My research lies at the interface between computer perception (which builds artificial systems for understanding images, sounds and videos), neuroscience (which tries to understand the brain) and machine-learning (which provides a theoretical framework for learning from data). The goal is to develop systems that solve important problems, drawing inspiration from the brain. For example, figuring out how many sound sources there are in an acoustic scene and what the individual contributions from each source are. There are medical and engineering applications of this work, such as in cochlear implants for the deaf. Importantly, the behaviour of these systems can also be compared to neural processing in the brain in order to better understand what the brain is doing.

## Publications

#### Partitioned Variational Inferece: A Framework for Probabilistic Federated Learning

Matthew Ashman, Thang D. Bui, Cuong V. Nguyen, Efstratios Markou, Adrian Weller, Siddharth Swaroop, Richard E. Turner, 2022.

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The proliferation of computing devices has brought about an opportunity to deploy machine learning models on new problem domains using previously inaccessible data. Traditional algorithms for training such models often require data to be stored on a single machine with compute performed by a single node, making them unsuitable for decentralised training on multiple devices. This deficiency has motivated the development of federated learning algorithms, which allow multiple data owners to train collaboratively and use a shared model whilst keeping local data private. However, many of these algorithms focus on obtaining point estimates of model parameters, rather than probabilistic estimates capable of capturing model uncertainty, which is essential in many applications. Variational inference (VI) has become the method of choice for fitting many modern probabilistic models. In this paper we introduce partitioned variational inference (PVI), a general framework for performing VI in the federated setting. We develop new supporting theory for PVI, demonstrating a number of properties that make it an attractive choice for practitioners; use PVI to unify a wealth of fragmented, yet related literature; and provide empirical results that showcase the effectiveness of PVI in a variety of federated settings.

#### Sparse Gaussian process variational autoencoders

Matthew Ashman, Jonny So, Will Tebbutt, Vincent Fortuin, Michael Pearce, Richard E. Turner, 2020.

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Large, multi-dimensional spatio-temporal datasets are omnipresent in modern science and engineering. An effective framework for handling such data are Gaussian process deep generative models (GP-DGMs), which employ GP priors over the latent variables of DGMs. Existing approaches for performing inference in GP-DGMs do not support sparse GP approximations based on inducing points, which are essential for the computational efficiency of GPs, nor do they handle missing data – a natural occurrence in many spatio-temporal datasets – in a principled manner. We address these shortcomings with the development of the sparse Gaussian process variational autoencoder (SGP-VAE), characterised by the use of partial inference networks for parameterising sparse GP approximations. Leveraging the benefits of amortised variational inference, the SGP-VAE enables inference in multi-output sparse GPs on previously unobserved data with no additional training. The SGP-VAE is evaluated in a variety of experiments where it outperforms alternative approaches including multi-output GPs and structured VAEs.

#### On sparsity and overcompleteness in image models

Pietro Berkes, Richard E. Turner, Maneesh Sahani, 2008. (In nips20). Edited by J. C. Platt, D. Koller, Y. Singer, S. Roweis. mit.

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Computational models of visual cortex, and in particular those based on sparse coding, have enjoyed much recent attention. Despite this currency, the question of how sparse or how over-complete a sparse representation should be, has gone without principled answer. Here, we use Bayesian model-selection methods to address these questions for a sparse-coding model based on a Student-t prior. Having validated our methods on toy data, we find that natural images are indeed best modelled by extremely sparse distributions; although for the Student-t prior, the associated optimal basis size is only modestly over-complete.

#### A Structured Model of Video Reproduces Primary Visual Cortical Organisation

Pietro Berkes, Richard E. Turner, Maneesh Sahani, 09 2009. (PLoS Computational Biology). Public Library of Science. **DOI**: 10.1371/journal.pcbi.1000495.

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The visual system must learn to infer the presence of objects and features in the world from the images it encounters, and as such it must, either implicitly or explicitly, model the way these elements interact to create the image. Do the response properties of cells in the mammalian visual system reflect this constraint? To address this question, we constructed a probabilistic model in which the identity and attributes of simple visual elements were represented explicitly and learnt the parameters of this model from unparsed, natural video sequences. After learning, the behaviour and grouping of variables in the probabilistic model corresponded closely to functional and anatomical properties of simple and complex cells in the primary visual cortex (V1). In particular, feature identity variables were activated in a way that resembled the activity of complex cells, while feature attribute variables responded much like simple cells. Furthermore, the grouping of the attributes within the model closely parallelled the reported anatomical grouping of simple cells in cat V1. Thus, this generative model makes explicit an interpretation of complex and simple cells as elements in the segmentation of a visual scene into basic independent features, along with a parametrisation of their moment-by-moment appearances. We speculate that such a segmentation may form the initial stage of a hierarchical system that progressively separates the identity and appearance of more articulated visual elements, culminating in view-invariant object recognition.

#### TaskNorm: rethinking batch normalization for meta-learning

John Bronskill, Jonathan Gordon, James Requeima, Sebastian Nowozin, Richard E. Turner, 2020. (In 37th International Conference on Machine Learning). Proceedings of Machine Learning Research.

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Modern meta-learning approaches for image classification rely on increasingly deep networks to achieve state-of-the-art performance, making batch normalization an essential component of meta-learning pipelines. However, the hierarchical nature of the meta-learning setting presents several challenges that can render conventional batch normalization ineffective, giving rise to the need to rethink normalization in this setting. We evaluate a range of approaches to batch normalization for meta-learning scenarios, and develop a novel approach that we call TASKNORM. Experiments on fourteen datasets demonstrate that the choice of batch normalization has a dramatic effect on both classification accuracy and training time for both gradient based and gradient free meta-learning approaches. Importantly, TASKNORM is found to consistently improve performance. Finally, we provide a set of best practices for normalization that will allow fair comparison of meta-learning algorithms.

#### Memory efficient meta-learning with large images

John Bronskill, Daniela Massiceti, Massimiliano Patacchiola, Katja Hofmann, Sebastian Nowozin, Richard E. Turner, 2021. (In Advances in Neural Information Processing Systems 35).

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Meta learning approaches to few-shot classification are computationally efficient at test time, requiring just a few optimization steps or single forward pass to learn a new task, but they remain highly memory-intensive to train. This limitation arises because a task’s entire support set, which can contain up to 1000 images, must be processed before an optimization step can be taken. Harnessing the performance gains offered by large images thus requires either parallelizing the meta-learner across multiple GPUs, which may not be available, or trade-offs between task and image size when memory constraints apply. We improve on both options by proposing LITE, a general and memory efficient episodic training scheme that enables meta-training on large tasks composed of large images on a single GPU. We achieve this by observing that the gradients for a task can be decomposed into a sum of gradients over the task’s training images. This enables us to perform a forward pass on a task’s entire training set but realize significant memory savings by back-propagating only a random subset of these images which we show is an unbiased approximation of the full gradient. We use LITE to train meta-learners and demonstrate new state-of-the-art accuracy on the real-world ORBIT benchmark and 3 of the 4 parts of the challenging VTAB+ MD benchmark relative to leading meta-learners. LITE also enables meta-learners to be competitive with transfer learning approaches but at a fraction of the test-time computational cost, thus serving as a counterpoint to the recent narrative that transfer learning is all you need for few-shot classification.

