A graph generative model defines a distribution over graphs. One type of generative model is constructed by autoregressive neural networks, which sequentially add nodes and edges to generate a graph. However, the likelihood of a graph under the autoregressive model is intractable, as there are numerous sequences leading to the given graph; this makes maximum likelihood estimation challenging. Instead, in this work we derive the exact joint probability over the graph and the node ordering of the sequential process. From the joint, we approximately marginalize out the node orderings and compute a lower bound on the log-likelihood using variational inference. We train graph generative models by maximizing this bound, without using the ad-hoc node orderings of previous methods. Our experiments show that the log-likelihood bound is significantly tighter than the bound of previous schemes. Moreover, the models fitted with the proposed algorithm can generate high-quality graphs that match the structures of target graphs not seen during training. We have made our code publicly available at \hyperref[https://github.com/tufts-ml/graph-generation-vi]{https://github.com/tufts-ml/graph-generation-vi}.
Constrained clustering has gained significant attention in the field of machine learning as it can leverage prior information on a growing amount of only partially labeled data. Following recent advances in deep generative models, we propose a novel framework for constrained clustering that is intuitive, interpretable, and can be trained efficiently in the framework of stochastic gradient variational inference. By explicitly integrating domain knowledge in the form of probabilistic relations, our proposed model (DC-GMM) uncovers the underlying distribution of data conditioned on prior clustering preferences, expressed as pairwise constraints. These constraints guide the clustering process towards a desirable partition of the data by indicating which samples should or should not belong to the same cluster. We provide extensive experiments to demonstrate that DC-GMM shows superior clustering performances and robustness compared to state-of-the-art deep constrained clustering methods on a wide range of data sets. We further demonstrate the usefulness of our approach on two challenging real-world applications.
Effectively modeling phenomena present in highly nonlinear dynamical systems whilst also accurately quantifying uncertainty is a challenging task, which often requires problem-specific techniques. We present a novel, domain-agnostic approach to tackling this problem, using compositions of physics-informed random features, derived from ordinary differential equations. The architecture of our model leverages recent advances in approximate inference for deep Gaussian processes, such as layer-wise weight-space approximations which allow us to incorporate random Fourier features, and stochastic variational inference for approximate Bayesian inference. We provide evidence that our model is capable of capturing highly nonlinear behaviour in real-world multivariate time series data. In addition, we find that our approach achieves comparable performance to a number of other probabilistic models on benchmark regression tasks.
In this work, we study the problem of clustering survival data $-$ a challenging and so far under-explored task. We introduce a novel semi-supervised probabilistic approach to cluster survival data by leveraging recent advances in stochastic gradient variational inference. In contrast to previous work, our proposed method employs a deep generative model to uncover the underlying distribution of both the explanatory variables and censored survival times. We compare our model to the related work on clustering and mixture models for survival data in comprehensive experiments on a wide range of synthetic, semi-synthetic, and real-world datasets, including medical imaging data. Our method performs better at identifying clusters and is competitive at predicting survival times. Relying on novel generative assumptions, the proposed model offers a holistic perspective on clustering survival data and holds a promise of discovering subpopulations whose survival is regulated by different generative mechanisms.
This paper introduces a method for the nonparametric Bayesian learning of nonlinear operators, through the use of the Volterra series with kernels represented using Gaussian processes (GPs), which we term the nonparametric Volterra kernels model (NVKM). When the input function to the operator is unobserved and has a GP prior, the NVKM constitutes a powerful method for both single and multiple output regression, and can be viewed as a nonlinear and nonparametric latent force model. When the input function is observed, the NVKM can be used to perform Bayesian system identification. We use recent advances in efficient sampling of explicit functions from GPs to map process realisations through the Volterra series without resorting to numerical integration, allowing scalability through doubly stochastic variational inference, and avoiding the need for Gaussian approximations of the output processes. We demonstrate the performance of the model for both multiple output regression and system identification using standard benchmarks.
