We provide new connections between two distinct federated learning approaches based on (i) ADMM and (ii) Variational Bayes (VB), and propose new variants by combining their complementary strengths. Specifically, we show that the dual variables in ADMM naturally emerge through the "site" parameters used in VB with isotropic Gaussian covariances. Using this, we derive two versions of ADMM from VB that use flexible covariances and functional regularisation, respectively. Through numerical experiments, we validate the improvements obtained in performance. The work shows connection between two fields that are believed to be fundamentally different and combines them to improve federated learning.

Variational Learning (VL) has recently gained popularity for training deep neural networks. Part of its empirical success can be explained by theories such as PAC-Bayes bounds, minimum description length and marginal likelihood, but little has been done to unravel the implicit regularization in play. Here, we analyze the implicit regularization of VL through the Edge of Stability (EoS) framework. EoS has previously been used to show that gradient descent can find flat solutions and we extend this result to show that VL can find even flatter solutions. This result is obtained by controlling the shape of the variational posterior as well as the number of posterior samples used during training. The derivation follows in a similar fashion as in the standard EoS literature for deep learning, by first deriving a result for a quadratic problem and then extending it to deep neural networks. We empirically validate these findings on a wide variety of large networks, such as ResNet and ViT, to find that the theoretical results closely match the empirical ones. Ours is the first work to analyze the EoS dynamics of~VL.

Despite recent progress, continual learning still does not match the performance of batch training. To avoid catastrophic forgetting, we need to build compact memory of essential past knowledge, but no clear solution has yet emerged, even for shallow neural networks with just one or two layers. In this paper, we present a new method to build compact memory for logistic regression. Our method is based on a result by Khan and Swaroop [2021] who show the existence of optimal memory for such models. We formulate the search for the optimal memory as Hessian-matching and propose a probabilistic PCA method to estimate them. Our approach can drastically improve accuracy compared to Experience Replay. For instance, on Split-ImageNet, we get 60% accuracy compared to 30% obtained by replay with memory-size equivalent to 0.3% of the data size. Increasing the memory size to 2% further boosts the accuracy to 74%, closing the gap to the batch accuracy of 77.6% on this task. Our work opens a new direction for building compact memory that can also be useful in the future for continual deep learning.

Models trained on different datasets can be merged by a weighted-averaging of their parameters, but why does it work and when can it fail? Here, we connect the inaccuracy of weighted-averaging to mismatches in the gradients and propose a new uncertainty-based scheme to improve the performance by reducing the mismatch. The connection also reveals implicit assumptions in other schemes such as averaging, task arithmetic, and Fisher-weighted averaging. Our new method gives consistent improvements for large language models and vision transformers, both in terms of performance and robustness to hyperparameters.

Conformal prediction (CP) for regression can be challenging, especially when the output distribution is heteroscedastic, multimodal, or skewed. Some of the issues can be addressed by estimating a distribution over the output, but in reality, such approaches can be sensitive to estimation error and yield unstable intervals. Here, we circumvent the challenges by converting regression to a classification problem and then use CP for classification to obtain CP sets for regression. To preserve the ordering of the continuous-output space, we design a new loss function and present necessary modifications to the CP classification techniques. Empirical results on many benchmarks show that this simple approach gives surprisingly good results on many practical problems.

In the current landscape of deep learning research, there is a predominant emphasis on achieving high predictive accuracy in supervised tasks involving large image and language datasets. However, a broader perspective reveals a multitude of overlooked metrics, tasks, and data types, such as uncertainty, active and continual learning, and scientific data, that demand attention. Bayesian deep learning (BDL) constitutes a promising avenue, offering advantages across these diverse settings. This paper posits that BDL can elevate the capabilities of deep learning. It revisits the strengths of BDL, acknowledges existing challenges, and highlights some exciting research avenues aimed at addressing these obstacles. Looking ahead, the discussion focuses on possible ways to combine large-scale foundation models with BDL to unlock their full potential.