#### Scalable Exact Inference in Multi-Output Gaussian Processes

Wessel Bruinsma, Eric Perim, Will Tebbutt, J. Scott Hosking, Arno Solin, Richard E. Turner, 2020. (In 37th International Conference on Machine Learning). Proceedings of Machine Learning Research.

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Multi-output Gaussian processes (MOGPs) leverage the flexibility and interpretability of GPs while capturing structure across outputs, which is desirable, for example, in spatio-temporal modelling. The key problem with MOGPs is their computational scaling O(n^3 p^3), which is cubic in the number of both inputs n (e.g., time points or locations) and outputs p. For this reason, a popular class of MOGPs assumes that the data live around a low-dimensional linear subspace, reducing the complexity to O(n^3 m^3). However, this cost is still cubic in the dimensionality of the subspace m, which is still prohibitively expensive for many applications. We propose the use of a sufficient statistic of the data to accelerate inference and learning in MOGPs with orthogonal bases. The method achieves linear scaling in m in practice, allowing these models to scale to large m without sacrificing significant expressivity or requiring approximation. This advance opens up a wide range of real-world tasks and can be combined with existing GP approximations in a plug-and-play way. We demonstrate the efficacy of the method on various synthetic and real-world data sets.

#### The Gaussian Neural Process

Wessel P. Bruinsma, James Requeima, Andrew Y. K. Foong, Jonathan Gordon, Richard E. Turner, 2021. (In 3rd Symposium on Advances in Approximate Bayesian Inference).

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Neural Processes (NPs; Garnelo et al., 2018a,b) are a rich class of models for meta-learning that map data sets directly to predictive stochastic processes. We provide a rigorous analysis of the standard maximum-likelihood objective used to train conditional NPs. Moreover, we propose a new member to the Neural Process family called the Gaussian Neural Process (GNP), which models predictive correlations, incorporates translation equivariance, provides universal approximation guarantees, and demonstrates encouraging performance.

#### Modelling Non-Smooth Signals with Complex Spectral Structure

Wessel P. Bruinsma, Martin Tegnér, Richard E. Turner, 2022. (In 25th International Conference on Artificial Intelligence and Statistics).

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The Gaussian Process Convolution Model (GPCM; Tobar et al., 2015a) is a model for signals with complex spectral structure. A significant limitation of the GPCM is that it assumes a rapidly decaying spectrum: it can only model smooth signals. Moreover, inference in the GPCM currently requires (1) a mean-field assumption, resulting in poorly calibrated uncertainties, and (2) a tedious variational optimisation of large covariance matrices. We redesign the GPCM model to induce a richer distribution over the spectrum with relaxed assumptions about smoothness: the Causal Gaussian Process Convolution Model (CGPCM) introduces a causality assumption into the GPCM, and the Rough Gaussian Process Convolution Model (RGPCM) can be interpreted as a Bayesian nonparametric generalisation of the fractional Ornstein–Uhlenbeck process. We also propose a more effective variational inference scheme, going beyond the mean-field assumption: we design a Gibbs sampler which directly samples from the optimal variational solution, circumventing any variational optimisation entirely. The proposed variations of the GPCM are validated in experiments on synthetic and real-world data, showing promising results.

#### Deep Gaussian Processes for Regression using Approximate Expectation Propagation

Thang D. Bui, Daniel Hernández-Lobato, José Miguel Hernández-Lobato, Yingzhen Li, Richard E. Turner, June 2016. (In 33rd International Conference on Machine Learning). New York, USA.

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Deep Gaussian processes (DGPs) are multi-layer hierarchical generalisations of Gaussian processes (GPs) and are formally equivalent to neural networks with multiple, infinitely wide hidden layers. DGPs are nonparametric probabilistic models and as such are arguably more flexible, have a greater capacity to generalise, and provide better calibrated uncertainty estimates than alternative deep models. This paper develops a new approximate Bayesian learning scheme that enables DGPs to be applied to a range of medium to large scale regression problems for the first time. The new method uses an approximate Expectation Propagation procedure and a novel and efficient extension of the probabilistic backpropagation algorithm for learning. We evaluate the new method for non-linear regression on eleven real-world datasets, showing that it always outperforms GP regression and is almost always better than state-of-the-art deterministic and sampling-based approximate inference methods for Bayesian neural networks. As a by-product, this work provides a comprehensive analysis of six approximate Bayesian methods for training neural networks.

#### Streaming sparse Gaussian process approximations

Thang D. Bui, Cuong V. Nguyen, Richard E. Turner, December 2017. (In Advances in Neural Information Processing Systems 31). Long Beach, California, USA.

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Sparse approximations for Gaussian process models provide a suite of methods that enable these models to be deployed in large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which the posterior distribution over function values and the hyperparameters are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. This paper develops a new principled framework for deploying Gaussian process probabilistic models in the streaming setting, providing principled methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is experimentally validated using synthetic and real-world datasets.

**Comment:** The first two authors contributed equally.

#### Tree-structured Gaussian Process Approximations

Thang D. Bui, Richard E. Turner, 2014. (In Advances in Neural Information Processing Systems 28). Edited by Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence, K.Q. Weinberger. Curran Associates, Inc..

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Gaussian process regression can be accelerated by constructing a small pseudo-dataset to summarize the observed data. This idea sits at the heart of many approximation schemes, but such an approach requires the number of pseudo-datapoints to be scaled with the range of the input space if the accuracy of the approximation is to be maintained. This presents problems in time-series settings or in spatial datasets where large numbers of pseudo-datapoints are required since computation typically scales quadratically with the pseudo-dataset size. In this paper we devise an approximation whose complexity grows linearly with the number of pseudo-datapoints. This is achieved by imposing a tree or chain structure on the pseudo-datapoints and calibrating the approximation using a Kullback-Leibler (KL) minimization. Inference and learning can then be performed efficiently using the Gaussian belief propagation algorithm. We demonstrate the validity of our approach on a set of challenging regression tasks including missing data imputation for audio and spatial datasets. We trace out the speed-accuracy trade-off for the new method and show that the frontier dominates those obtained from a large number of existing approximation techniques.

#### A Unifying Framework for Gaussian Process Pseudo-Point Approximations using Power Expectation Propagation

Thang D. Bui, Josiah Yan, Richard E. Turner, 2017. (Journal of Machine Learning Research).

