When a missing process depends on the missing values themselves, it needs to be explicitly modelled and taken into account while doing likelihood-based inference. We present an approach for building and fitting deep latent variable models (DLVMs) in cases where the missing process is dependent on the missing data. Specifically, a deep neural network enables us to flexibly model the conditional distribution of the missingness pattern given the data. This allows for incorporating prior information about the type of missingness (e.g. self-censoring) into the model. Our inference technique, based on importance-weighted variational inference, involves maximising a lower bound of the joint likelihood. Stochastic gradients of the bound are obtained by using the reparameterisation trick both in latent space and data space. We show on various kinds of data sets and missingness patterns that explicitly modelling the missing process can be invaluable.
Active Inference (AIF) is an emerging framework in the brain sciences which suggests that biological agents act to minimise a variational bound on model evidence. Control-as-Inference (CAI) is a framework within reinforcement learning which casts decision making as a variational inference problem. While these frameworks both consider action selection through the lens of variational inference, their relationship remains unclear. Here, we provide a formal comparison between them and demonstrate that the primary difference arises from how value is incorporated into their respective generative models. In the context of this comparison, we highlight several ways in which these frameworks can inform one another.
Temporal point process (TPP) models combined with recurrent neural networks provide a powerful framework for modeling continuous-time event data. While such models are flexible, they are inherently sequential and therefore cannot benefit from the parallelism of modern hardware. By exploiting the recent developments in the field of normalizing flows, we design TriTPP -- a new class of non-recurrent TPP models, where both sampling and likelihood computation can be done in parallel. TriTPP matches the flexibility of RNN-based methods but permits orders of magnitude faster sampling. This enables us to use the new model for variational inference in continuous-time discrete-state systems. We demonstrate the advantages of the proposed framework on synthetic and real-world datasets.
The model uncertainty obtained by variational Bayesian inference with Monte Carlo dropout is prone to miscalibration. In this paper, different logit scaling methods are extended to dropout variational inference to recalibrate model uncertainty. Expected uncertainty calibration error (UCE) is presented as a metric to measure miscalibration. The effectiveness of recalibration is evaluated on CIFAR-10/100 and SVHN for recent CNN architectures. Experimental results show that logit scaling considerably reduce miscalibration by means of UCE. Well-calibrated uncertainty enables reliable rejection of uncertain predictions and robust detection of out-of-distribution data.
Continuously learning to solve unseen tasks with limited experience has been extensively pursued in meta-learning and continual learning, but with restricted assumptions such as accessible task distributions, independently and identically distributed tasks, and clear task delineations. However, real-world physical tasks frequently violate these assumptions, resulting in performance degradation. This paper proposes a continual online model-based reinforcement learning approach that does not require pre-training to solve task-agnostic problems with unknown task boundaries. We maintain a mixture of experts to handle nonstationarity, and represent each different type of dynamics with a Gaussian Process to efficiently leverage collected data and expressively model uncertainty. We propose a transition prior to account for the temporal dependencies in streaming data and update the mixture online via sequential variational inference. Our approach reliably handles the task distribution shift by generating new models for never-before-seen dynamics and reusing old models for previously seen dynamics. In experiments, our approach outperforms alternative methods in non-stationary tasks, including classic control with changing dynamics and decision making in different driving scenarios.
We present a probabilistic model where the latent variable respects both the distances and the topology of the modeled data. The model leverages the Riemannian geometry of the generated manifold to endow the latent space with a well-defined stochastic distance measure, which is modeled locally as Nakagami distributions. These stochastic distances are sought to be as similar as possible to observed distances along a neighborhood graph through a censoring process. The model is inferred by variational inference based on observations of pairwise distances. We demonstrate how the new model can encode invariances in the learned manifolds.