We give extensive empirical evidence against the common belief that variational learning is ineffective for large neural networks. We show that an optimizer called Improved Variational Online Newton (IVON) consistently matches or outperforms Adam for training large networks such as GPT-2 and ResNets from scratch. IVON's computational costs are nearly identical to Adam but its predictive uncertainty is better. We show several new use cases of IVON where we improve finetuning and model merging in Large Language Models, accurately predict generalization error, and faithfully estimate sensitivity to data. We find overwhelming evidence that variational learning is effective. Code is available at https://github.com/team-approx-bayes/ivon.

The Bayesian Learning Rule provides a framework for generic algorithm design but can be difficult to use for three reasons. First, it requires a specific parameterization of exponential family. Second, it uses gradients which can be difficult to compute. Third, its update may not always stay on the manifold. We address these difficulties by proposing an extension based on Lie-groups where posteriors are parametrized through transformations of an arbitrary base distribution and updated via the group’s exponential map. This simplifies all three difficulties for many cases, providing flexible parametrizations through group’s action, simple gradient computation through reparameterization, and updates that always stay on the manifold. We use the new learning rule to derive a new algorithm for deep learning with desirable biologically-plausible attributes to learn sparse features. Our work opens a new frontier for the design of new algorithms by exploiting lie-group structures.

Sequential learning with Gaussian processes (GPs) is challenging when access to past data is limited, for example, in continual and active learning. In such cases, errors can accumulate over time due to inaccuracies in the posterior, hyperparameters, and inducing points, making accurate learning challenging. Here, we present a method to keep all such errors in check using the recently proposed dual sparse variational GP. Our method enables accurate inference for generic likelihoods and improves learning by actively building and updating a memory of past data. We demonstrate its effectiveness in several applications involving Bayesian optimization, active learning, and continual learning.

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free $2^\text{nd}$-order optimizers for deep learning in low precision settings.

Understanding model’s sensitivity to its training data is crucial but can also be challenging and costly, especially during training. To simplify such issues, we present the Memory-Perturbation Equation (MPE) which relates model's sensitivity to perturbation in its training data. Derived using Bayesian principles, the MPE unifies existing sensitivity measures, generalizes them to a wide-variety of models and algorithms, and unravels useful properties regarding sensitivities. Our empirical results show that sensitivity estimates obtained during training can be used to faithfully predict generalization on unseen test data. The proposed equation is expected to be useful for future research on robust and adaptive learning.

Neural Processes (NPs) are appealing due to their ability to perform fast adaptation based on a context set. This set is encoded by a latent variable, which is often assumed to follow a simple distribution. However, in real-word settings, the context set may be drawn from richer distributions having multiple modes, heavy tails, etc. In this work, we provide a framework that allows NPs’ latent variable to be given a rich prior defined by a graphical model. These distributional assumptions directly translate into an appropriate aggregation strategy for the context set. Moreover, we describe a message-passing procedure that still allows for end-to-end optimization with stochastic gradients. We demonstrate the generality of our framework by using mixture and Student-t assumptions that yield improvements in function modelling and test-time robustness.

We show that many machine-learning algorithms are specific instances of a single algorithm called the Bayesian learning rule. The rule, derived from Bayesian principles, yields a wide-range of algorithms from fields such as optimization, deep learning, and graphical models. This includes classical algorithms such as ridge regression, Newton's method, and Kalman filter, as well as modern deep-learning algorithms such as stochastic-gradient descent, RMSprop, and Dropout. The key idea in deriving such algorithms is to approximate the posterior using candidate distributions estimated by using natural gradients. Different candidate distributions result in different algorithms and further approximations to natural gradients give rise to variants of those algorithms. Our work not only unifies, generalizes, and improves existing algorithms, but also helps us design new ones.

