Making predictions robust is an important challenge. A separate challenge in federated learning (FL) is to reduce the number of communication rounds, particularly since doing so reduces performance in heterogeneous data settings. To tackle both issues, we take a Bayesian perspective on the problem of learning a global model. We show how the global predictive posterior can be approximated using client predictive posteriors. This is unlike other works which aggregate the local model space posteriors into the global model space posterior, and are susceptible to high approximation errors due to the posterior's high dimensional multimodal nature. In contrast, our method performs the aggregation on the predictive posteriors, which are typically easier to approximate owing to the low-dimensionality of the output space. We present an algorithm based on this idea, which performs MCMC sampling at each client to obtain an estimate of the local posterior, and then aggregates these in one round to obtain a global ensemble model. Through empirical evaluation on several classification and regression tasks, we show that despite using one round of communication, the method is competitive with other FL techniques, and outperforms them on heterogeneous settings. The code is publicly available at https://github.com/hasanmohsin/FedPredSpace_1Round.
Discrete state spaces represent a major computational challenge to statistical inference, since the computation of normalisation constants requires summation over large or possibly infinite sets, which can be impractical. This paper addresses this computational challenge through the development of a novel generalised Bayesian inference procedure suitable for discrete intractable likelihood. Inspired by recent methodological advances for continuous data, the main idea is to update beliefs about model parameters using a discrete Fisher divergence, in lieu of the problematic intractable likelihood. The result is a generalised posterior that can be sampled from using standard computational tools, such as Markov chain Monte Carlo, circumventing the intractable normalising constant. The statistical properties of the generalised posterior are analysed, with sufficient conditions for posterior consistency and asymptotic normality established. In addition, a novel and general approach to calibration of generalised posteriors is proposed. Applications are presented on lattice models for discrete spatial data and on multivariate models for count data, where in each case the methodology facilitates generalised Bayesian inference at low computational cost.
Bayesian inference for phylogenetics is a gold standard for computing distributions of phylogenies. It faces the challenging problem of. moving throughout the high-dimensional space of trees. However, hyperbolic space offers a low dimensional representation of tree-like data. In this paper, we embed genomic sequences into hyperbolic space and perform hyperbolic Markov Chain Monte Carlo for Bayesian inference. The posterior probability is computed by decoding a neighbour joining tree from proposed embedding locations. We empirically demonstrate the fidelity of this method on eight data sets. The sampled posterior distribution recovers the splits and branch lengths to a high degree. We investigated the effects of curvature and embedding dimension on the Markov Chain's performance. Finally, we discuss the prospects for adapting this method to navigate tree space with gradients.
Machine learning methods on graphs have proven useful in many applications due to their ability to handle generally structured data. The framework of Gaussian Markov Random Fields (GMRFs) provides a principled way to define Gaussian models on graphs by utilizing their sparsity structure. We propose a flexible GMRF model for general graphs built on the multi-layer structure of Deep GMRFs, originally proposed for lattice graphs only. By designing a new type of layer we enable the model to scale to large graphs. The layer is constructed to allow for efficient training using variational inference and existing software frameworks for Graph Neural Networks. For a Gaussian likelihood, close to exact Bayesian inference is available for the latent field. This allows for making predictions with accompanying uncertainty estimates. The usefulness of the proposed model is verified by experiments on a number of synthetic and real world datasets, where it compares favorably to other both Bayesian and deep learning methods.
Calculation of Bayesian posteriors and model evidences typically requires numerical integration. Bayesian quadrature (BQ), a surrogate-model-based approach to numerical integration, is capable of superb sample efficiency, but its lack of parallelisation has hindered its practical applications. In this work, we propose a parallelised (batch) BQ method, employing techniques from kernel quadrature, that possesses an empirically exponential convergence rate. Additionally, just as with Nested Sampling, our method permits simultaneous inference of both posteriors and model evidence. Samples from our BQ surrogate model are re-selected to give a sparse set of samples, via a kernel recombination algorithm, requiring negligible additional time to increase the batch size. Empirically, we find that our approach significantly outperforms the sampling efficiency of both state-of-the-art BQ techniques and Nested Sampling in various real-world datasets, including lithium-ion battery analytics.
