This tutorial paper focuses on the variants of the bottleneck problem taking an information theoretic perspective and discusses practical methods to solve it, as well as its connection to coding and learning aspects. The intimate connections of this setting to remote source-coding under logarithmic loss distortion measure, information combining, common reconstruction, the Wyner-Ahlswede-Korner problem, the efficiency of investment information, as well as, generalization, variational inference, representation learning, autoencoders, and others are highlighted. We discuss its extension to the distributed information bottleneck problem with emphasis on the Gaussian model and highlight the basic connections to the uplink Cloud Radio Access Networks (CRAN) with oblivious processing. For this model, the optimal trade-offs between relevance (i.e., information) and complexity (i.e., rates) in the discrete and vector Gaussian frameworks is determined. In the concluding outlook, some interesting problems are mentioned such as the characterization of the optimal inputs ("features") distributions under power limitations maximizing the "relevance" for the Gaussian information bottleneck, under "complexity" constraints.
Markov Logic Networks (MLNs), which elegantly combine logic rules and probabilistic graphical models, can be used to address many knowledge graph problems. However, inference in MLN is computationally intensive, making the industrial-scale application of MLN very difficult. In recent years, graph neural networks (GNNs) have emerged as efficient and effective tools for large-scale graph problems. Nevertheless, GNNs do not explicitly incorporate prior logic rules into the models, and may require many labeled examples for a target task. In this paper, we explore the combination of MLNs and GNNs, and use graph neural networks for variational inference in MLN. We propose a GNN variant, named ExpressGNN, which strikes a nice balance between the representation power and the simplicity of the model. Our extensive experiments on several benchmark datasets demonstrate that ExpressGNN leads to effective and efficient probabilistic logic reasoning.
Most combinations of NLP tasks and language varieties lack in-domain examples for supervised training because of the paucity of annotated data. How can neural models make sample-efficient generalizations from task-language combinations with available data to low-resource ones? In this work, we propose a Bayesian generative model for the space of neural parameters. We assume that this space can be factorized into latent variables for each language and each task. We infer the posteriors over such latent variables based on data from seen task-language combinations through variational inference. This enables zero-shot classification on unseen combinations at prediction time. For instance, given training data for named entity recognition (NER) in Vietnamese and for part-of-speech (POS) tagging in Wolof, our model can perform accurate predictions for NER in Wolof. In particular, we experiment with a typologically diverse sample of 33 languages from 4 continents and 11 families, and show that our model yields comparable or better results than state-of-the-art, zero-shot cross-lingual transfer methods. Moreover, we demonstrate that approximate Bayesian model averaging results in smoother predictive distributions, whose entropy inversely correlates with accuracy. Hence, the proposed framework also offers robust estimates of prediction uncertainty. Our code is located at github.com/cambridgeltl/parameter-factorization
Much research has been devoted to the problem of estimating treatment effects from observational data; however, most methods assume that the observed variables only contain confounders, i.e., variables that affect both the treatment and the outcome. Unfortunately, this assumption is frequently violated in real-world applications, since some variables only affect the treatment but not the outcome, and vice versa. Moreover, in many cases only the proxy variables of the underlying confounding factors can be observed. In this work, we first show the importance of differentiating confounding factors from instrumental and risk factors for both average and conditional average treatment effect estimation, and then we propose a variational inference approach to simultaneously infer latent factors from the observed variables, disentangle the factors into three disjoint sets corresponding to the instrumental, confounding, and risk factors, and use the disentangled factors for treatment effect estimation. Experimental results demonstrate the effectiveness of the proposed method on a wide range of synthetic, benchmark, and real-world datasets.
This short note reviews so-called Natural Gradient Descent (NGD) for multivariate Gaussians. The Fisher Information Matrix (FIM) is derived for several different parameterizations of Gaussians. Careful attention is paid to the symmetric nature of the covariance matrix when calculating derivatives. We show that there are some advantages to choosing a parameterization comprising the mean and inverse covariance matrix and provide a simple NGD update that accounts for the symmetric (and sparse) nature of the inverse covariance matrix.