Marginal-likelihood based model-selection, even though promising, is rarely used in deep learning due to estimation difficulties. Instead, most approaches rely on validation data, which may not be readily available. In this work, we present a scalable marginal-likelihood estimation method to select both hyperparameters and network architectures, based on the training data alone. Some hyperparameters can be estimated online during training, simplifying the procedure. Our marginal-likelihood estimate is based on Laplace’s method and Gauss-Newton approximations to the Hessian, and it outperforms cross-validation and manual tuning on standard regression and image classification datasets, especially in terms of calibration and out-of-distribution detection. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable (e.g., in nonstationary settings).

Natural-gradient descent (NGD) on structured parameter spaces (e.g., low-rank covariances) is computationally challenging due to difficult Fisher-matrix computations. We address this issue by using \emph{local-parameter coordinates} to obtain a flexible and efficient NGD method that works well for a wide-variety of structured parameterizations. We show four applications where our method (1) generalizes the exponential natural evolutionary strategy, (2) recovers existing Newton-like algorithms, (3) yields new structured second-order algorithms, and (4) gives new algorithms to learn covariances of Gaussian and Wishart-based distributions. We show results on a range of problems from deep learning, variational inference, and evolution strategies. Our work opens a new direction for scalable structured geometric methods.

Sparse variational Gaussian process (SVGP) methods are a common choice for non-conjugate Gaussian process inference because of their computational benefits. In this paper, we improve their computational efficiency by using a dual parameterization where each data example is assigned dual parameters, similarly to site parameters used in expectation propagation. Our dual parameterization speeds-up inference using natural gradient descent, and provides a tighter evidence lower bound for hyperparameter learning. The approach has the same memory cost as the current SVGP methods, but it is faster and more accurate.

Humans and animals have a natural ability to quickly adapt to their surroundings, but machine-learning models, when subjected to changes, often require a complete retraining from scratch. We present Knowledge-adaptation priors (K-priors) to reduce the cost of retraining by enabling quick and accurate adaptation for a wide-variety of tasks and models. This is made possible by a combination of weight and function-space priors to reconstruct the gradients of the past, which recovers and generalizes many existing, but seemingly-unrelated, adaptation strategies. Training with simple first-order gradient methods can often recover the exact retrained model to an arbitrary accuracy by choosing a sufficiently large memory of the past data. Empirical results show that adaptation with K-priors achieves performance similar to full retraining, but only requires training on a handful of past examples.

Deep Gaussian Processes (DGPs) are multi-layer, flexible extensions of Gaussian Processes but their training remains challenging. Most existing methods for inference in DGPs use sparse approximation which require optimization over a large number of inducing inputs and their locations across layers. In this paper, we simplify the training by setting the locations to a fixed subset of data and sampling the inducing inputs from a variational distribution. This reduces the trainable parameters and computation cost without any performance degradation, as demonstrated by our empirical results on regression data sets. Our modifications simplify and stabilize DGP training methods while making them amenable to sampling schemes such as leverage score and determinantal point processes.

Exact recovery of tensor decomposition (TD) methods is a desirable property in both unsupervised learning and scientific data analysis. The numerical defects of TD methods, however, limit their practical applications on real-world data. As an alternative, convex tensor decomposition (CTD) was proposed to alleviate these problems, but its exact-recovery property is not properly addressed so far. To this end, we focus on latent convex tensor decomposition (LCTD), a practically widely-used CTD model, and rigorously prove a sufficient condition for its exact-recovery property. Furthermore, we show that such property can be also achieved by a more general model than LCTD. In the new model, we generalize the classic tensor (un-)folding into reshuffling operation, a more flexible mapping to relocate the entries of the matrix into a tensor. Armed with the reshuffling operations and exact-recovery property, we explore a totally novel application for (generalized) LCTD, i.e., image steganography. Experimental results on synthetic data validate our theory, and results on image steganography show that our method outperforms the state-of-the-art methods.