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Gaussian processes (GPs) are flexible distributions over functions that enable high-level assumptions about unknown functions to be encoded in a parsimonious, flexible and general way. Although elegant, the application of GPs is limited by computational and analytical intractabilities that arise when data are sufficiently numerous or when employing non-Gaussian models. Consequently, a wealth of GP approximation schemes have been developed over the last 15 years to address these key limitations. Many of these schemes employ a small set of pseudo data points to summarise the actual data. In this paper we develop a new pseudo-point approximation framework using Power Expectation Propagation (Power EP) that unifies a large number of these pseudo-point approximations. Unlike much of the previous venerable work in this area, the new framework is built on standard methods for approximate inference (variational free-energy, EP and Power EP methods) rather than employing approximations to the probabilistic generative model itself. In this way all of the approximation is performed at `inference time' rather than at`

modelling time’, resolving awkward philosophical and empirical questions that trouble previous approaches. Crucially, we demonstrate that the new framework includes new pseudo-point approximation methods that outperform current approaches on regression and classification tasks.

#### The Geometry of Random Features

Krzysztof Choromanski, Mark Rowland, Tamas Sarlos, Vikas Sindhwani, Richard E. Turner, Adrian Weller, April 2018. (In 21st International Conference on Artificial Intelligence and Statistics). Playa Blanca, Lanzarote, Canary Islands.

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We present an in-depth examination of the effectiveness of radial basis function kernel (beyond Gaussian) estimators based on orthogonal random feature maps. We show that orthogonal estimators outperform state-of-the-art mechanisms that use iid sampling under weak conditions for tails of the associated Fourier distributions. We prove that for the case of many dimensions, the superiority of the orthogonal transform can be accurately measured by a property we define called the charm of the kernel, and that orthogonal random features provide optimal (in terms of mean squared error) kernel estimators. We provide the first theoretical results which explain why orthogonal random features outperform unstructured on downstream tasks such as kernel ridge regression by showing that orthogonal random features provide kernel algorithms with better spectral properties than the previous state-of-the-art. Our results enable practitioners more generally to estimate the benefits from applying orthogonal transforms.

#### Structured evolution with compact architectures for scalable policy optimization

Krzysztof Choromanski, Mark Rowland, Vikas Sindhwani, Richard Turner, Adrian Weller, July 2018. (In 35th International Conference on Machine Learning). Stockholm Sweden.

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We present a new method of blackbox optimization via gradient approximation with the use of structured random orthogonal matrices, providing more accurate estimators than baselines and with provable theoretical guarantees. We show that this algorithm can be successfully applied to learn better quality compact policies than those using standard gradient estimation techniques. The compact policies we learn have several advantages over unstructured ones, including faster training algorithms and faster inference. These benefits are important when the policy is deployed on real hardware with limited resources. Further, compact policies provide more scalable architectures for derivative-free optimization (DFO) in high dimensional spaces. We show that most robotics tasks from the OpenAI Gym can be solved using neural networks with less than 300 parameters, with almost linear time complexity of the inference phase, with up to 13x fewer parameters relative to the Evolution Strategies (ES) algorithm introduced by Salimans et al. (2017). We do not need heuristics such as fitness shaping to learn good quality policies, resulting in a simple and theoretically motivated training mechanism.

#### How Tight Can PAC-Bayes Be in the Small Data Regime?

Andrew Y. K. Foong, Wessel P. Bruinsma, David R. Burt, Richard E. Turner, 2021. (In Advances in Neural Information Processing Systems 34). Curran Associates, Inc..

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In this paper, we investigate the question: Given a small number of datapoints, for example N = 30, how tight can PAC-Bayes and test set bounds be made? For such small datasets, test set bounds adversely affect generalisation performance by withholding data from the training procedure. In this setting, PAC-Bayes bounds are especially attractive, due to their ability to use all the data to simultaneously learn a posterior and bound its generalisation risk. We focus on the case of i.i.d. data with a bounded loss and consider the generic PAC-Bayes theorem of Germain et al. While their theorem is known to recover many existing PAC-Bayes bounds, it is unclear what the tightest bound derivable from their framework is. For a fixed learning algorithm and dataset, we show that the tightest possible bound coincides with a bound considered by Catoni; and, in the more natural case of distributions over datasets, we establish a lower bound on the best bound achievable in expectation. Interestingly, this lower bound recovers the Chernoff test set bound if the posterior is equal to the prior. Moreover, to illustrate how tight these bounds can be, we study synthetic one-dimensional classification tasks in which it is feasible to meta-learn both the prior and the form of the bound to numerically optimise for the tightest bounds possible. We ind that in this simple, controlled scenario, PAC-Bayes bounds are competitive with comparable, commonly used Chernoff test set bounds. However, the sharpest test set bounds still lead to better guarantees on the generalisation error than the PAC-Bayes bounds we consider.

#### Meta-Learning Stationary Stochastic Process Prediction With Convolutional Neural Processes

Andrew Y. K. Foong, Wessel P. Bruinsma, Jonathan Gordon, Yann Dubois, James Requeima, Richard E. Turner, 2020. (In Advances in Neural Information Processing Systems 33). Curran Associates, Inc..

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Stationary stochastic processes (SPs) are a key component of many probabilistic models, such as those for off-the-grid spatio-temporal data. They enable the statistical symmetry of underlying physical phenomena to be leveraged, thereby aiding generalization. Prediction in such models can be viewed as a translation equivariant map from observed data sets to predictive SPs, emphasizing the intimate relationship between stationarity and equivariance. Building on this, we propose the Convolutional Neural Process (ConvNP), which endows Neural Processes (NPs) with translation equivariance and extends convolutional conditional NPs to allow for dependencies in the predictive distribution. The latter enables ConvNPs to be deployed in settings which require coherent samples, such as Thompson sampling or conditional image completion. Moreover, we propose a new maximum-likelihood objective to replace the standard ELBO objective in NPs, which conceptually simplifies the framework and empirically improves performance. We demonstrate the strong performance and generalization capabilities of ConvNPs on 1D regression, image completion, and various tasks with real-world spatio-temporal data.

#### On the Expressiveness of Approximate Inference in Bayesian Neural Networks

Andrew Foong, David Burt, Yingzhen Li, Richard Turner, 2020. (In Advances in Neural Information Processing Systems 34).

#### Bayesian neural network priors revisited

Vincent Fortuin, Adrià Garriga-Alonso, Sebastian W. Ober, Florian Wenzel, Gunnar Rätsch, Richard E. Turner, Mark van der Wilk, Laurence Aitchison, 2022. (In 10th International Conference on Learning Representations).

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Isotropic Gaussian priors are the de facto standard for modern Bayesian neural network inference. However, it is unclear whether these priors accurately reflect our true beliefs about the weight distributions or give optimal performance. To find better priors, we study summary statistics of neural network weights in networks trained using stochastic gradient descent (SGD). We find that convolutional neural network (CNN) and ResNet weights display strong spatial correlations, while fully connected networks (FCNNs) display heavy-tailed weight distributions. We show that building these observations into priors can lead to improved performance on a variety of image classification datasets. Surprisingly, these priors mitigate the cold posterior effect in FCNNs, but slightly increase the cold posterior effect in ResNets.

#### Improving the Gaussian Process Sparse Spectrum Approximation by Representing Uncertainty in Frequency Inputs

Yarin Gal, Richard Turner, 2015. (In Proceedings of the 32nd International Conference on Machine Learning (ICML-15)).