Extracting low-dimensional summary statistics from large datasets is essential for efficient (likelihood-free) inference. We characterize different classes of summaries and demonstrate their importance for correctly analysing dimensionality reduction algorithms. We demonstrate that minimizing the expected posterior entropy (EPE) under the prior predictive distribution of the model subsumes many existing methods. They are equivalent to or are special or limiting cases of minimizing the EPE. We offer a unifying framework for obtaining informative summaries, provide concrete recommendations for practitioners, and propose a practical method to obtain high-fidelity summaries whose utility we demonstrate for both benchmark and practical examples.
Bayesian models have many desirable properties, most notable is their ability to generalize from limited data and to properly estimate the uncertainty in their predictions. However, these benefits come at a steep computational cost as Bayesian inference, in most cases, is computationally intractable. One popular approach to alleviate this problem is using a Monte-Carlo estimation with an ensemble of models sampled from the posterior. However, this approach still comes at a significant computational cost, as one needs to store and run multiple models at test time. In this work, we investigate how to best distill an ensemble's predictions using an efficient model. First, we argue that current approaches that simply return distribution over predictions cannot compute important properties, such as the covariance between predictions, which can be valuable for further processing. Second, in many limited data settings, all ensemble members achieve nearly zero training loss, namely, they produce near-identical predictions on the training set which results in sub-optimal distilled models. To address both problems, we propose a novel and general distillation approach, named Functional Ensemble Distillation (FED), and we investigate how to best distill an ensemble in this setting. We find that learning the distilled model via a simple augmentation scheme in the form of mixup augmentation significantly boosts the performance. We evaluated our method on several tasks and showed that it achieves superior results in both accuracy and uncertainty estimation compared to current approaches.
Along with Markov chain Monte Carlo (MCMC) methods, variational inference (VI) has emerged as a central computational approach to large-scale Bayesian inference. Rather than sampling from the true posterior $\pi$, VI aims at producing a simple but effective approximation $\hat \pi$ to $\pi$ for which summary statistics are easy to compute. However, unlike the well-studied MCMC methodology, algorithmic guarantees for VI are still relatively less well-understood. In this work, we propose principled methods for VI, in which $\hat \pi$ is taken to be a Gaussian or a mixture of Gaussians, which rest upon the theory of gradient flows on the Bures--Wasserstein space of Gaussian measures. Akin to MCMC, it comes with strong theoretical guarantees when $\pi$ is log-concave.
Markov decision processes (MDPs) are formal models commonly used in sequential decision-making. MDPs capture the stochasticity that may arise, for instance, from imprecise actuators via probabilities in the transition function. However, in data-driven applications, deriving precise probabilities from (limited) data introduces statistical errors that may lead to unexpected or undesirable outcomes. Uncertain MDPs (uMDPs) do not require precise probabilities but instead use so-called uncertainty sets in the transitions, accounting for such limited data. Tools from the formal verification community efficiently compute robust policies that provably adhere to formal specifications, like safety constraints, under the worst-case instance in the uncertainty set. We continuously learn the transition probabilities of an MDP in a robust anytime-learning approach that combines a dedicated Bayesian inference scheme with the computation of robust policies. In particular, our method (1) approximates probabilities as intervals, (2) adapts to new data that may be inconsistent with an intermediate model, and (3) may be stopped at any time to compute a robust policy on the uMDP that faithfully captures the data so far. Furthermore, our method is capable of adapting to changes in the environment. We show the effectiveness of our approach and compare it to robust policies computed on uMDPs learned by the UCRL2 reinforcement learning algorithm in an experimental evaluation on several benchmarks.
Addressing the challenge of scaling-up epidemiological inference to complex and heterogeneous models, we introduce Poisson Approximate Likelihood (PAL) methods. In contrast to the popular ODE approach to compartmental modelling, in which a large population limit is used to motivate a deterministic model, PALs are derived from approximate filtering equations for finite-population, stochastic compartmental models, and the large population limit drives consistency of maximum PAL estimators. Our theoretical results appear to be the first likelihood-based parameter estimation consistency results which apply to a broad class of partially observed stochastic compartmental models and address the large population limit. PALs are simple to implement, involving only elementary arithmetic operations and no tuning parameters, and fast to evaluate, requiring no simulation from the model and having computational cost independent of population size. Through examples we demonstrate how PALs can be used to: fit an age-structured model of influenza, taking advantage of automatic differentiation in Stan; compare over-dispersion mechanisms in a model of rotavirus by embedding PALs within sequential Monte Carlo; and evaluate the role of unit-specific parameters in a meta-population model of measles.