The Bayesian learning rule is a natural-gradient variational inference method, which not only contains many existing learning algorithms as special cases but also enables the design of new algorithms. Unfortunately, when variational parameters lie in an open constraint set, the rule may not satisfy the constraint and requires line-searches which could slow down the algorithm. In this work, we address this issue for positive-definite constraints by proposing an improved rule that naturally handles the constraints. Our modification is obtained by using Riemannian gradient methods, and is valid when the approximation attains a block-coordinate natural parameterization (e.g., Gaussian distributions and their mixtures). Our method outperforms existing methods without any significant increase in computation. Our work makes it easier to apply the rule in the presence of positive-definite constraints in parameter spaces.

Neural networks with binary weights are computation-efficient and hardware-friendly, but their training is challenging because it involves a discrete optimization problem. Surprisingly, ignoring the discrete nature of the problem and using gradient-based methods, such as the Straight-Through Estimator, still works well in practice. This raises the question: are there principled approaches which justify such methods? In this paper, we propose such an approach using the Bayesian learning rule. The rule, when applied to estimate a Bernoulli distribution over the binary weights, results in an algorithm which justifies some of the algorithmic choices made by the previous approaches. The algorithm not only obtains state-of-the-art performance, but also enables uncertainty estimation and continual learning to avoid catastrophic forgetting. Our work provides a principled approach for training binary neural networks which also justifies and extends existing approaches.

Learning from demonstrations can be challenging when the quality of demonstrations is diverse, and even more so when the quality is unknown and there is no additional information to estimate the quality. We propose a new method for imitation learning in such scenarios. We show that simple quality-estimation approaches might fail due to compounding error, and fix this issue by jointly estimating both the quality and reward using a variational approach. Our method is easy to implement within reinforcement-learning frameworks and also achieves state-of-the-art performance on continuous-control benchmarks.Our work enables scalable and data-efficient imitation learning under more realistic settings than before.

Continually learning new skills is important for intelligent systems, yet standard deep learning methods suffer from catastrophic forgetting of the past. Recent works address this with weight regularisation. Functional regularisation, although computationally expensive, is expected to perform better, but rarely does so in practice. In this paper, we fix this issue by using a new functional-regularisation approach that utilises a few memorable past examples crucial to avoid forgetting. By using a Gaussian Process formulation of deep networks, our approach enables training in weight-space while identifying both the memorable past and a functional prior. Our method achieves state-of-the-art performance on standard benchmarks and opens a new direction for life-long learning where regularisation and memory-based methods are naturally combined.

Bayesian inference provides an attractive online-learning framework to analyze sequential data, and offers generalization guarantees which hold even with model mismatch and adversaries. Unfortunately, exact Bayesian inference is rarely feasible in practice and approximation methods are usually employed, but do such methods preserve the generalization properties of Bayesian inference ? In this paper, we show that this is indeed the case for some variational inference (VI) algorithms. We consider a few existing online, tempered VI algorithms, as well as a new algorithm, and derive their generalization bounds. Our theoretical result relies on the convexity of the variational objective, but we argue that the result should hold more generally and present empirical evidence in support of this. Our work in this paper presents theoretical justifications in favor of online algorithms relying on approximate Bayesian methods.

Natural-gradient methods enable fast and simple algorithms for variational inference, but due to computational difficulties, their use is mostly limited to minimal exponential-family (EF) approximations. In this paper, we extend their application to estimate structured approximations such as mixtures of EF distributions. Such approximations can fit complex, multimodal posterior distributions and are generally more accurate than unimodal EF approximations. By using a minimal conditional-EF representation of such approximations, we derive simple natural-gradient updates. Our empirical results demonstrate a faster convergence of our natural-gradient method compared to black-box gradient-based methods. Our work expands the scope of natural gradients for Bayesian inference and makes them more widely applicable than before.