Hawkes processes are a class of point processes that have the ability to model the self- and mutual-exciting phenomena. Although the classic Hawkes processes cover a wide range of applications, their expressive ability is limited due to three key hypotheses: parametric, linear and homogeneous. Recent work has attempted to address these limitations separately. This work aims to overcome all three assumptions simultaneously by proposing the flexible state-switching Hawkes processes: a flexible, nonlinear and nonhomogeneous variant where a state process is incorporated to interact with the point processes. The proposed model empowers Hawkes processes to be applied to time-varying systems. For inference, we utilize the latent variable augmentation technique to design two efficient Bayesian inference algorithms: Gibbs sampler and mean-field variational inference, with analytical iterative updates to estimate the posterior. In experiments, our model achieves superior performance compared to the state-of-the-art competitors.
We propose probabilistic task modelling -- a generative probabilistic model for collections of tasks used in meta-learning. The proposed model combines variational auto-encoding and latent Dirichlet allocation to model each task as a mixture of Gaussian distribution in an embedding space. Such modelling provides an explicit representation of a task through its task-theme mixture. We present an efficient approximation inference technique based on variational inference method for empirical Bayes parameter estimation. We perform empirical evaluations to validate the task uncertainty and task distance produced by the proposed method through correlation diagrams of the prediction accuracy on testing tasks. We also carry out experiments of task selection in meta-learning to demonstrate how the task relatedness inferred from the proposed model help to facilitate meta-learning algorithms.
This paper focuses on $\alpha$-divergence minimisation methods for Variational Inference. More precisely, we are interested in algorithms optimising the mixture weights of any given mixture model, without any information on the underlying distribution of its mixture components parameters. The Power Descent, defined for all $\alpha \neq 1$, is one such algorithm and we establish in our work the full proof of its convergence towards the optimal mixture weights when $\alpha <1$. Since the $\alpha$-divergence recovers the widely-used forward Kullback-Leibler when $\alpha \to 1$, we then extend the Power Descent to the case $\alpha = 1$ and show that we obtain an Entropic Mirror Descent. This leads us to investigate the link between Power Descent and Entropic Mirror Descent: first-order approximations allow us to introduce the Renyi Descent, a novel algorithm for which we prove an $O(1/N)$ convergence rate. Lastly, we compare numerically the behavior of the unbiased Power Descent and of the biased Renyi Descent and we discuss the potential advantages of one algorithm over the other.
Bayesian model averaging, obtained as the expectation of a likelihood function by a posterior distribution, has been widely used for prediction, evaluation of uncertainty, and model selection. Various approaches have been developed to efficiently capture the information in the posterior distribution; one such approach is the optimization of a set of models simultaneously with interaction to ensure the diversity of the individual models in the same way as ensemble learning. A representative approach is particle variational inference (PVI), which uses an ensemble of models as an empirical approximation for the posterior distribution. PVI iteratively updates each model with a repulsion force to ensure the diversity of the optimized models. However, despite its promising performance, a theoretical understanding of this repulsion and its association with the generalization ability remains unclear. In this paper, we tackle this problem in light of PAC-Bayesian analysis. First, we provide a new second-order Jensen inequality, which has the repulsion term based on the loss function. Thanks to the repulsion term, it is tighter than the standard Jensen inequality. Then, we derive a novel generalization error bound and show that it can be reduced by enhancing the diversity of models. Finally, we derive a new PVI that optimizes the generalization error bound directly. Numerical experiments demonstrate that the performance of the proposed PVI compares favorably with existing methods in the experiment.
A critical challenge faced by supervised word sense disambiguation (WSD) is the lack of large annotated datasets with sufficient coverage of words in their diversity of senses. This inspired recent research on few-shot WSD using meta-learning. While such work has successfully applied meta-learning to learn new word senses from very few examples, its performance still lags behind its fully supervised counterpart. Aiming to further close this gap, we propose a model of semantic memory for WSD in a meta-learning setting. Semantic memory encapsulates prior experiences seen throughout the lifetime of the model, which aids better generalization in limited data settings. Our model is based on hierarchical variational inference and incorporates an adaptive memory update rule via a hypernetwork. We show our model advances the state of the art in few-shot WSD, supports effective learning in extremely data scarce (e.g. one-shot) scenarios and produces meaning prototypes that capture similar senses of distinct words.
We study the role of information complexity in privacy leakage about an attribute of an adversary's interest, which is not known a priori to the system designer. Considering the supervised representation learning setup and using neural networks to parameterize the variational bounds of information quantities, we study the impact of the following factors on the amount of information leakage: information complexity regularizer weight, latent space dimension, the cardinalities of the known utility and unknown sensitive attribute sets, the correlation between utility and sensitive attributes, and a potential bias in a sensitive attribute of adversary's interest. We conduct extensive experiments on Colored-MNIST and CelebA datasets to evaluate the effect of information complexity on the amount of intrinsic leakage.