It is a common practice in the machine learning community to assume that the observed data are noise-free in the input attributes. Nevertheless, scenarios with input noise are common in real problems, as measurements are never perfectly accurate. If this input noise is not taken into account, a supervised machine learning method is expected to perform sub-optimally. In this paper, we focus on multi-class classification problems and use Gaussian processes (GPs) as the underlying classifier. Motivated by a data set coming from the astrophysics domain, we hypothesize that the observed data may contain noise in the inputs. Therefore, we devise several multi-class GP classifiers that can account for input noise. Such classifiers can be efficiently trained using variational inference to approximate the posterior distribution of the latent variables of the model. Moreover, in some situations, the amount of noise can be known before-hand. If this is the case, it can be readily introduced in the proposed methods. This prior information is expected to lead to better performance results. We have evaluated the proposed methods by carrying out several experiments, involving synthetic and real data. These include several data sets from the UCI repository, the MNIST data set and a data set coming from astrophysics. The results obtained show that, although the classification error is similar across methods, the predictive distribution of the proposed methods is better, in terms of the test log-likelihood, than the predictive distribution of a classifier based on GPs that ignores input noise.
Marginal maximum likelihood (MML) estimation is the preferred approach to fitting item response theory models in psychometrics due to the MML estimator's consistency, normality, and efficiency as the sample size tends to infinity. However, state-of-the-art MML estimation procedures such as the Metropolis-Hastings Robbins-Monro (MH-RM) algorithm as well as approximate MML estimation procedures such as variational inference (VI) are computationally time-consuming when the sample size and the number of latent factors are very large. In this work, we investigate a deep learning-based VI algorithm for exploratory item factor analysis (IFA) that is computationally fast even in large data sets with many latent factors. The proposed approach applies a deep artificial neural network model called an importance-weighted autoencoder (IWAE) for exploratory IFA. The IWAE approximates the MML estimator using an importance sampling technique wherein increasing the number of importance-weighted (IW) samples drawn during fitting improves the approximation, typically at the cost of decreased computational efficiency. We provide a real data application that recovers results aligning with psychological theory across random starts. Via simulation studies, we show that the IWAE yields more accurate estimates as either the sample size or the number of IW samples increases (although factor correlation and intercepts estimates exhibit some bias) and obtains similar results to MH-RM in less time. Our simulations also suggest that the proposed approach performs similarly to and is potentially faster than constrained joint maximum likelihood estimation, a fast procedure that is consistent when the sample size and the number of items simultaneously tend to infinity.
We present a pixel-based deep active inference algorithm (PixelAI) inspired by human body perception and action. Our algorithm combines the free-energy principle from neuroscience, rooted in variational inference, with deep convolutional decoders to scale the algorithm to directly deal with raw visual input and provide online adaptive inference. Our approach is validated by studying body perception and action in a simulated and a real Nao robot. Results show that our approach allows the robot to perform 1) dynamical body estimation of its arm using only monocular camera images and 2) autonomous reaching to "imagined" arm poses in the visual space. This suggests that robot and human body perception and action can be efficiently solved by viewing both as an active inference problem guided by ongoing sensory input.
Single-site Markov Chain Monte Carlo (MCMC) is a variant of MCMC in which a single coordinate in the state space is modified in each step. Structured relational models are a good candidate for this style of inference. In the single-site context, second order methods become feasible because the typical cubic costs associated with these methods is now restricted to the dimension of each coordinate. Our work, which we call Newtonian Monte Carlo (NMC), is a method to improve MCMC convergence by analyzing the first and second order gradients of the target density to determine a suitable proposal density at each point. Existing first order gradient-based methods suffer from the problem of determining an appropriate step size. Too small a step size and it will take a large number of steps to converge, while a very large step size will cause it to overshoot the high density region. NMC is similar to the Newton-Raphson update in optimization where the second order gradient is used to automatically scale the step size in each dimension. However, our objective is to find a parameterized proposal density rather than the maxima. As a further improvement on existing first and second order methods, we show that random variables with constrained supports don't need to be transformed before taking a gradient step. We demonstrate the efficiency of NMC on a number of different domains. For statistical models where the prior is conjugate to the likelihood, our method recovers the posterior quite trivially in one step. However, we also show results on fairly large non-conjugate models, where NMC performs better than adaptive first order methods such as NUTS or other inexact scalable inference methods such as Stochastic Variational Inference or bootstrapping.
Current variational inference methods for hierarchical Bayesian nonparametric models can neither characterize the correlation structure among latent variables due to the mean-field setting, nor infer the true posterior dimension because of the universal truncation. To overcome these limitations, we propose the conditional and adaptively truncated variational inference method (CATVI) by maximizing the nonparametric evidence lower bound and integrating Monte Carlo into the variational inference framework. CATVI enjoys several advantages over traditional methods, including a smaller divergence between variational and true posteriors, reduced risk of underfitting or overfitting, and improved prediction accuracy. Empirical studies on three large datasets reveal that CATVI applied in Bayesian nonparametric topic models substantially outperforms competing models, providing lower perplexity and clearer topic-words clustering.