Inference in Gaussian process (GP) models is computationally challenging for large data, and often difficult to approximate with a small number of inducing points. We explore an alternative approximation that employs stochastic inference networks for a flexible inference. Unfortunately, for such networks, minibatch training is difficult to be able to learn meaningful correlations over function outputs for a large dataset. We propose an algorithm that enables such training by tracking a stochastic, functional mirror-descent algorithm. At each iteration, this only requires considering a finite number of input locations, resulting in a scalable and easy-to-implement algorithm. Empirical results show comparable and, sometimes, superior performance to existing sparse variational GP methods.

Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that certain Gaussian posterior approximations for Bayesian DNNs are equivalent to GP posteriors. This enables us to relate solutions and iterations of a deep-learning algorithm to GP inference. As a result, we can obtain a GP kernel and a nonlinear feature map while training a DNN. Surprisingly, the resulting kernel is the neural tangent kernel. We show kernels obtained on real datasets and demonstrate the use of the GP marginal likelihood to tune hyperparameters of DNNs. Our work aims to facilitate further research on combining DNNs and GPs in practical settings.

Bayesian methods promise to fix many shortcomings of deep learning, but they are impractical and rarely match the performance of standard methods, let alone improve them. In this paper, we demonstrate practical training of deep networks with natural-gradient variational inference. By applying techniques such as batch normalisation, data augmentation, and distributed training, we achieve similar performance in about the same number of epochs as the Adam optimiser, even on large datasets such as ImageNet. Importantly, the benefits of Bayesian principles are preserved: predictive probabilities are well-calibrated, uncertainties on out-of-distribution data are improved, and continual-learning performance is boosted. This work enables practical deep learning while preserving benefits of Bayesian principles. A PyTorch implementation is available as a plug-and-play optimiser.

Actor-critic methods can achieve incredible performance on difficult reinforcement learning problems, but they are also prone to instability. This is partly due to the interaction between the actor and critic during learning, e.g., an inaccurate step taken by one of them might adversely affect the other and destabilize the learning. To avoid such issues, we propose to regularize the learning objective of the actor by penalizing the temporal difference (TD) error of the critic. This improves stability by avoiding large steps in the actor update whenever the critic is highly inaccurate. The resulting method, which we call the TD-regularized actor-critic method, is a simple plug-and-play approach to improve stability and overall performance of the actor-critic methods. Evaluations on standard benchmarks confirm this. Source code can be found at https://github.com/sparisi/td-reg .

Analyzing the underlying structure of multiple time-sequences provides insights into the understanding of social networks and human activities. In this work, we present the Bayesian nonparametric Poisson process allocation (BaNPPA), a latent-function model for time-sequences, which automatically infers the number of latent functions. We model the intensity of each sequence as an infinite mixture of latent functions, each of which is obtained using a function drawn from a Gaussian process. We show that a technical challenge for the inference of such mixture models is the unidentifiability of the weights of the latent functions. We propose to cope with the issue by regulating the volume of each latent function within a variational inference algorithm. Our algorithm is computationally efficient and scales well to large data sets. We demonstrate the usefulness of our proposed model through experiments on both synthetic and real-world data sets.

Uncertainty computation in deep learning is essential to design robust and reliable systems. Variational inference (VI) is a promising approach for such computation, but requires more effort to implement and execute compared to maximum-likelihood methods. In this paper, we propose new natural-gradient algorithms to reduce such efforts for Gaussian mean-field VI. Our algorithms can be implemented within the Adam optimizer by perturbing the network weights during gradient evaluations, and uncertainty estimates can be cheaply obtained by using the vector that adapts the learning rate. This requires lower memory, computation, and implementation effort than existing VI methods, while obtaining uncertainty estimates of comparable quality. Our empirical results confirm this and further suggest that the weight-perturbation in our algorithm could be useful for exploration in reinforcement learning and stochastic optimization.