When working with multimodal Bayesian posterior distributions, Markov chain Monte Carlo (MCMC) algorithms have difficulty moving between modes, and default variational or mode-based approximate inferences will understate posterior uncertainty. And, even if the most important modes can be found, it is difficult to evaluate their relative weights in the posterior. Here we propose an approach using parallel runs of MCMC, variational, or mode-based inference to hit as many modes or separated regions as possible and then combine these using Bayesian stacking, a scalable method for constructing a weighted average of distributions. The result from stacking efficiently samples from multimodal posterior distribution, minimizes cross validation prediction error, and represents the posterior uncertainty better than variational inference, but it is not necessarily equivalent, even asymptotically, to fully Bayesian inference. We present theoretical consistency with an example where the stacked inference approximates the true data generating process from the misspecified model and a non-mixing sampler, from which the predictive performance is better than full Bayesian inference, hence the multimodality can be considered a blessing rather than a curse under model misspecification. We demonstrate practical implementation in several model families: latent Dirichlet allocation, Gaussian process regression, hierarchical regression, horseshoe variable selection, and neural networks.
A popular theory of perceptual processing holds that the brain learns both a generative model of the world and a paired recognition model using variational Bayesian inference. Most hypotheses of how the brain might learn these models assume that neurons in a population are conditionally independent given their common inputs. This simplification is likely not compatible with the type of local recurrence observed in the brain. Seeking an alternative that is compatible with complex inter-dependencies yet consistent with known biology, we argue here that the cortex may learn with an adversarial algorithm. Many observable symptoms of this approach would resemble known neural phenomena, including wake/sleep cycles and oscillations that vary in magnitude with surprise, and we describe how further predictions could be tested. We illustrate the idea on recurrent neural networks trained to model image and video datasets. This framework for learning brings variational inference closer to neuroscience and yields multiple testable hypotheses.
Recent research has seen several advances relevant to black-box VI, but the current state of automatic posterior inference is unclear. One such advance is the use of normalizing flows to define flexible posterior densities for deep latent variable models. Another direction is the integration of Monte-Carlo methods to serve two purposes; first, to obtain tighter variational objectives for optimization, and second, to define enriched variational families through sampling. However, both flows and variational Monte-Carlo methods remain relatively unexplored for black-box VI. Moreover, on a pragmatic front, there are several optimization considerations like step-size scheme, parameter initialization, and choice of gradient estimators, for which there are no clear guidance in the existing literature. In this paper, we postulate that black-box VI is best addressed through a careful combination of numerous algorithmic components. We evaluate components relating to optimization, flows, and Monte-Carlo methods on a benchmark of 30 models from the Stan model library. The combination of these algorithmic components significantly advances the state-of-the-art "out of the box" variational inference.
We solve the problem of 6-DoF localisation and 3D dense reconstruction in spatial environments as approximate Bayesian inference in a deep state-space model. Our approach leverages both learning and domain knowledge from multiple-view geometry and rigid-body dynamics. This results in an expressive predictive model of the world, often missing in current state-of-the-art visual SLAM solutions. The combination of variational inference, neural networks and a differentiable raycaster ensures that our model is amenable to end-to-end gradient-based optimisation. We evaluate our approach on realistic unmanned aerial vehicle flight data, nearing the performance of state-of-the-art visual-inertial odometry systems. We demonstrate the applicability of the model to generative prediction and planning.
We examine the problem of controlling divergences for latent space regularisation in variational autoencoders. Specifically, when aiming to reconstruct example $x\in\mathbb{R}^{m}$ via latent space $z\in\mathbb{R}^{n}$ ($n\leq m$), while balancing this against the need for generalisable latent representations. We present a regularisation mechanism based on the skew-geometric Jensen-Shannon divergence $\left(\textrm{JS}^{\textrm{G}_{\alpha}}\right)$. We find a variation in $\textrm{JS}^{\textrm{G}_{\alpha}}$, motivated by limiting cases, which leads to an intuitive interpolation between forward and reverse KL in the space of both distributions and divergences. We motivate its potential benefits for VAEs through low-dimensional examples, before presenting quantitative and qualitative results. Our experiments demonstrate that skewing our variant of $\textrm{JS}^{\textrm{G}_{\alpha}}$, in the context of $\textrm{JS}^{\textrm{G}_{\alpha}}$-VAEs, leads to better reconstruction and generation when compared to several baseline VAEs. Our approach is entirely unsupervised and utilises only one hyperparameter which can be easily interpreted in latent space.
