Federated learning methods enable model training across distributed data sources without data leaving their original locations and have gained increasing interest in various fields. However, existing approaches are limited, excluding many structured probabilistic models. We present a general and elegant solution based on structured variational inference, widely used in Bayesian machine learning, adapted for the federated setting. Additionally, we provide a communication-efficient variant analogous to the canonical FedAvg algorithm. The proposed algorithms' effectiveness is demonstrated, and their performance is compared with hierarchical Bayesian neural networks and topic models.
Bayesian Neural Networks (BNNs) provide a probabilistic interpretation for deep learning models by imposing a prior distribution over model parameters and inferring a posterior distribution based on observed data. The model sampled from the posterior distribution can be used for providing ensemble predictions and quantifying prediction uncertainty. It is well-known that deep learning models with lower sharpness have better generalization ability. However, existing posterior inferences are not aware of sharpness/flatness in terms of formulation, possibly leading to high sharpness for the models sampled from them. In this paper, we develop theories, the Bayesian setting, and the variational inference approach for the sharpness-aware posterior. Specifically, the models sampled from our sharpness-aware posterior, and the optimal approximate posterior estimating this sharpness-aware posterior, have better flatness, hence possibly possessing higher generalization ability. We conduct experiments by leveraging the sharpness-aware posterior with state-of-the-art Bayesian Neural Networks, showing that the flat-seeking counterparts outperform their baselines in all metrics of interest.
The advances in variational inference are providing promising paths in Bayesian estimation problems. These advances make variational phylogenetic inference an alternative approach to Markov Chain Monte Carlo methods for approximating the phylogenetic posterior. However, one of the main drawbacks of such approaches is modelling the prior through fixed distributions, which could bias the posterior approximation if they are distant from the current data distribution. In this paper, we propose an approach and an implementation framework to relax the rigidity of the prior densities by learning their parameters using a gradient-based method and a neural network-based parameterization. We applied this approach for branch lengths and evolutionary parameters estimation under several Markov chain substitution models. The results of performed simulations show that the approach is powerful in estimating branch lengths and evolutionary model parameters. They also show that a flexible prior model could provide better results than a predefined prior model. Finally, the results highlight that using neural networks improves the initialization of the optimization of the prior density parameters.
Traditional neural networks are simple to train but they typically produce overconfident predictions. In contrast, Bayesian neural networks provide good uncertainty quantification but optimizing them is time consuming due to the large parameter space. This paper proposes to combine the advantages of both approaches by performing Variational Inference in the Final layer Output space (VIFO), because the output space is much smaller than the parameter space. We use neural networks to learn the mean and the variance of the probabilistic output. Using the Bayesian formulation we incorporate collapsed variational inference with VIFO which significantly improves the performance in practice. On the other hand, like standard, non-Bayesian models, VIFO enjoys simple training and one can use Rademacher complexity to provide risk bounds for the model. Experiments show that VIFO provides a good tradeoff in terms of run time and uncertainty quantification, especially for out of distribution data.
Noisy-OR Bayesian Networks (BNs) are a family of probabilistic graphical models which express rich statistical dependencies in binary data. Variational inference (VI) has been the main method proposed to learn noisy-OR BNs with complex latent structures (Jaakkola & Jordan, 1999; Ji et al., 2020; Buhai et al., 2020). However, the proposed VI approaches either (a) use a recognition network with standard amortized inference that cannot induce ``explaining-away''; or (b) assume a simple mean-field (MF) posterior which is vulnerable to bad local optima. Existing MF VI methods also update the MF parameters sequentially which makes them inherently slow. In this paper, we propose parallel max-product as an alternative algorithm for learning noisy-OR BNs with complex latent structures and we derive a fast stochastic training scheme that scales to large datasets. We evaluate both approaches on several benchmarks where VI is the state-of-the-art and show that our method (a) achieves better test performance than Ji et al. (2020) for learning noisy-OR BNs with hierarchical latent structures on large sparse real datasets; (b) recovers a higher number of ground truth parameters than Buhai et al. (2020) from cluttered synthetic scenes; and (c) solves the 2D blind deconvolution problem from Lazaro-Gredilla et al. (2021) and variant - including binary matrix factorization - while VI catastrophically fails and is up to two orders of magnitude slower.