Bayesian inference plays an important role in advancing machine learning, but faces computational challenges when applied to complex models such as deep neural networks. Variational inference circumvents these challenges by formulating Bayesian inference as an optimization problem and solving it using gradient-based optimization. In this paper, we argue in favor of natural-gradient approaches which, unlike their gradient-based counterparts, can improve convergence by exploiting the information geometry of the solutions. We show how to derive fast yet simple natural-gradient updates by using a duality associated with exponential-family distributions. An attractive feature of these methods is that, by using natural-gradients, they are able to extract accurate local approximations for individual model components. We summarize recent results for Bayesian deep learning showing the superiority of natural-gradient approaches over their gradient counterparts.

Uncertainty estimation in large deep-learning models is a computationally challenging task, where it is difficult to form even a Gaussian approximation to the posterior distribution. In such situations, existing methods usually resort to a diagonal approximation of the covariance matrix despite the fact that these matrices are known to give poor uncertainty estimates. To address this issue, we propose a new stochastic, low-rank, approximate natural-gradient (SLANG) method for variational inference in large deep models. Our method estimates a “diagonal plus low-rank” structure based solely on back-propagated gradients of the network log-likelihood. This requires strictly less gradient computations than methods that compute the gradient of the whole variational objective. Empirical evaluations on standard benchmarks confirm that SLANG enables faster and more accurate estimation of uncertainty than mean-field methods, and performs comparably to state-of-the-art methods.

Variational inference is computationally challenging in models that contain both conjugate and non-conjugate terms. Methods specifically designed for conjugate models, even though computationally efficient, find it difficult to deal with non-conjugate terms. On the other hand, stochastic-gradient methods can handle the non-conjugate terms but they usually ignore the conjugate structure of the model which might result in slow convergence. In this paper, we propose a new algorithm called Conjugate-computation Variational Inference (CVI) which brings the best of the two worlds together – it uses conjugate computations for the conjugate terms and employs stochastic gradients for the rest. We derive this algorithm by using a stochastic mirror-descent method in the mean-parameter space, and then expressing each gradient step as a variational inference in a conjugate model. We demonstrate our algorithm’s applicability to a large class of models and establish its convergence. Our experimental results show that our method converges much faster than the methods that ignore the conjugate structure of the model.

Permission systems are the main defense that mobile platforms, such as Android and iOS, offer to users to protect their private data from prying apps. However, due to the tension between usability and control, such systems have several limitations that often force users to overshare sensitive data. We address some of these limitations with SmarPer, an advanced permission mechanism for Android. To address the rigidity of current permission systems and their poor matching of users' privacy preferences, SmarPer relies on contextual information and machine learning methods to predict permission decisions at runtime. Note that the goal of SmarPer is to mimic the users' decisions, not to make privacy-preserving decisions per se. Using our SmarPer implementation, we collected 8,521 runtime permission decisions from 41 participants in real conditions. With this unique data set, we show that using an efficient Bayesian linear regression model results in a mean correct classification rate of 80% (±3%). This represents a mean relative reduction of approximately 50% in the number of incorrect decisions when compared with a user-defined static permission policy, i.e., the model used in current permission systems. SmarPer also focuses on the suboptimal trade-off between privacy and utility, instead of only "allow" or "deny" type of decisions, SmarPer also offers an "obfuscate" option where users can still obtain utility by revealing partial information to apps. We implemented obfuscation techniques in SmarPer for different data types and evaluated them during our data collection campaign. Our results show that 73% of the participants found obfuscation useful and it accounted for almost a third of the total number of decisions. In short, we are the first to show, using a large dataset of real in situ permission decisions, that it is possible to learn users' unique decision patterns at runtime using contextual information while supporting data obfuscation, this is an important step towards automating the management of permissions in smartphones.