Quantification of uncertainty in point cloud matching is critical in many tasks such as pose estimation, sensor fusion, and grasping. Iterative closest point (ICP) is a commonly used pose estimation algorithm which provides a point estimate of the transformation between two point clouds. There are many sources of uncertainty in this process that may arise due to sensor noise, ambiguous environment, and occlusion. However, for safety critical problems such as autonomous driving, a point estimate of the pose transformation is not sufficient as it does not provide information about the multiple solutions. Current probabilistic ICP methods usually do not capture all sources of uncertainty and may provide unreliable transformation estimates which can have a detrimental effect in state estimation or decision making tasks that use this information. In this work we propose a new algorithm to align two point clouds that can precisely estimate the uncertainty of ICP's transformation parameters. We develop a Stein variational inference framework with gradient based optimization of ICP's cost function. The method provides a non-parametric estimate of the transformation, can model complex multi-modal distributions, and can be effectively parallelized on a GPU. Experiments using 3D kinect data as well as sparse indoor/outdoor LiDAR data show that our method is capable of efficiently producing accurate pose uncertainty estimates.
Valuation problems, such as feature interpretation, data valuation and model valuation for ensembles, become increasingly more important in many machine learning applications. Such problems are commonly solved by well-known game-theoretic criteria, such as Shapley value or Banzhaf value. In this work, we present a novel energy-based treatment for cooperative games, with a theoretical justification by the maximum entropy framework. Surprisingly, by conducting variational inference of the energy-based model, we recover various game-theoretic valuation criteria through conducting one-step fixed point iteration for maximizing the mean-field ELBO objective. This observation also verifies the rationality of existing criteria, as they are all attempting to decouple the correlations among the players through the mean-field approach. By running fixed point iteration for multiple steps, we achieve a trajectory of the valuations, among which we define the valuation with the best conceivable decoupling error as the Variational Index. We prove that under uniform initializations, these variational valuations all satisfy a set of game-theoretic axioms. We experimentally demonstrate that the proposed Variational Index enjoys lower decoupling error and better valuation performance on certain synthetic and real-world valuation problems.
Blind image deblurring is an important yet very challenging problem in low-level vision. Traditional optimization based methods generally formulate this task as a maximum-a-posteriori estimation or variational inference problem, whose performance highly relies on the handcraft priors for both the latent image and the blur kernel. In contrast, recent deep learning methods generally learn, from a large collection of training images, deep neural networks (DNNs) directly mapping the blurry image to the clean one or to the blur kernel, paying less attention to the physical degradation process of the blurry image. In this paper, we present a deep variational Bayesian framework for blind image deblurring. Under this framework, the posterior of the latent clean image and blur kernel can be jointly estimated in an amortized inference fashion with DNNs, and the involved inference DNNs can be trained by fully considering the physical blur model, together with the supervision of data driven priors for the clean image and blur kernel, which is naturally led to by the evidence lower bound objective. Comprehensive experiments are conducted to substantiate the effectiveness of the proposed framework. The results show that it can not only achieve a promising performance with relatively simple networks, but also enhance the performance of existing DNNs for deblurring.
Modern time series data often display complex nonlinear dependencies along with irregular regime-switching behaviors. These features present technical challenges in modeling, inference, and in offering insightful understanding into the underlying stochastic phenomena. To tackle these challenges, we introduce a novel modeling framework known as the Deep Switching State Space Model (DS$^3$M). This framework is engineered to make accurate forecasts for such time series while adeptly identifying the irregular regimes hidden within the dynamics. These identifications not only have significant economic ramifications but also contribute to a deeper understanding of the underlying phenomena. In DS$^3$M, the architecture employs discrete latent variables to represent regimes and continuous latent variables to account for random driving factors. By melding a Recurrent Neural Network (RNN) with a nonlinear Switching State Space Model (SSSM), we manage to capture the nonlinear dependencies and irregular regime-switching behaviors, governed by a Markov chain and parameterized using multilayer perceptrons. We validate the effectiveness and regime identification capabilities of DS$^3$M through short- and long-term forecasting tests on a wide array of simulated and real-world datasets, spanning sectors such as healthcare, economics, traffic, meteorology, and energy. Experimental results reveal that DS$^3$M outperforms several state-of-the-art models in terms of forecasting accuracy, while providing meaningful regime identifications.