The computation of Wasserstein gradient direction is essential for posterior sampling problems and scientific computing. The approximation of the Wasserstein gradient with finite samples requires solving a variational problem. We study the variational problem in the family of two-layer networks with squared-ReLU activations, towards which we derive a semi-definite programming (SDP) relaxation. This SDP can be viewed as an approximation of the Wasserstein gradient in a broader function family including two-layer networks. By solving the convex SDP, we obtain the optimal approximation of the Wasserstein gradient direction in this class of functions. Numerical experiments including PDE-constrained Bayesian inference and parameter estimation in COVID-19 modeling demonstrate the effectiveness of the proposed method.
Can non-programmers annotate natural language utterances with complex programs that represent their meaning? We introduce APEL, a framework in which non-programmers select among candidate programs generated by a seed semantic parser (e.g., Codex). Since they cannot understand the candidate programs, we ask them to select indirectly by examining the programs' input-ouput examples. For each utterance, APEL actively searches for a simple input on which the candidate programs tend to produce different outputs. It then asks the non-programmers only to choose the appropriate output, thus allowing us to infer which program is correct and could be used to fine-tune the parser. As a first case study, we recruited human non-programmers to use APEL to re-annotate SPIDER, a text-to-SQL dataset. Our approach achieved the same annotation accuracy as the original expert annotators (75%) and exposed many subtle errors in the original annotations.
Constructive studies on symbol emergence systems seek to investigate computational models that can better explain human language evolution, the creation of symbol systems, and the construction of internal representations. This study provides a new model for emergent communication, which is based on a probabilistic generative model (PGM) instead of a discriminative model based on deep reinforcement learning. We define the Metropolis-Hastings (MH) naming game by generalizing previously proposed models. It is not a referential game with explicit feedback, as assumed by many emergent communication studies. Instead, it is a game based on joint attention without explicit feedback. Mathematically, the MH naming game is proved to be a type of MH algorithm for an integrative PGM that combines two agents that play the naming game. From this viewpoint, symbol emergence is regarded as decentralized Bayesian inference, and semiotic communication is regarded as inter-personal cross-modal inference. This notion leads to the collective predictive coding hypothesis} regarding language evolution and, in general, the emergence of symbols. We also propose the inter-Gaussian mixture model (GMM)+ variational autoencoder (VAE), a deep generative model for emergent communication based on the MH naming game. The model has been validated on MNIST and Fruits 360 datasets. Experimental findings demonstrate that categories are formed from real images observed by agents, and signs are correctly shared across agents by successfully utilizing both of the observations of agents via the MH naming game. Furthermore, scholars verified that visual images were recalled from signs uttered by agents. Notably, emergent communication without supervision and reward feedback improved the performance of the unsupervised representation learning of agents.
Molt is an essential life-history event in birds and many mammals, as maintenance of feathers and fur is critical for survival. Despite this molt remains an understudied life-history event. Non-standard statistical techniques are required to estimate the phenology of molt from observations of plumage or pelage state, and existing molt phenology models have strict sampling requirements which can be difficult to meet under real-world conditions. We present an extended modelling framework that can accommodate features of real-world molt datasets, such re-encounters of individuals, misclassified molt states, and/or molt state-dependent sampling bias. We demonstrate that such features can lead to biased inferences when using existing molt phenology models, and show that our model extensions can improve inferences about molt phenology under a wide range of sampling conditions. We hope that our novel modelling framework removes barriers for modelling molt phenology data from realworld datasets and thereby further facilitates the uptake of appropriate statistical methods for such data. Although we focus on molt, the modelling framework is applicable to other phenological processes which can be recorded using either ordered categories or approximately linear progress scores.
Measuring contributions is a classical problem in cooperative game theory where the Shapley value is the most well-known solution concept. In this paper, we establish the convergence property of the Shapley value in parametric Bayesian learning games where players perform a Bayesian inference using their combined data, and the posterior-prior KL divergence is used as the characteristic function. We show that for any two players, under some regularity conditions, their difference in Shapley value converges in probability to the difference in Shapley value of a limiting game whose characteristic function is proportional to the log-determinant of the joint Fisher information. As an application, we present an online collaborative learning framework that is asymptotically Shapley-fair. Our result enables this to be achieved without any costly computations of posterior-prior KL divergences. Only a consistent estimator of the Fisher information is needed. The effectiveness of our framework is demonstrated with experiments using real-world data.