We consider online predictions of vote results, where regions across a country vote on an issue under discussion. Such online predictions before and during the day of the vote are useful to media agencies, polling institutes, and political parties, e.g., to identify regions that are crucial in determining the national outcome of a vote. We analyze a unique dataset from Switzerland. The dataset contains 281 votes from 2352 regions over a period of 34 years. We make several contributions towards improving online predictions. First, we show that these votes exhibit a bi-clustering of the vote results, i.e., regions that are spatially close tend to vote similarly, and issues that discuss similar topics show similar global voting patterns. Second, we develop models that can exploit this bi-clustering, as well as the features associated with the votes and regions. Third, we show that, when combining vote results and features together, Bayesian methods are essential to obtaining good performance. Our results show that Bayesian methods give better estimates of the hyperparameters than non-Bayesian methods such as cross-validation. The resulting models generalize well to many different tasks, produce robust predictions, and are easily interpretable.

Learning about users’ utilities from preference, discrete choice or implicit feedback data is of integral importance in e-commerce, targeted advertising and web search. Due to the sparsity and diffuse nature of data, Bayesian approaches hold much promise, yet most prior work does not scale up to realistic data sizes. We shed light on why inference for such settings is computationally difficult for standard machine learning methods, most of which focus on predicting explicit ratings only. To simplify the difficulty, we present a novel expectation maximization algorithm, driven by expectation propagation approximate inference, which scales to very large datasets without requiring strong factorization assumptions. Our utility model uses both latent bilinear collaborative filtering and non-parametric Gaussian process (GP) regression. In experiments on large real-world datasets, our method gives substantially better results than either matrix factorization or GPs in isolation, and converges significantly faster.

Latent Gaussian models (LGMs) are widely used in statistics and machine learning. Bayesian inference in non-conjugate LGM is difficult due to intractable integrals involving the Gaussian prior and non-conjugate likelihoods. Algorithms based on Variational Gaussian (VG) approximations are widely employed since they strike a favorable balance between accuracy, generality, speed, and ease of use. However, the structure of optimization problems associated with them remains poorly understood, and standard solvers take too long to converge. In this paper, we derive a novel dual variational inference approach, which exploits the convexity property of the VG approximations. The implications of our approach is that we obtain an algorithm that solves a convex optimization problem, reduces the number of variational parameters, and converges much faster than previous methods. Using real world data, we demonstrate these advantages on a variety of LGMs including Gaussian process classification and latent Gaussian Markov random fields.

We present a new variational inference algorithm for Gaussian processes with non-conjugate likelihood functions. This includes binary and multi-class classification, as well as ordinal regression. Our method constructs a convex lower bound, which can be optimized by using an efficient fixed point update method. We then show empirically that our new approach is much faster than existing methods without any degradation in performance.

We propose a new variational EM algorithm for fitting factor analysis models with mixed continuous and categorical observations. The algorithm is based on a simple quadratic bound to the log-sum-exp function. In the special case of fully observed binary data, the bound we propose is significantly faster than previous variational methods. We show that EM is significantly more robust in the presence of missing data compared to treating the latent factors as parameters, which is the approach used by exponential family PCA and other related matrix-factorization methods. A further benefit of the variational approach is that it can easily be extended to the case of mixtures of factor analyzers, as we show. We present results on synthetic and real data sets demonstrating several desirable properties of our proposed method.

In this paper we make several contributions towards accelerating approximate Bayesian structural inference for non-decomposable GGMs. Our first contribution is to show how to efficiently compute a BIC or Laplace approximation to the marginal likelihood of non-decomposable graphs using convex methods for precision matrix estimation. This optimization technique can be used as a fast scoring function inside standard Stochastic Local Search (SLS) for generating posterior samples. Our second contribution is a novel framework for efficiently generating large sets of high-quality graph topologies without performing local search. This graph proposal method, which we call Neighborhood Fusion" (NF), samples candidate Markov blankets at each node using sparse regression techniques. Our final contribution is a hybrid method combining the complementary strengths of NF and SLS. Experimental results in structural recovery and prediction tasks demonstrate that NF and hybrid NF/SLS out-perform state-of-the-art local search methods, on both synthetic and real-world datasets, when realistic computational limits are imposed."