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Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the properties of our approximate inference, compare it to alternative ones, and extend it to the distributed and stochastic domains. Our approximation captures complex functions better than standard approaches and avoids over-fitting.

#### Meta-learning probabilistic inference for prediction

Jonathan Gordon, John Bronskill, Matthias Bauer, Sebastian Nowozin, Richard Turner, April 2019. (In 7th International Conference on Learning Representations). New Orleans.

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This paper introduces a new framework for data efficient and versatile learning. Specifically: 1) We develop ML-PIP, a general framework for Meta-Learning approximate Probabilistic Inference for Prediction. ML-PIP extends existing probabilistic interpretations of meta-learning to cover a broad class of methods. 2) We introduce , an instance of the framework employing a flexible and versatile amortization network that takes few-shot learning datasets as inputs, with arbitrary numbers of shots, and outputs a distribution over task-specific parameters in a single forward pass. Versa substitutes optimization at test time with forward passes through inference networks, amortizing the cost of inference and relieving the need for second derivatives during training. 3) We evaluate on benchmark datasets where the method sets new state-of-the-art results, and can handle arbitrary number of shots, and for classification, arbitrary numbers of classes at train and test time. The power of the approach is then demonstrated through a challenging few-shot ShapeNet view reconstruction task.

#### Convolutional Conditional Neural Processes

Jonathan Gordon, Wessel Bruinsma, Andrew Y. K. Foong, James Requeima, Yann Dubois, Richard Turner, April 2020. (In 8th International Conference on Learning Representations). Adis Ababa.

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We introduce the Convolutional Conditional Neural Process (ConvCNP), a new member of the Neural Process family that models translation equivariance in the data. Translation equivariance is an important inductive bias for many learning problems including time series modelling, spatial data, and images. The model embeds data sets into an infinite-dimensional function space, as opposed to finite-dimensional vector spaces. To formalize this notion, we extend the theory of neural representations of sets to include functional representations, and demonstrate that any translation-equivariant embedding can be represented using a convolutional deep-set. We evaluate ConvCNPs in several settings, demonstrating that they achieve state-of-the-art performance compared to existing NPs. We demonstrate that building in translation equivariance enables zero-shot generalization to challenging, out-of-domain tasks.

#### Neural Adaptive Sequential Monte Carlo

Shixiang Gu, Zoubin Ghahramani, Richard E. Turner, Dec 2015. (In Advances in Neural Information Processing Systems 29). Montréal CANADA.

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Sequential Monte Carlo (SMC), or particle filtering, is a popular class of methods for sampling from an intractable target distribution using a sequence of simpler intermediate distributions. Like other importance sampling-based methods, performance is critically dependent on the proposal distribution: a bad proposal can lead to arbitrarily inaccurate estimates of the target distribution. This paper presents a new method for automatically adapting the proposal using an approximation of the Kullback-Leibler divergence between the true posterior and the proposal distribution. The method is very flexible, applicable to any parameterised proposal distribution and it supports online and batch variants. We use the new framework to adapt powerful proposal distributions with rich parameterisations based upon neural networks leading to Neural Adaptive Sequential Monte Carlo (NASMC). Experiments indicate that NASMC significantly improves inference in a non-linear state space model outperforming adaptive proposal methods including the Extended Kalman and Unscented Particle Filters. Experiments also indicate that improved inference translates into improved parameter learning when NASMC is used as a subroutine of Particle Marginal Metropolis Hastings. Finally we show that NASMC is able to train a neural network-based deep recurrent generative model achieving results that compete with the state-of-the-art for polymorphic music modelling. NASMC can be seen as bridging the gap between adaptive SMC methods and the recent work in scalable, black-box variational inference.

#### Q-Prop: Sample-Efficient Policy Gradient with An Off-Policy Critic

Shixiang Gu, Timothy Lillicrap, Zoubin Ghahramani, Richard E. Turner, Sergey Levine, April 2017. (In 5th International Conference on Learning Representations). Toulon France.

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Model-free deep reinforcement learning (RL) methods have been successful in a wide variety of simulated domains. However, a major obstacle facing deep RL in the real world is their high sample complexity. Batch policy gradient methods offer stable learning, but at the cost of high variance, which often requires large batches. TD-style methods, such as off-policy actor-critic and Q-learning, are more sample-efficient but biased, and often require costly hyperparameter sweeps to stabilize. In this work, we aim to develop methods that combine the stability of policy gradients with the efficiency of off-policy RL. We present Q-Prop, a policy gradient method that uses a Taylor expansion of the off-policy critic as a control variate. Q-Prop is both sample efficient and stable, and effectively combines the benefits of on-policy and off-policy methods. We analyze the connection between Q-Prop and existing model-free algorithms, and use control variate theory to derive two variants of Q-Prop with conservative and aggressive adaptation. We show that conservative Q-Prop provides substantial gains in sample efficiency over trust region policy optimization (TRPO) with generalized advantage estimation (GAE), and improves stability over deep deterministic policy gradient (DDPG), the state-of-the-art on-policy and off-policy methods, on OpenAI Gym’s MuJoCo continuous control environments.

#### Interpolated Policy Gradient: Merging On-Policy and Off-Policy Policy Gradient Estimation for Deep Reinforcement Learning

Shixiang Gu, Timothy Lillicrap, Zoubin Ghahramani, Richard E. Turner, Bernhard Schölkopf, Sergey Levine, Dec 2017. (In Advances in Neural Information Processing Systems 31). Long Beach USA.

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Off-policy model-free deep reinforcement learning methods using previously collected data can improve sample efficiency over on-policy policy gradient techniques. On the other hand, on-policy algorithms are often more stable and easier to use. This paper examines, both theoretically and empirically, approaches to merging on- and off-policy updates for deep reinforcement learning. Theoretical results show that off-policy updates with a value function estimator can be interpolated with on-policy policy gradient updates whilst still satisfying performance bounds. Our analysis uses control variate methods to produce a family of policy gradient algorithms, with several recently proposed algorithms being special cases of this family. We then provide an empirical comparison of these techniques with the remaining algorithmic details fixed, and show how different mixing of off-policy gradient estimates with on-policy samples contribute to improvements in empirical performance. The final algorithm provides a generalization and unification of existing deep policy gradient techniques, has theoretical guarantees on the bias introduced by off-policy updates, and improves on the state-of-the-art model-free deep RL methods on a number of OpenAI Gym continuous control benchmarks.

#### Black-Box Alpha Divergence Minimization

José Miguel Hernández-Lobato, Yingzhen Li, Mark Rowland, Thang D. Bui, Daniel Hernández-Lobato, Richard E. Turner, June 2016. (In 33rd International Conference on Machine Learning). New York USA.