Short-term forecasting of solar photovoltaic energy (PV) production is important for powerplant management. Ideally these forecasts are equipped with error bars, so that downstream decisions can account for uncertainty. To produce predictions with error bars in this setting, we consider Gaussian processes (GPs) for modelling and predicting solar photovoltaic energy production in the UK. A standard application of GP regression on the PV timeseries data is infeasible due to the large data size and non-Gaussianity of PV readings. However, this is made possible by leveraging recent advances in scalable GP inference, in particular, by using the state-space form of GPs, combined with modern variational inference techniques. The resulting model is not only scalable to large datasets but can also handle continuous data streams via Kalman filtering.
Generative flow networks (GFlowNets) are amortized variational inference algorithms that are trained to sample from unnormalized target distributions over compositional objects. A key limitation of GFlowNets until this time has been that they are restricted to discrete spaces. We present a theory for generalized GFlowNets, which encompasses both existing discrete GFlowNets and ones with continuous or hybrid state spaces, and perform experiments with two goals in mind. First, we illustrate critical points of the theory and the importance of various assumptions. Second, we empirically demonstrate how observations about discrete GFlowNets transfer to the continuous case and show strong results compared to non-GFlowNet baselines on several previously studied tasks. This work greatly widens the perspectives for the application of GFlowNets in probabilistic inference and various modeling settings.
Generative graph self-supervised learning (SSL) aims to learn node representations by reconstructing the input graph data. However, most existing methods focus on unsupervised learning tasks only and very few work has shown its superiority over the state-of-the-art graph contrastive learning (GCL) models, especially on the classification task. While a very recent model has been proposed to bridge the gap, its performance on unsupervised learning tasks is still unknown. In this paper, to comprehensively enhance the performance of generative graph SSL against other GCL models on both unsupervised and supervised learning tasks, we propose the SeeGera model, which is based on the family of self-supervised variational graph auto-encoder (VGAE). Specifically, SeeGera adopts the semi-implicit variational inference framework, a hierarchical variational framework, and mainly focuses on feature reconstruction and structure/feature masking. On the one hand, SeeGera co-embeds both nodes and features in the encoder and reconstructs both links and features in the decoder. Since feature embeddings contain rich semantic information on features, they can be combined with node embeddings to provide fine-grained knowledge for feature reconstruction. On the other hand, SeeGera adds an additional layer for structure/feature masking to the hierarchical variational framework, which boosts the model generalizability. We conduct extensive experiments comparing SeeGera with 9 other state-of-the-art competitors. Our results show that SeeGera can compare favorably against other state-of-the-art GCL methods in a variety of unsupervised and supervised learning tasks.
Model-based methods have recently shown great potential for off-policy evaluation (OPE); offline trajectories induced by behavioral policies are fitted to transitions of Markov decision processes (MDPs), which are used to rollout simulated trajectories and estimate the performance of policies. Model-based OPE methods face two key challenges. First, as offline trajectories are usually fixed, they tend to cover limited state and action space. Second, the performance of model-based methods can be sensitive to the initialization of their parameters. In this work, we propose the variational latent branching model (VLBM) to learn the transition function of MDPs by formulating the environmental dynamics as a compact latent space, from which the next states and rewards are then sampled. Specifically, VLBM leverages and extends the variational inference framework with the recurrent state alignment (RSA), which is designed to capture as much information underlying the limited training data, by smoothing out the information flow between the variational (encoding) and generative (decoding) part of VLBM. Moreover, we also introduce the branching architecture to improve the model's robustness against randomly initialized model weights. The effectiveness of the VLBM is evaluated on the deep OPE (DOPE) benchmark, from which the training trajectories are designed to result in varied coverage of the state-action space. We show that the VLBM outperforms existing state-of-the-art OPE methods in general.
Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules and constitute an important step towards the modeling of distributions of many interacting molecules.