Density deconvolution is the task of estimating a probability density function given only noise-corrupted samples. We can fit a Gaussian mixture model to the underlying density by maximum likelihood if the noise is normally distributed, but would like to exploit the superior density estimation performance of normalizing flows and allow for arbitrary noise distributions. Since both adjustments lead to an intractable likelihood, we resort to amortized variational inference. We demonstrate some problems involved in this approach, however, experiments on real data demonstrate that flows can already out-perform Gaussian mixtures for density deconvolution.
Despite their popularity, to date, the application of normalizing flows on categorical data stays limited. The current practice of using dequantization to map discrete data to a continuous space is inapplicable as categorical data has no intrinsic order. Instead, categorical data have complex and latent relations that must be inferred, like the synonymy between words. In this paper, we investigate \emph{Categorical Normalizing Flows}, that is normalizing flows for categorical data. By casting the encoding of categorical data in continuous space as a variational inference problem, we jointly optimize the continuous representation and the model likelihood. Using a factorized decoder, we introduce an inductive bias to model any interactions in the normalizing flow. As a consequence, we do not only simplify the optimization compared to having a joint decoder, but also make it possible to scale up to a large number of categories that is currently impossible with discrete normalizing flows. Based on Categorical Normalizing Flows, we propose GraphCNF a permutation-invariant generative model on graphs. GraphCNF implements a three step approach modeling the nodes, edges and adjacency matrix stepwise to increase efficiency. On molecule generation, GraphCNF outperforms both one-shot and autoregressive flow-based state-of-the-art.
Deep Generative Networks (DGNs) with probabilistic modeling of their output and latent space are currently trained via Variational Autoencoders (VAEs). In the absence of a known analytical form for the posterior and likelihood expectation, VAEs resort to approximations, including (Amortized) Variational Inference (AVI) and Monte-Carlo (MC) sampling. We exploit the Continuous Piecewise Affine (CPA) property of modern DGNs to derive their posterior and marginal distributions as well as the latter's first moments. These findings enable us to derive an analytical Expectation-Maximization (EM) algorithm that enables gradient-free DGN learning. We demonstrate empirically that EM training of DGNs produces greater likelihood than VAE training. Our findings will guide the design of new VAE AVI that better approximate the true posterior and open avenues to apply standard statistical tools for model comparison, anomaly detection, and missing data imputation.
We study PAC-Bayesian generalization bounds for Multilayer Perceptrons (MLPs) with the cross entropy loss. Above all, we introduce probabilistic explanations for MLPs in two aspects: (i) MLPs formulate a family of Gibbs distributions, and (ii) minimizing the cross-entropy loss for MLPs is equivalent to Bayesian variational inference, which establish a solid probabilistic foundation for studying PAC-Bayesian bounds on MLPs. Furthermore, based on the Evidence Lower Bound (ELBO), we prove that MLPs with the cross entropy loss inherently guarantee PAC- Bayesian generalization bounds, and minimizing PAC-Bayesian generalization bounds for MLPs is equivalent to maximizing the ELBO. Finally, we validate the proposed PAC-Bayesian generalization bound on benchmark datasets.
Neural Stochastic Differential Equations (NSDEs) model the drift and diffusion functions of a stochastic process as neural networks. While NSDEs are known to make accurate predictions, their uncertainty quantification properties have been remained unexplored so far. We report the empirical finding that obtaining well-calibrated uncertainty estimations from NSDEs is computationally prohibitive. As a remedy, we develop a computationally affordable deterministic scheme which accurately approximates the transition kernel, when dynamics is governed by a NSDE. Our method introduces a bidimensional moment matching algorithm: vertical along the neural net layers and horizontal along the time direction, which benefits from an original combination of effective approximations. Our deterministic approximation of the transition kernel is applicable to both training and prediction. We observe in multiple experiments that the uncertainty calibration quality of our method can be matched by Monte Carlo sampling only after introducing high computational cost. Thanks to the numerical stability of deterministic training, our method also improves prediction accuracy.