A variety of lifted inference algorithms, which exploit model symmetry to reduce computational cost, have been proposed to render inference tractable in probabilistic relational models. Most existing lifted inference algorithms operate only over discrete domains or continuous domains with restricted potential functions, e.g., Gaussian. We investigate two approximate lifted variational approaches that are applicable to hybrid domains and expressive enough to capture multi-modality. We demonstrate that the proposed variational methods are both scalable and can take advantage of approximate model symmetries, even in the presence of a large amount of continuous evidence. We demonstrate that our approach compares favorably against existing message-passing based approaches in a variety of settings. Finally, we present a sufficient condition for the Bethe approximation to yield a non-trivial estimate over the marginal polytope.
Hidden Markov Model (HMM) combined with Gaussian Process (GP) emission can be effectively used to estimate the hidden state with a sequence of complex input-output relational observations. Especially when the spectral mixture (SM) kernel is used for GP emission, we call this model as a hybrid HMM-GPSM. This model can effectively model the sequence of time-series data. However, because of a large number of parameters for the SM kernel, this model can not effectively be trained with a large volume of data having (1) long sequence for state transition and 2) a large number of time-series dataset in each sequence. This paper proposes a scalable learning method for HMM-GPSM. To effectively train the model with a long sequence, the proposed method employs a Stochastic Variational Inference (SVI) approach. Also, to effectively process a large number of data point each time-series data, we approximate the SM kernel using Reparametrized Random Fourier Feature (R-RFF). The combination of these two techniques significantly reduces the training time. We validate the proposed learning method in terms of its hidden-sate estimation accuracy and computation time using large-scale synthetic and real data sets with missing values.
The adjoint sensitivity method scalably computes gradients of solutions to ordinary differential equations. We generalize this method to stochastic differential equations, allowing time-efficient and constant-memory computation of gradients with high-order adaptive solvers. Specifically, we derive a stochastic differential equation whose solution is the gradient, a memory-efficient algorithm for caching noise, and conditions under which numerical solutions converge. In addition, we combine our method with gradient-based stochastic variational inference for latent stochastic differential equations. We use our method to fit stochastic dynamics defined by neural networks, achieving competitive performance on a 50-dimensional motion capture dataset.
Autonomous agents can learn by imitating teacher demonstrations of the intended behavior. Hierarchical control policies are ubiquitously useful for such learning, having the potential to break down structured tasks into simpler sub-tasks, thereby improving data efficiency and generalization. In this paper, we propose a variational inference method for imitation learning of a control policy represented by parametrized hierarchical procedures (PHP), a program-like structure in which procedures can invoke sub-procedures to perform sub-tasks. Our method discovers the hierarchical structure in a dataset of observation-action traces of teacher demonstrations, by learning an approximate posterior distribution over the latent sequence of procedure calls and terminations. Samples from this learned distribution then guide the training of the hierarchical control policy. We identify and demonstrate a novel benefit of variational inference in the context of hierarchical imitation learning: in decomposing the policy into simpler procedures, inference can leverage acausal information that is unused by other methods. Training PHP with variational inference outperforms LSTM baselines in terms of data efficiency and generalization, requiring less than half as much data to achieve a 24% error rate in executing the bubble sort algorithm, and to achieve no error in executing Karel programs.
Learning in Gaussian Process models occurs through the adaptation of hyperparameters of the mean and the covariance function. The classical approach entails maximizing the marginal likelihood yielding fixed point estimates (an approach called \textit{Type II maximum likelihood} or ML-II). An alternative learning procedure is to infer the posterior over hyperparameters in a hierarchical specification of GPs we call \textit{Fully Bayesian Gaussian Process Regression} (GPR). This work considers two approximation schemes for the intractable hyperparameter posterior: 1) Hamiltonian Monte Carlo (HMC) yielding a sampling-based approximation and 2) Variational Inference (VI) where the posterior over hyperparameters is approximated by a factorized Gaussian (mean-field) or a full-rank Gaussian accounting for correlations between hyperparameters. We analyze the predictive performance for fully Bayesian GPR on a range of benchmark data sets.