In recent years, particle-based variational inference (ParVI) methods such as Stein variational gradient descent (SVGD) have grown in popularity as scalable methods for Bayesian inference. Unfortunately, the properties of such methods invariably depend on hyperparameters such as the learning rate, which must be carefully tuned by the practitioner in order to ensure convergence to the target measure at a suitable rate. In this paper, we introduce a suite of new particle-based methods for scalable Bayesian inference based on coin betting, which are entirely learning-rate free. We illustrate the performance of our approach on a range of numerical examples, including several high-dimensional models and datasets, demonstrating comparable performance to other ParVI algorithms with no need to tune a learning rate.
Recent developments using End-to-End Deep Learning models have been shown to have near or better performance than state of the art Recurrent Neural Networks (RNNs) on Automatic Speech Recognition tasks. These models tend to be lighter weight and require less training time than traditional RNN-based approaches. However, these models take frequentist approach to weight training. In theory, network weights are drawn from a latent, intractable probability distribution. We introduce BayesSpeech for end-to-end Automatic Speech Recognition. BayesSpeech is a Bayesian Transformer Network where these intractable posteriors are learned through variational inference and the local reparameterization trick without recurrence. We show how the introduction of variance in the weights leads to faster training time and near state-of-the-art performance on LibriSpeech-960.
Topic models and all their variants analyse text by learning meaningful representations through word co-occurrences. As pointed out by Williamson et al. (2010), such models implicitly assume that the probability of a topic to be active and its proportion within each document are positively correlated. This correlation can be strongly detrimental in the case of documents created over time, simply because recent documents are likely better described by new and hence rare topics. In this work we leverage recent advances in neural variational inference and present an alternative neural approach to the dynamic Focused Topic Model. Indeed, we develop a neural model for topic evolution which exploits sequences of Bernoulli random variables in order to track the appearances of topics, thereby decoupling their activities from their proportions. We evaluate our model on three different datasets (the UN general debates, the collection of NeurIPS papers, and the ACL Anthology dataset) and show that it (i) outperforms state-of-the-art topic models in generalization tasks and (ii) performs comparably to them on prediction tasks, while employing roughly the same number of parameters, and converging about two times faster. Source code to reproduce our experiments is available online.
Recently, graph-based models designed for downstream tasks have significantly advanced research on graph neural networks (GNNs). GNN baselines based on neural message-passing mechanisms such as GCN and GAT perform worse as the network deepens. Therefore, numerous GNN variants have been proposed to tackle this performance degradation problem, including many deep GNNs. However, a unified framework is still lacking to connect these existing models and interpret their effectiveness at a high level. In this work, we focus on deep GNNs and propose a novel view for understanding them. We establish a theoretical framework via inference on a probabilistic graphical model. Given the fixed point equation (FPE) derived from the variational inference on the Markov random fields, the deep GNNs, including JKNet, GCNII, DGCN, and the classical GNNs, such as GCN, GAT, and APPNP, can be regarded as different approximations of the FPE. Moreover, given this framework, more accurate approximations of FPE are brought, guiding us to design a more powerful GNN: coupling graph neural network (CoGNet). Extensive experiments are carried out on citation networks and natural language processing downstream tasks. The results demonstrate that the CoGNet outperforms the SOTA models.
The Gaussian process state-space model (GPSSM) has garnered considerable attention over the past decade. However, the standard GP with a preliminary kernel, such as the squared exponential kernel or Mat\'{e}rn kernel, that is commonly used in GPSSM studies, limits the model's representation power and substantially restricts its applicability to complex scenarios. To address this issue, we propose a new class of probabilistic state-space models called TGPSSMs, which leverage a parametric normalizing flow to enrich the GP priors in the standard GPSSM, enabling greater flexibility and expressivity. Additionally, we present a scalable variational inference algorithm that offers a flexible and optimal structure for the variational distribution of latent states. The proposed algorithm is interpretable and computationally efficient due to the sparse GP representation and the bijective nature of normalizing flow. Moreover, we incorporate a constrained optimization framework into the algorithm to enhance the state-space representation capabilities and optimize the hyperparameters, leading to superior learning and inference performance. Experimental results on synthetic and real datasets corroborate that the proposed TGPSSM outperforms several state-of-the-art methods. The accompanying source code is available at \url{https://github.com/zhidilin/TGPSSM}.