The goal of bayesian approach used in variational inference is to minimize the KL divergence between variational distribution and unknown posterior distribution. This is done by maximizing the Evidence Lower Bound (ELBO). A neural network is used to parametrize these distributions using Stochastic Gradient Descent. This work extends the work done by others by deriving the variational inference models. We show how SGD can be applied on bayesian neural networks by gradient estimation techniques. For validation, we have tested our model on 5 UCI datasets and the metrics chosen for evaluation are Root Mean Square Error (RMSE) error and negative log likelihood. Our work considerably beats the previous state of the art approaches for regression using bayesian neural networks.
A common problem in neuroscience is to elucidate the collective neural representations of behaviorally important variables such as head direction, spatial location, upcoming movements, or mental spatial transformations. Often, these latent variables are internal constructs not directly accessible to the experimenter. Here, we propose a new probabilistic latent variable model to simultaneously identify the latent state and the way each neuron contributes to its representation in an unsupervised way. In contrast to previous models which assume Euclidean latent spaces, we embrace the fact that latent states often belong to symmetric manifolds such as spheres, tori, or rotation groups of various dimensions. We therefore propose the manifold Gaussian process latent variable model (mGPLVM), where neural responses arise from (i) a shared latent variable living on a specific manifold, and (ii) a set of non-parametric tuning curves determining how each neuron contributes to the representation. Cross-validated comparisons of models with different topologies can be used to distinguish between candidate manifolds, and variational inference enables quantification of uncertainty. We demonstrate the validity of the approach on several synthetic datasets, as well as on calcium recordings from the ellipsoid body of Drosophila melanogaster and extracellular recordings from the mouse anterodorsal thalamic nucleus. These circuits are both known to encode head direction, and mGPLVM correctly recovers the ring topology expected from neural populations representing a single angular variable.
In this work, we introduce kernels with random Fourier features in the meta-learning framework to leverage their strong few-shot learning ability. We propose meta variational random features (MetaVRF) to learn adaptive kernels for the base-learner, which is developed in a latent variable model by treating the random feature basis as the latent variable. We formulate the optimization of MetaVRF as a variational inference problem by deriving an evidence lower bound under the meta-learning framework. To incorporate shared knowledge from related tasks, we propose a context inference of the posterior, which is established by an LSTM architecture. The LSTM-based inference network can effectively integrate the context information of previous tasks with task-specific information, generating informative and adaptive features. The learned MetaVRF can produce kernels of high representational power with a relatively low spectral sampling rate and also enables fast adaptation to new tasks. Experimental results on a variety of few-shot regression and classification tasks demonstrate that MetaVRF delivers much better, or at least competitive, performance compared to existing meta-learning alternatives.
A spectral mixture (SM) kernel is a flexible kernel used to model any stationary covariance function. Although it is useful in modeling data, the learning of the SM kernel is generally difficult because optimizing a large number of parameters for the SM kernel typically induces an over-fitting, particularly when a gradient-based optimization is used. Also, a longer training time is required. To improve the training, we propose an approximate Bayesian inference for the SM kernel. Specifically, we employ the variational distribution of the spectral points to approximate SM kernel with a random Fourier feature. We optimize the variational parameters by applying a sampling-based variational inference to the derived evidence lower bound (ELBO) estimator constructed from the approximate kernel. To improve the inference, we further propose two additional strategies: (1) a sampling strategy of spectral points to estimate the ELBO estimator reliably and thus its associated gradient, and (2) an approximate natural gradient to accelerate the convergence of the parameters. The proposed inference combined with two strategies accelerates the convergence of the parameters and leads to better optimal parameters.