This paper presents an efficient variational inference framework for deriving a family of structured gaussian process regression network (SGPRN) models. The key idea is to incorporate auxiliary inducing variables in latent functions and jointly treats both the distributions of the inducing variables and hyper-parameters as variational parameters. Then we propose structured variable distributions and marginalize latent variables, which enables the decomposability of a tractable variational lower bound and leads to stochastic optimization. Our inference approach is able to model data in which outputs do not share a common input set with a computational complexity independent of the size of the inputs and outputs and thus easily handle datasets with missing values. We illustrate the performance of our method on synthetic data and real datasets and show that our model generally provides better imputation results on missing data than the state-of-the-art. We also provide a visualization approach for time-varying correlation across outputs in electrocoticography data and those estimates provide insight to understand the neural population dynamics.
This paper is concerned about a learning algorithm for a probabilistic model of spiking neural networks (SNNs). Jimenez Rezende & Gerstner (2014) proposed a stochastic variational inference algorithm to train SNNs with hidden neurons. The algorithm updates the variational distribution using the score function gradient estimator, whose high variance often impedes the whole learning algorithm. This paper presents an alternative gradient estimator for SNNs based on the path-wise gradient estimator. The main technical difficulty is a lack of a general method to differentiate a realization of an arbitrary point process, which is necessary to derive the path-wise gradient estimator. We develop a differentiable point process, which is the technical highlight of this paper, and apply it to derive the path-wise gradient estimator for SNNs. We investigate the effectiveness of our gradient estimator through numerical simulation.
The main challenge in Bayesian models is to determine the posterior for the model parameters. Already, in models with only one or few parameters, the analytical posterior can only be determined in special settings. In Bayesian neural networks, variational inference is widely used to approximate difficult-to-compute posteriors by variational distributions. Usually, Gaussians are used as variational distributions (Gaussian-VI) which limits the quality of the approximation due to their limited flexibility. Transformation models on the other hand are flexible enough to fit any distribution. Here we present transformation model-based variational inference (TM-VI) and demonstrate that it allows to accurately approximate complex posteriors in models with one parameter and also works in a mean-field fashion for multi-parameter models like neural networks.
We adopt the interpretability offered by a parametric, Hawkes-process-inspired conditional probability mass function for the marks and apply variational inference techniques to derive a general and scalable inferential framework for marked point processes. The framework can handle both exchangeable and non-exchangeable event sequences with minimal tuning and without any pre-training. This contrasts with many parametric and non-parametric state-of-the-art methods that typically require pre-training and/or careful tuning, and can only handle exchangeable event sequences. The framework's competitive computational and predictive performance against other state-of-the-art methods are illustrated through real data experiments. Its attractiveness for large-scale applications is demonstrated through a case study involving all events occurring in an English Premier League season.
Variational auto-encoders (VAEs) are a powerful approach to unsupervised learning. They enable scalable approximate posterior inference in latent-variable models using variational inference (VI). A VAE posits a variational family parameterized by a deep neural network called an encoder that takes data as input. This encoder is shared across all the observations, which amortizes the cost of inference. However the encoder of a VAE has the undesirable property that it maps a given observation and a semantics-preserving transformation of it to different latent representations. This "inconsistency" of the encoder lowers the quality of the learned representations, especially for downstream tasks, and also negatively affects generalization. In this paper, we propose a regularization method to enforce consistency in VAEs. The idea is to minimize the Kullback-Leibler (KL) divergence between the variational distribution when conditioning on the observation and the variational distribution when conditioning on a random semantic-preserving transformation of this observation. This regularization is applicable to any VAE. In our experiments we apply it to four different VAE variants on several benchmark datasets and found it always improves the quality of the learned representations but also leads to better generalization. In particular, when applied to the Nouveau Variational Auto-Encoder (NVAE), our regularization method yields state-of-the-art performance on MNIST and CIFAR-10. We also applied our method to 3D data and found it learns representations of superior quality as measured by accuracy on a downstream classification task.