Variational inference is an approximation framework for Bayesian inference that seeks to improve quantified uncertainty in predictions by optimizing a simplified distribution over parameters to stand in for the full posterior. Capturing model variations that remain consistent with training data enables more robust predictions by reducing parameter sensitivity. This work introduces a fixed-point optimization for variational inference that is applicable when every feasible log density can be expressed as a linear combination of functions from a given basis. In such cases, the optimizer becomes a fixed-point of projective integral updates. When the basis spans univariate quadratics in each parameter, feasible densities are Gaussian and the projective integral updates yield quasi-Newton variational Bayes (QNVB). Other bases and updates are also possible. As these updates require high-dimensional integration, this work first proposes an efficient quasirandom quadrature sequence for mean-field distributions. Each iterate of the sequence contains two evaluation points that combine to correctly integrate all univariate quadratics and, if the mean-field factors are symmetric, all univariate cubics. More importantly, averaging results over short subsequences achieves periodic exactness on a much larger space of multivariate quadratics. The corresponding variational updates require 4 loss evaluations with standard (not second-order) backpropagation to eliminate error terms from over half of all multivariate quadratic basis functions. This integration technique is motivated by first proposing stochastic blocked mean-field quadratures, which may be useful in other contexts. A PyTorch implementation of QNVB allows for better control over model uncertainty during training than competing methods. Experiments demonstrate superior generalizability for multiple learning problems and architectures.
Variational inference has been widely used in machine learning literature to fit various Bayesian models. In network analysis, this method has been successfully applied to solve the community detection problems. Although these results are promising, their theoretical support is only for relatively dense networks, an assumption that may not hold for real networks. In addition, it has been shown recently that the variational loss surface has many saddle points, which may severely affect its performance, especially when applied to sparse networks. This paper proposes a simple way to improve the variational inference method by hard thresholding the posterior of the community assignment after each iteration. Using a random initialization that correlates with the true community assignment, we show that the proposed method converges and can accurately recover the true community labels, even when the average node degree of the network is bounded. Extensive numerical study further confirms the advantage of the proposed method over the classical variational inference and another state-of-the-art algorithm.
We study the foundations of variational inference, which frames posterior inference as an optimisation problem, for probabilistic programming. The dominant approach for optimisation in practice is stochastic gradient descent. In particular, a variant using the so-called reparameterisation gradient estimator exhibits fast convergence in a traditional statistics setting. Unfortunately, discontinuities, which are readily expressible in programming languages, can compromise the correctness of this approach. We consider a simple (higher-order, probabilistic) programming language with conditionals, and we endow our language with both a measurable and a smoothed (approximate) value semantics. We present type systems which establish technical pre-conditions. Thus we can prove stochastic gradient descent with the reparameterisation gradient estimator to be correct when applied to the smoothed problem. Besides, we can solve the original problem up to any error tolerance by choosing an accuracy coefficient suitably. Empirically we demonstrate that our approach has a similar convergence as a key competitor, but is simpler, faster, and attains orders of magnitude reduction in work-normalised variance.
Visual localization allows autonomous robots to relocalize when losing track of their pose by matching their current observation with past ones. However, ambiguous scenes pose a challenge for such systems, as repetitive structures can be viewed from many distinct, equally likely camera poses, which means it is not sufficient to produce a single best pose hypothesis. In this work, we propose a probabilistic framework that for a given image predicts the arbitrarily shaped posterior distribution of its camera pose. We do this via a novel formulation of camera pose regression using variational inference, which allows sampling from the predicted distribution. Our method outperforms existing methods on localization in ambiguous scenes. Code and data will be released at https://github.com/efreidun/vapor.
Variational inference uses optimization, rather than integration, to approximate the marginal likelihood, and thereby the posterior, in a Bayesian model. Thanks to advances in computational scalability made in the last decade, variational inference is now the preferred choice for many high-dimensional models and large datasets. This tutorial introduces variational inference from the parametric perspective that dominates these recent developments, in contrast to the mean-field perspective commonly found in other introductory texts.