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Black-box alpha (BB-α) is a new approximate inference method based on the minimization of α-divergences. BB-α scales to large datasets because it can be implemented using stochastic gradient descent. BB-α can be applied to complex probabilistic models with little effort since it only requires as input the likelihood function and its gradients. These gradients can be easily obtained using automatic differentiation. By changing the divergence parameter α, the method is able to interpolate between variational Bayes (VB) (α→ 0) and an algorithm similar to expectation propagation (EP) (α = 1). Experiments on probit regression and neural network regression and classification problems show that BB-αwith non-standard settings of α, such as α = 0.5, usually produces better predictions than with α→ 0 (VB) or α = 1 (EP).

#### Sequence Tutor: Conservative fine-tuning of sequence generation models with KL-control

Natasha Jaques, Shixiang Gu, Dzmitry Bahdanau, Jose Miguel Hernndez Lobato, Richard E. Turner, Douglas Eck, Aug 2017. (In 34th International Conference on Machine Learning). Sydney AUSTRALIA.

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This paper proposes a general method for improving the structure and quality of sequences generated by a recurrent neural network (RNN), while maintaining information originally learned from data, as well as sample diversity. An RNN is first pre-trained on data using maximum likelihood estimation (MLE), and the probability distribution over the next token in the sequence learned by this model is treated as a prior policy. Another RNN is then trained using reinforcement learning (RL) to generate higher-quality outputs that account for domain-specific incentives while retaining proximity to the prior policy of the MLE RNN. To formalize this objective, we derive novel off-policy RL methods for RNNs from KL-control. The effectiveness of the approach is demonstrated on two applications; 1) generating novel musical melodies, and 2) computational molecular generation. For both problems, we show that the proposed method improves the desired properties and structure of the generated sequences, while maintaining information learned from data.

**Comment:** [MIT Technology Review] [Video]

#### Kernel Learning for Explainable Climate Science

Vidhi Lalchand, Kenza Tazi, Talay M Cheema, Richard E Turner, Scott Hosking, 2022. (In 16th Bayesian Modelling Applications Workshop at UAI, 2022).

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The Upper Indus Basin, Himalayas provides water for 270 million people and countless ecosystems. However, precipitation, a key component to hydrological modelling, is poorly understood in this area. A key challenge surrounding this uncertainty comes from the complex spatial-temporal distribution of precipitation across the basin. In this work we propose Gaussian processes with structured non-stationary kernels to model precipitation patterns in the UIB. Previous attempts to quantify or model precipitation in the Hindu Kush Karakoram Himalayan region have often been qualitative or include crude assumptions and simplifications which cannot be resolved at lower resolutions. This body of research also provides little to no error propagation. We account for the spatial variation in precipitation with a non-stationary Gibbs kernel parameterised with an input dependent lengthscale. This allows the posterior function samples to adapt to the varying precipitation patterns inherent in the distinct underlying topography of the Indus region. The input dependent lengthscale is governed by a latent Gaussian process with a stationary squared-exponential kernel to allow the function level hyperparameters to vary smoothly. In ablation experiments we motivate each component of the proposed kernel by demonstrating its ability to model the spatial covariance, temporal structure and joint spatio-temporal reconstruction. We benchmark our model with a stationary Gaussian process and a Deep Gaussian processes.

#### A role for amplitude modulation phase relationships in speech rhythm perception

Victoria Leong, Michael A Stone, Richard E Turner, Usha Goswami, 2014. (Journal of the Acoustical Society of America). Acoustical Society of America.

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Prosodic rhythm in speech [the alternation of “Strong” (S) and “weak” (w) syllables] is cued, among others, by slow rates of amplitude modulation (AM) within the speech envelope. However, it is unclear exactly which envelope modulation rates and statistics are the most important for the rhythm percept. Here, the hypothesis that the phase relationship between “Stress” rate ( 2 Hz) and “Syllable” rate ( 4 Hz) AMs provides a perceptual cue for speech rhythm is tested. In a rhythm judgment task, adult listeners identified AM tone-vocoded nursery rhyme sentences that carried either trochaic (S-w) or iambic patterning (w-S). Manipulation of listeners’ rhythm perception was attempted by parametrically phase-shifting the Stress AM and Syllable AM in the vocoder. It was expected that a 1π radian phase-shift (half a cycle) would reverse the perceived rhythm pattern (i.e., trochaic -> iambic) whereas a 2πradian shift (full cycle) would retain the perceived rhythm pattern (i.e., trochaic -> trochaic). The results confirmed these predictions. Listeners judgments of rhythm systematically followed Stress-Syllable AM phase-shifts, but were unaffected by phase-shifts between the Syllable AM and the Sub-beat AM ( 14 Hz) in a control condition. It is concluded that the Stress-Syllable AM phase relationship is an envelope-based modulation statistic that supports speech rhythm perception.

#### Stochastic Expectation Propagation

Yingzhen Li, José Miguel Hernández-Lobato, Richard E. Turner, Dec 2015. (In Advances in Neural Information Processing Systems 28). Montréal CANADA.

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Expectation propagation (EP) is a deterministic approximation algorithm that is often used to perform approximate Bayesian parameter learning. EP approximates the full intractable posterior distribution through a set of local-approximations that are iteratively refined for each datapoint. EP can offer analytic and computational advantages over other approximations, such as Variational Inference (VI), and is the method of choice for a number of models. The local nature of EP appears to make it an ideal candidate for performing Bayesian learning on large models in large-scale datasets settings. However, EP has a crucial limitation in this context: the number approximating factors needs to increase with the number of data-points, N, which often entails a prohibitively large memory overhead. This paper presents an extension to EP, called stochastic expectation propagation (SEP), that maintains a global posterior approximation (like VI) but updates it in a local way (like EP ). Experiments on a number of canonical learning problems using synthetic and real-world datasets indicate that SEP performs almost as well as full EP, but reduces the memory consumption by a factor of N. SEP is therefore ideally suited to performing approximate Bayesian learning in the large model, large dataset setting.

#### Rényi Divergence Variational Inference

Yingzhen Li, Richard E. Turner, Dec 2016. (In Advances in Neural Information Processing Systems 29). Barcelona SPAIN.

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This paper introduces the variational Rényi bound (VR) that extends traditional variational inference to Rényi’s alpha-divergences. This new family of variational methods unifies a number of existing approaches, and enables a smooth interpolation from the evidence lower-bound to the log (marginal) likelihood that is controlled by the value of alpha that parametrises the divergence. The reparameterization trick, Monte Carlo approximation and stochastic optimisation methods are deployed to obtain a tractable and unified framework for optimisation. We further consider negative alpha values and propose a novel variational inference method as a new special case in the proposed framework. Experiments on Bayesian neural networks and variational auto-encoders demonstrate the wide applicability of the VR bound.

#### Gradient Estimators for Implicit Models

Yingzhen Li, Richard E. Turner, May 2018. (In Sixth International Conference on Learning Representations). Vancouver CANADA.

