either match or improve the best-known complexities for solving structured min-max problems. Our results reinforce the correspondence between variance reduction in variational inequalities and minimizat…
This paper introduces the Variational Determinant Estimator (VDE), a variational extension of the recently proposed determinant estimator discovered by arXiv:2005.06553v2. Our estimator significantly reduces the variance even for low sample sizes by combining (importance-weighted) variational inference and a family of normalizing flows which allow density estimation on hyperspheres. In the ideal case of a tight variational bound, the VDE becomes a zero variance estimator, and a single sample is sufficient for an exact (log) determinant estimate.
Google Brain（谷歌大脑）；MILA（蒙特利尔学习算法研究所）；DeepMind（深度思维）
Estimating and optimizing Mutual Information (MI) is core to many problems in machine learning; however, bounding MI in high dimensions is challenging. To establish tractable and scalable objectives, recent work has turned to variational bounds parameterized by neural networks, but the relationships and tradeoffs between these bounds remains unclear. In this work, we unify these recent developments in a single framework. We find that the existing variational lower bounds degrade when the MI is large, exhibiting either high bias or high variance. To address this problem, we introduce a continuu…
University of Sussex（萨塞克斯大学）；University of Bristol（布里斯托尔大学）
We develop variational Laplace for Bayesian neural networks (BNNs) which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is (in essence) the log-likelihood, plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a-posteriori inference and standard sampling-based variational inference, despite using the same variational approximat…
We develop variational Laplace for Bayesian neural networks (BNNs) which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. The Variational Laplace objective is simple to evaluate, as it is (in essence) the log-likelihood, plus weight-decay, plus a squared-gradient regularizer. Variational Laplace gave better test performance and expected calibration errors than maximum a-posteriori inference and standard sampling-based variational inference, despite using the same variational approximat…
Variational approaches based on neural networks are showing promise for estimating mutual information (MI) between high dimensional variables. However, they can be difficult to use in practice due to poorly understood bias/variance tradeoffs. We theoretically show that, under some conditions, estimators such as MINE exhibit variance that could grow exponentially with the true amount of underlying MI. We also empirically demonstrate that existing estimators fail to satisfy basic self-consistency properties of MI, such as data processing and additivity under independence. Based on a unified pers…
International Institute of Information Technology, Hyderabad（海得拉巴国际信息技术研究所）
Mutual information (MI) is an information-theoretic measure of dependency between two random variables. Several methods to estimate MI, from samples of two random variables with unknown underlying probability distributions have been proposed in the literature. Recent methods realize parametric probability distributions or critic as a neural network to approximate unknown density ratios. The approximated density ratios are used to estimate different variational lower bounds of MI. While these methods provide reliable estimation when the true MI is low, they produce high variance estimates in ca…
Sorbonne Université（索邦大学）；CNRS（法国国家科学研究中心）；Électricité de France R&D（法国电力研发中心）
We consider the problem of time series forecasting in an adaptive setting. We focus on the inference of state-space models under unknown and potentially time-varying noise variances. We introduce an augmented model in which the variances are represented as auxiliary gaussian latent variables in a tracking mode. As variances are nonnegative, a transformation is chosen and applied to these latent variables. The inference relies on the online variational Bayesian methodology, which consists in minimizing a Kullback-Leibler divergence at each time step. We observe that the minimum of the Kullback-…
We prove that the evidence lower bound (ELBO) employed by variational auto-encoders (VAEs) admits non-trivial solutions having constant posterior variances under certain mild conditions, removing the need to learn variances in the encoder. The proof follows from an unexpected journey through an array of topics: the closed form optimal decoder for Gaussian VAEs, a proof that the decoder is always smooth, a proof that the ELBO at its stationary points is equal to the exact log evidence, and the posterior variance is merely part of a stochastic estimator of the decoder Hessian. The penalty incurr…
We observe that the standard log likelihood training objective for a Recurrent Neural Network (RNN) model of time series data is equivalent to a variational Bayesian training objective, given the proper choice of generative and inference models. This perspective may motivate extensions to both RNNs and variational Bayesian models. We propose one such extension, where multiple particles are used for the hidden state of an RNN, allowing a natural representation of uncertainty or multimodality.
SARAH-based Variance-reduced Algorithm for Stochastic Finite-sum Cocoercive Variational Inequalities
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；HSE University（高等经济大学）；IITP RAS（俄罗斯科学院信息传输问题研究所）；Caucasus Mathematical Center, Adyghe State University（阿迪格国立大学高加索数学中心）
University of Vienna（维也纳大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔国立理工学院）；LIG（格勒诺布尔信息学实验室）；Maastricht University（马斯特里赫特大学）
We develop a new stochastic algorithm with variance reduction for solving pseudo-monotone stochastic variational inequalities. Our method builds on Tseng's forward-backward-forward (FBF) algorithm, which is known in the deterministic literature to be a valuable alternative to Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by pseudo-monotone, Lipschitz continuous operators. The main computational advantage of Tseng's algorithm is that it relies only on a single projection step and two independent queries of a stochastic oracle. Our…
University of California, Irvine（加州大学欧文分校）；Canadian Institute for Advanced Research（加拿大高等研究院）；University of Amsterdam（阿姆斯特丹大学）
We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the minibatch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gau…
In this article, we highlight what appears to be major issue of Variational Autoencoders, evinced from an extensive experimentation with different network architectures and datasets: the variance of generated data is significantly lower than that of training data. Since generative models are usually evaluated with metrics such as the Frechet Inception Distance (FID) that compare the distributions of (features of) real versus generated images, the variance loss typically results in degraded scores. This problem is particularly relevant in a two stage setting, where we use a second VAE to sample…
As deep Variational Auto-Encoder (VAE) frameworks become more widely used for modeling biomolecular simulation data, we emphasize the capability of the VAE architecture to concurrently maximize the timescale of the latent space while inferring a reduced coordinate, which assists in finding slow processes as according to the variational approach to conformational dynamics. We additionally provide evidence that the VDE framework (Hernández et al., 2017), which uses this autocorrelation loss along with a time-lagged reconstruction loss, obtains a variationally optimized latent coordinate in compa…
Amazon Web Services（亚马逊网络服务公司）；University of Southern California（南加州大学）；Los Angeles, CA, USA（美国加利福尼亚州洛杉矶）
Estimation of uncertainty in deep learning models is of vital importance, especially in medical imaging, where reliance on inference without taking into account uncertainty could lead to misdiagnosis. Recently, the probabilistic Variational AutoEncoder (VAE) has become a popular model for anomaly detection in applications such as lesion detection in medical images. The VAE is a generative graphical model that is used to learn the data distribution from samples and then generate new samples from this distribution. By training on normal samples, the VAE can be used to detect inputs that deviate …