Metric-based meta-learning has attracted a lot of attention due to its effectiveness and efficiency in few-shot learning. Recent studies show that metric scaling plays a crucial role in the performance of metric-based meta-learning algorithms. However, there still lacks a principled method for learning the metric scaling parameter automatically. In this paper, we recast metric-based meta-learning from a Bayesian perspective and develop a variational metric scaling framework for learning a proper metric scaling parameter. Firstly, we propose a stochastic variational method to learn a single global scaling parameter. To better fit the embedding space to a given data distribution, we extend our method to learn a dimensional scaling vector to transform the embedding space. Furthermore, to learn task-specific embeddings, we generate task-dependent dimensional scaling vectors with amortized variational inference. Our method is end-to-end without any pre-training and can be used as a simple plug-and-play module for existing metric-based meta-algorithms. Experiments on mini-ImageNet show that our methods can be used to consistently improve the performance of existing metric-based meta-algorithms including prototypical networks and TADAM. The source code can be downloaded from https://github.com/jiaxinchen666/variational-scaling.
The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings using relatively few data points and achieves user-controlled sparse levels that are comparable to other methods such as compressive sensing. The proposed approach is illustrated on two numerical examples, a synthetic response function that is explored for validation purposes and a low-carbon steel plate with random Young's modulus and random loading, which is modeled by stochastic finite element with 38 input random variables.
In this paper, we are interested in unsupervised (unknown noise) audio-visual speech enhancement based on variational autoencoders (VAEs), where the probability distribution of clean speech spectra is simulated using an encoder-decoder architecture. The trained generative model (decoder) is then combined with a noise model at test time to estimate the clean speech. In the speech enhancement phase (test time), the initialization of the latent variables, which describe the generative process of clean speech via decoder, is crucial, as the overall inference problem is non-convex. This is usually done by using the output of the trained encoder where the noisy audio and clean visual data are given as input. Current audio-visual VAE models do not provide an effective initialization because the two modalities are tightly coupled (concatenated) in the associated architectures. To overcome this issue, inspired by mixture models, we introduce the mixture of inference networks variational autoencoder (MIN-VAE). Two encoder networks input, respectively, audio and visual data, and the posterior of the latent variables is modeled as a mixture of two Gaussian distributions output from each encoder network. The mixture variable is also latent, and therefore the inference of learning the optimal balance between the audio and visual inference networks is unsupervised as well. By training a shared decoder, the overall network learns to adaptively fuse the two modalities. Moreover, at test time, the visual encoder, which takes (clean) visual data, is used for initialization. A variational inference approach is derived to train the proposed generative model. Thanks to the novel inference procedure and the robust initialization, the proposed MIN-VAE exhibits superior performance on speech enhancement than using the standard audio-only as well as audio-visual counterparts.
Posterior inference in directed graphical models is commonly done using a probabilistic encoder (a.k.a inference model) conditioned on the input. Often this inference model is trained jointly with the probabilistic decoder (a.k.a generator model). If probabilistic encoder encounters complexities during training (e.g. suboptimal complxity or parameterization), then learning reaches a suboptimal objective; a phenomena commonly called inference suboptimality. In Variational Inference (VI), optimizing the ELBo using Stochastic Variational Inference (SVI) can eliminate the inference suboptimality (as demonstrated in this paper), however, this solution comes at a substantial computational cost when inference needs to be done on new data points. Essentially, a long sequential chain of gradient updates is required to fully optimize approximate posteriors. In this paper, we present an approach called Pseudo-Encoded Stochastic Variational Inference (PE-SVI), to reduce the inference complexity of SVI during test time. Our approach relies on finding a suitable initial start point for gradient operations, which naturally reduces the required gradient steps. Furthermore, this initialization allows for adopting larger step sizes (compared to random initialization used in SVI), which further reduces the inference time complexity. PE-SVI reaches the same ELBo objective as SVI using less than one percent of required steps, on average.
In many data analysis tasks, it is beneficial to learn representations where each dimension is statistically independent and thus disentangled from the others. If data generating factors are also statistically independent, disentangled representations can be formed by Bayesian inference of latent variables. We examine a generalization of the Variational Autoencoder (VAE), $\beta$-VAE, for learning such representations using variational inference. $\beta$-VAE enforces conditional independence of its bottleneck neurons controlled by its hyperparameter $\beta$. This condition is in general not compatible with the statistical independence of latents. By providing analytical and numerical arguments, we show that this incompatibility leads to a non-monotonic inference performance in $\beta$-VAE with a finite optimal $\beta$.