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Implicit models, which allow for the generation of samples but not for point-wise evaluation of probabilities, are omnipresent in real-world problems tackled by machine learning and a hot topic of current research. Some examples include data simulators that are widely used in engineering and scientific research, generative adversarial networks (GANs) for image synthesis, and hot-off-the-press approximate inference techniques relying on implicit distributions. The majority of existing approaches to learning implicit models rely on approximating the intractable distribution or optimisation objective for gradient- based optimisation, which is liable to produce inaccurate updates and thus poor models. This paper alleviates the need for such approximations by proposing the Stein gradient estimator, which directly estimates the score function of the implicitly defined distribution. The efficacy of the proposed estimator is empirically demonstrated by examples that include meta-learning for approximate inference and entropy regularised GANs that provide improved sample diversities.

#### Occlusive Components Analysis

Jörg Lücke, Richard E. Turner, Maneesh Sahani, Marc Henniges, 2009. (In Advances in Neural Information Processing Systems 22). Edited by Y Bengio, D Schuurmans, J Lafferty, C K I Williams, A Culotta. mit.

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We study unsupervised learning in a probabilistic generative model for occlusion. The model uses two types of latent variables: one indicates which objects are present in the image, and the other how they are ordered in depth. This depth order then determines how the positions and appearances of the objects present, specified in the model parameters, combine to form the image. We show that the object parameters can be learnt from an unlabelled set of images in which objects occlude one another. Exact maximum-likelihood learning is intractable. However, we show that tractable approximations to Expectation Maximization (EM) can be found if the training images each contain only a small number of objects on average. In numerical experiments it is shown that these approximations recover the correct set of object parameters. Experiments on a novel version of the bars test using colored bars, and experiments on more realistic data, show that the algorithm performs well in extracting the generating causes. Experiments based on the standard bars benchmark test for object learning show that the algorithm performs well in comparison to other recent component extraction approaches. The model and the learning algorithm thus connect research on occlusion with the research field of multiple-causes component extraction methods.

#### Practical Conditional Neural Processes via Tractable Dependent Predictions

Stratis Markou, James Requeima, Wessel P. Bruinsma, Anna Vaughan, Richard E. Turner, 2022. (In 10th International Conference on Learning Representations).

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Conditional Neural Processes (CNPs; Garnelo et al., 2018) are meta-learning models which leverage the flexibility of deep learning to produce well-calibrated predictions and naturally handle off-the-grid and missing data. CNPs scale to large datasets and train with ease. Due to these features, CNPs appear well-suited to tasks from environmental sciences or healthcare. Unfortunately, CNPs do not produce correlated predictions, making them fundamentally inappropriate for many estimation and decision making tasks. Predicting heat waves or floods, for example, requires modelling dependencies in temperature or precipitation over time and space. Existing approaches which model output dependencies, such as Neural Processes (NPs; Garnelo et al., 2018b) or the FullConvGNP (Bruinsma et al., 2021), are either complicated to train or prohibitively expensive. What is needed is an approach which provides dependent predictions, but is simple to train and computationally tractable. In this work, we present a new class of Neural Process models that make correlated predictions and support exact maximum likelihood training that is simple and scalable. We extend the proposed models by using invertible output transformations, to capture non-Gaussian output distributions. Our models can be used in downstream estimation tasks which require dependent function samples. By accounting for output dependencies, our models show improved predictive performance on a range of experiments with synthetic and real data.

#### On Sparse Variational methods and the Kullback-Leibler divergence between stochastic processes

Alexander G D G Matthews, James Hensman, Richard E. Turner, Zoubin Ghahramani, May 2016. (In 19th International Conference on Artificial Intelligence and Statistics). Cadiz, Spain.

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The variational framework for learning inducing variables (Titsias, 2009a) has had a large impact on the Gaussian process literature. The framework may be interpreted as minimizing a rigorously defined Kullback-Leibler divergence between the approximating and posterior processes. To our knowledge this connection has thus far gone unremarked in the literature. In this paper we give a substantial generalization of the literature on this topic. We give a new proof of the result for infinite index sets which allows inducing points that are not data points and likelihoods that depend on all function values. We then discuss augmented index sets and show that, contrary to previous works, marginal consistency of augmentation is not enough to guarantee consistency of variational inference with the original model. We then characterize an extra condition where such a guarantee is obtainable. Finally we show how our framework sheds light on interdomain sparse approximations and sparse approximations for Cox processes.

#### Gaussian process behaviour in wide deep neural networks

Alexander G. D. G. Matthews, Jiri Hron, Mark Rowland, Richard E. Turner, Zoubin Ghahramani, 2018. (ICLR).

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Whilst deep neural networks have shown great empirical success, there is still much work to be done to understand their theoretical properties. In this paper, we study the relationship between random, wide, fully connected, feedforward networks with more than one hidden layer and Gaussian processes with a recursive kernel definition. We show that, under broad conditions, as we make the architecture increasingly wide, the implied random function converges in distribution to a Gaussian process, formalising and extending existing results by Neal (1996) to deep networks. To evaluate convergence rates empirically, we use maximum mean discrepancy. We then compare finite Bayesian deep networks from the literature to Gaussian processes in terms of the key predictive quantities of interest, finding that in some cases the agreement can be very close. We discuss the desirability of Gaussian process behaviour and review non-Gaussian alternative models from the literature.

#### Sample-then-optimise posterior sampling for Bayesian linear models

Alexander G. D. G. Matthews, Jiri Hron, Richard E. Turner, Zoubin Ghahramani, 2017. (AABI (NeurIPS workshop)).

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In modern machine learning it is common to train models which have an extremely high intrinsic capacity. The results obtained are often i nitialization dependent, are different for disparate optimizers and in some cases have no explicit regularization. This raises difficult questions about generalization. A natural approach to questions of generalization is a Bayesian one. There is therefore a growing literature attempting to understand how Bayesian posterior inference could emerge from the complexity of modern practice, even without having such a procedure as the stated goal. In this work we consider a simple special case where exact Bayesian posterior sampling emerges from sampling (cf initialization) and then gradient descent. Specifically, for a Bayesian linear model, if we parameterize it in terms of a deterministic function of an isotropic normal prior, then the action of sampling from the prior followed by first order optimization of the squared loss will give a posterior sample. Although the assumptions are stronger than many real problems, it still exhibits the challenging properties of redundant model capacity and a lack of explicit regularizers, along with initialization and optimizer dependence. It is therefore an interesting controlled test case. Given its simplicity, the method itself may turn out to be of independent interest from our original goal.

#### Attacking Few-Shot Classifiers with Adversarial Support Poisoning

Elre T. Oldewage, John Bronskill, Richard E. Turner, 2021. (In A Blessing in Disguise: The Prospects and Perils of Adversarial Machine Learning, Workshop at ICML 2021).

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This paper examines the robustness of deployed few-shot meta-learning systems when they are fed an imperceptibly perturbed few-shot dataset, showing that the resulting predictions on test inputs can become worse than chance. This is achieved by developing a novel attack, Adversarial Support Poisoning or ASP, which crafts a poisoned set of examples. When even a small subset of malicious data points is inserted into the support set of a meta-learner, accuracy is significantly reduced. We evaluate the new attack on a variety of few-shot classification algorithms and scenarios, and propose a form of adversarial training that significantly improves robustness against both poisoning and evasion attacks.