Semantic-descriptor-based Generalized Zero-Shot Learning (GZSL) poses challenges in recognizing novel classes in the test phase. The development of generative models enables current GZSL techniques to probe further into the semantic-visual link, culminating in a two-stage form that includes a generator and a classifier. However, existing generation-based methods focus on enhancing the generator's effect while neglecting the improvement of the classifier. In this paper, we first analyze of two properties of the generated pseudo unseen samples: bias and homogeneity. Then, we perform variational Bayesian inference to back-derive the evaluation metrics, which reflects the balance of the seen and unseen classes. As a consequence of our derivation, the aforementioned two properties are incorporated into the classifier training as seen-unseen priors via logit adjustment. The Zero-Shot Logit Adjustment further puts semantic-based classifiers into effect in generation-based GZSL. Our experiments demonstrate that the proposed technique achieves state-of-the-art when combined with the basic generator, and it can improve various generative Zero-Shot Learning frameworks. Our codes are available on https://github.com/cdb342/IJCAI-2022-ZLA.
Models of human behavior for prediction and collaboration tend to fall into two categories: ones that learn from large amounts of data via imitation learning, and ones that assume human behavior to be noisily-optimal for some reward function. The former are very useful, but only when it is possible to gather a lot of human data in the target environment and distribution. The advantage of the latter type, which includes Boltzmann rationality, is the ability to make accurate predictions in new environments without extensive data when humans are actually close to optimal. However, these models fail when humans exhibit systematic suboptimality, i.e. when their deviations from optimal behavior are not independent, but instead consistent over time. Our key insight is that systematic suboptimality can be modeled by predicting policies, which couple action choices over time, instead of trajectories. We introduce the Boltzmann policy distribution (BPD), which serves as a prior over human policies and adapts via Bayesian inference to capture systematic deviations by observing human actions during a single episode. The BPD is difficult to compute and represent because policies lie in a high-dimensional continuous space, but we leverage tools from generative and sequence models to enable efficient sampling and inference. We show that the BPD enables prediction of human behavior and human-AI collaboration equally as well as imitation learning-based human models while using far less data.
Stein variational gradient descent (SVGD) is a general-purpose optimization-based sampling algorithm that has recently exploded in popularity, but is limited by two issues: it is known to produce biased samples, and it can be slow to converge on complicated distributions. A recently proposed stochastic variant of SVGD (sSVGD) addresses the first issue, producing unbiased samples by incorporating a special noise into the SVGD dynamics such that asymptotic convergence is guaranteed. Meanwhile, Stein variational Newton (SVN), a Newton-like extension of SVGD, dramatically accelerates the convergence of SVGD by incorporating Hessian information into the dynamics, but also produces biased samples. In this paper we derive, and provide a practical implementation of, a stochastic variant of SVN (sSVN) which is both asymptotically correct and converges rapidly. We demonstrate the effectiveness of our algorithm on a difficult class of test problems -- the Hybrid Rosenbrock density -- and show that sSVN converges using three orders of magnitude fewer gradient evaluations of the log likelihood than its stochastic SVGD counterpart. Our results show that sSVN is a promising approach to accelerating high-precision Bayesian inference tasks with modest-dimension, $d\sim\mathcal{O}(10)$.
We introduce a deep learning accelerated methodology to solve PDE-based Bayesian inverse problems with guaranteed accuracy. This is motivated by the ill-posed problem of inferring a spatio-temporal heat-flux parameter known as the Biot number given temperature data, however the methodology is generalisable to other settings. To accelerate Bayesian inference, we develop a novel training scheme that uses data to adaptively train a neural-network surrogate simulating the parametric forward model. By simultaneously identifying an approximate posterior distribution over the Biot number, and weighting a physics-informed training loss according to this, our approach approximates forward and inverse solution together without any need for external solves. Using a random Chebyshev series, we outline how to approximate a Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to sample from the posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss approaches zero. Additionally, we describe how this surrogate-accelerated HMC approach can be combined with traditional PDE solvers in a delayed-acceptance scheme to a-priori control the posterior accuracy. This overcomes a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved in turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high dimensions whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results are given using real and simulated data.
We propose a variational Bayesian inference procedure for online nonlinear system identification. For each output observation, a set of parameter posterior distributions is updated, which is then used to form a posterior predictive distribution for future outputs. We focus on the class of polynomial NARMAX models, which we cast into probabilistic form and represent in terms of a Forney-style factor graph. Inference in this graph is efficiently performed by a variational message passing algorithm. We show empirically that our variational Bayesian estimator outperforms an online recursive least-squares estimator, most notably in small sample size settings and low noise regimes, and performs on par with an iterative least-squares estimator trained offline.