#### Adversarial Attacks are a Surprisingly Strong Baseline for Poisoning Few-Shot Meta-Learners

Elre T. Oldewage, John Bronskill, Richard E. Turner, 2022. (In I Can’t Believe It’s Not Better, Workshop at Neurips 2022).

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This paper examines the robustness of deployed few-shot meta-learning systems when they are fed an imperceptibly perturbed few-shot dataset. We attack amortized meta-learners, which allows us to craft colluding sets of inputs that are tailored to fool the system’s learning algorithm when used as training data. Jointly crafted adversarial inputs might be expected to synergistically manipulate a classifier, allowing for very strong data-poisoning attacks that would be hard to detect. We show that in a white box setting, these attacks are very successful and can cause the target model’s predictions to become worse than chance. However, in opposition to the well-known transferability of adversarial examples in general, the colluding sets do not transfer well to different classifiers. We explore two hypotheses to explain this: ‘overfitting’ by the attack, and mismatch between the model on which the attack is generated and that to which the attack is transferred. Regardless of the mitigation strategies suggested by these hypotheses, the colluding inputs transfer no better than adversarial inputs that are generated independently in the usual way.

#### Practical Deep Learning with Bayesian Principles

Kazuki Osawa, Siddharth Swaroop, Anirudh Jain, Runa Eschenhagen, Richard E. Turner, Rio Yokota, Mohammad Emtiyaz Khan, 2019. (In Advances in Neural Information Processing Systems 33).

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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.

#### Challenges and Pitfalls of Bayesian Unlearning

Ambrish Rawat, James Requeima, Wessel Bruinsma, Richard Turner, 2022. (In ICML 2022 Workshop on Updatable Machine Learning (UpML)).

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Machine unlearning refers to the task of removing a subset of training data, thereby removing its contributions to a trained model. Approximate unlearning are one class of methods for this task which avoid the need to retrain the model from scratch on the retained data. Bayes’ rule can be used to cast approximate unlearning as an inference problem where the objective is to obtain the updated posterior by dividing out the likelihood of deleted data. However this has its own set of challenges as one often doesn’t have access to the exact posterior of the model parameters. In this work we examine the use of the Laplace approximation and Variational Inference to obtain the updated posterior. With a neural network trained for a regression task as the guiding example, we draw insights on the applicability of Bayesian unlearning in practical scenarios.

#### Fast and Flexible Multi-Task Classification using Conditional Neural Adaptive Processes

James Requeima, Jonathan Gordon, John Bronskill, Sebastian Nowozin, Richard E. Turner, 2019. (In Advances in Neural Information Processing Systems 33).

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The goal of this paper is to design image classification systems that, after an initial multi-task training phase, can automatically adapt to new tasks encountered at test time. We introduce a conditional neural process based approach to the multi-task classification setting for this purpose, and establish connections to the meta- and few-shot learning literature. The resulting approach, called CNAPs, comprises a classifier whose parameters are modulated by an adaptation network that takes the current task’s dataset as input. We demonstrate that CNAPs achieves state-of-the-art results on the challenging Meta-Dataset benchmark indicating high-quality transfer-learning. We show that the approach is robust, avoiding both over-fitting in low-shot regimes and under-fitting in high-shot regimes. Timing experiments reveal that CNAPs is computationally efficient at test-time as it does not involve gradient based adaptation. Finally, we show that trained models are immediately deployable to continual learning and active learning where they can outperform existing approaches that do not leverage transfer learning.

#### The Gaussian Process Autoregressive Regression Model (GPAR)

James Requeima, William Tebbutt, Wessel Bruinsma, Richard E. Turner, 2019. (In 22nd International Conference on Artificial Intelligence and Statistics). Proceedings of Machine Learning Research.

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Multi-output regression models must exploit dependencies between outputs to maximise predictive performance. The application of Gaussian processes (GPs) to this setting typically yields models that are computationally demanding and have limited representational power. We present the Gaussian Process Autoregressive Regression (GPAR) model, a scalable multi-output GP model that is able to capture nonlinear, possibly input-varying, dependencies between outputs in a simple and tractable way: the product rule is used to decompose the joint distribution over the outputs into a set of conditionals, each of which is modelled by a standard GP. GPAR’s efficacy is demonstrated on a variety of synthetic and real-world problems, outperforming existing GP models and achieving state-of-the-art performance on established benchmarks.

#### A causal perspective on domain adaptation

Mateo Rojas-Carulla, Bernhard Schölkopf, Richard Turner, Jonas Peters, 2015. (arXiv preprint arXiv:1507.05333)).

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From training data from several related domains (or tasks), methods of domain adaptation try to combine knowledge to improve performance. This paper discusses an approach to domain adaptation which is inspired by a causal interpretation of the multi-task problem. We assume that a covariate shift assumption holds true for a subset of predictor variables: the conditional of the target variable given this subset of predictors is invariant with respect to shifts in those predictors (covariates). We propose to learn the corresponding conditional expectation in the training domains and use it for estimation in the target domain. We further introduce a method which allows for automatic inference of the above subset in regression and classification. We study the performance of this approach in an adversarial setting, in the case where no additional examples are available in the test domain. If a labeled sample is available, we provide a method for using both the transferred invariant conditional and task specific information. We present results on synthetic data sets and a sentiment analysis problem.

#### Geometrically coupled Monte Carlo sampling

Mark Rowland, Krzysztof Choromanski, Francois Chalus, Aldo Pacchiano, Tamas Sarlos, Richard Turner, Adrian Weller, December 2018. (In Advances in Neural Information Processing Systems 32). Montreal Canada.

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Monte Carlo sampling in high-dimensional, low-sample settings is important in many machine learning tasks. We improve current methods for sampling in Euclidean spaces by avoiding independence, and instead consider ways to couple samples. We show fundamental connections to optimal transport theory, leading to novel sampling algorithms, and providing new theoretical grounding for existing strategies. We compare our new strategies against prior methods for improving sample efficiency, including quasi-Monte Carlo, by studying discrepancy. We explore our findings empirically, and observe benefits of our sampling schemes for reinforcement learning and generative modelling.

#### Combining pseudo-point and state space approximations for sum-separable Gaussian Processes

Will Tebbutt, Arno Solin, Richard E. Turner, 2021. (In Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence). Edited by Cassio de Campos, Marloes H. Maathuis. PMLR. Proceedings of Machine Learning Research.

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Gaussian processes (GPs) are important probabilistic tools for inference and learning in spatio-temporal modelling problems such as those in climate science and epidemiology. However, existing GP approximations do not simultaneously support large numbers of off-the-grid spatial data-points and long time-series which is a hallmark of many applications. Pseudo-point approximations, one of the gold-standard methods for scaling GPs to large data sets, are well suited for handling off-the-grid spatial data. However, they cannot handle long temporal observation horizons effectively reverting to cubic computational scaling in the time dimension. State space GP approximations are well suited to handling temporal data, if the temporal GP prior admits a Markov form, leading to linear complexity in the number of temporal observations, but have a cubic spatial cost and cannot handle off-the-grid spatial data. In this work we show that there is a simple and elegant way to combine pseudo-point methods with the state space GP approximation framework to get the best of both worlds. The approach hinges on a surprising conditional independence property which applies to space–time separable GPs. We demonstrate empirically that the combined approach is more scalable and applicable to a greater range of spatio-temporal problems than either method on its own.

#### Learning Stationary Time Series using Gaussian Process with Nonparametric Kernels

Felipe Tobar, Thang D. Bui, Richard E. Turner, Dec 2015. (In Advances in Neural Information Processing Systems 29). Montréal CANADA.

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We introduce the Gaussian Process Convolution Model (GPCM), a two-stage nonparametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametricwindow moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational freeenergy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals.

#### Modelling of Complex Signals using Gaussian Processes

Felipe Tobar, Richard E. Turner, 2015. (In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)).

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In complex-valued signal processing, estimation algorithms require complete knowledge (or accurate estimation) of the second order statistics, this makes Gaussian processes (GP) well suited for modelling complex signals, as they are designed in terms of covariance functions. Dealing with bivariate signals using GPs require four covariance matrices, or equivalently, two complex matrices. We propose a GP-based approach for modelling complex signals, whereby the second-order statistics are learnt through maximum likelihood; in particular, the complex GP approach allows for circularity coefficient estimation in a robust manner when the observed signal is corrupted by (circular) white noise. The proposed model is validated using climate signals, for both circular and noncircular cases. The results obtained open new possibilities for collaboration between the complex signal processing and Gaussian processes communities towards an appealing representation and statistical description of bivariate signals.

#### Magnetic Hamiltonian Monte Carlo

Nilesh Tripuraneni, Mark Rowland, Zoubin Ghahramani, Richard E. Turner, 2017. (In 34th International Conference on Machine Learning).

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Hamiltonian Monte Carlo (HMC) exploits Hamiltonian dynamics to construct efficient proposals for Markov chain Monte Carlo (MCMC). In this paper, we present a generalization of HMC which exploits non-canonical Hamiltonian dynamics. We refer to this algorithm as magnetic HMC, since in 3 dimensions a subset of the dynamics map onto the mechanics of a charged particle coupled to a magnetic field. We establish a theoretical basis for the use of non-canonical Hamiltonian dynamics in MCMC, and construct a symplectic, leapfrog-like integrator allowing for the implementation of magnetic HMC. Finally, we exhibit several examples where these non-canonical dynamics can lead to improved mixing of magnetic HMC relative to ordinary HMC.

#### Statistical Models for Natural Sounds

Richard E. Turner, 2010. Gatsby Computational Neuroscience Unit, UCL,

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It is important to understand the rich structure of natural sounds in order to solve important tasks, like automatic speech recognition, and to understand auditory processing in the brain. This thesis takes a step in this direction by characterising the statistics of simple natural sounds. We focus on the statistics because perception often appears to depend on them, rather than on the raw waveform. For example the perception of auditory textures, like running water, wind, fire and rain, depends on summary-statistics, like the rate of falling rain droplets, rather than on the exact details of the physical source. In order to analyse the statistics of sounds accurately it is necessary to improve a number of traditional signal processing methods, including those for amplitude demodulation, time-frequency analysis, and sub-band demodulation. These estimation tasks are ill-posed and therefore it is natural to treat them as Bayesian inference problems. The new probabilistic versions of these methods have several advantages. For example, they perform more accurately on natural signals and are more robust to noise, they can also fill-in missing sections of data, and provide error-bars. Furthermore, free-parameters can be learned from the signal. Using these new algorithms we demonstrate that the energy, sparsity, modulation depth and modulation time-scale in each sub-band of a signal are critical statistics, together with the dependencies between the sub-band modulators. In order to validate this claim, a model containing co-modulated coloured noise carriers is shown to be capable of generating a range of realistic sounding auditory textures. Finally, we explored the connection between the statistics of natural sounds and perception. We demonstrate that inference in the model for auditory textures qualitatively replicates the primitive grouping rules that listeners use to understand simple acoustic scenes. This suggests that the auditory system is optimised for the statistics of natural sounds.

#### Probabilistic Amplitude Demodulation

Richard E. Turner, M Sahani, 2007. (In 7th International Conference on Independent Component Analysis and Signal Separation).

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Auditory scene analysis is extremely challenging. One approach, perhaps that adopted by the brain, is to shape useful representations of sounds on prior knowledge about their statistical structure. For example, sounds with harmonic sections are common and so time-frequency representations are efficient. Most current representations concentrate on the shorter components. Here, we propose representations for structures on longer time-scales, like the phonemes and sentences of speech. We decompose a sound into a product of processes, each with its own characteristic time-scale. This demodulation cascade relates to classical amplitude demodulation, but traditional algorithms fail to realise the representation fully. A new approach, probabilistic amplitude demodulation, is shown to out-perform the established methods, and to easily extend to representation of a full demodulation cascade.

#### Modeling natural sounds with modulation cascade processes

Richard E. Turner, Maneesh Sahani, 2008. (In nips20). Edited by J. C. Platt, D. Koller, Y. Singer, S. Roweis. mit.

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Natural sounds are structured on many time-scales. A typical segment of speech, for example, contains features that span four orders of magnitude: Sentences (∼1s); phonemes (∼10−1 s); glottal pulses (∼ 10−2s); and formants (∼ 10−3s). The auditory system uses information from each of these time-scales to solve complicated tasks such as auditory scene analysis [1]. One route toward understanding how auditory processing accomplishes this analysis is to build neuroscience-inspired algorithms which solve similar tasks and to compare the properties of these algorithms with properties of auditory processing. There is however a discord: Current machine-audition algorithms largely concentrate on the shorter time-scale structures in sounds, and the longer structures are ignored. The reason for this is two-fold. Firstly, it is a difficult technical problem to construct an algorithm that utilises both sorts of information. Secondly, it is computationally demanding to simultaneously process data both at high resolution (to extract short temporal information) and for long duration (to extract long temporal information). The contribution of this work is to develop a new statistical model for natural sounds that captures structure across a wide range of time-scales, and to provide efficient learning and inference algorithms. We demonstrate the success of this approach on a missing data task.

#### Statistical inference for single- and multi-band probabilistic amplitude demodulation.

Richard E. Turner, Maneesh Sahani, 2010. (In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)).

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Amplitude demodulation is an ill-posed problem and so it is natural to treat it from a Bayesian viewpoint, inferring the most likely carrier and envelope under probabilistic constraints. One such treatment is Probabilistic Amplitude Demodulation (PAD), which, whilst computationally more intensive than traditional approaches, offers several advantages. Here we provide methods for estimating the uncertainty in the PAD-derived envelopes and carriers, and for learning free-parameters like the time-scale of the envelope. We show how the probabilistic approach can naturally handle noisy and missing data. Finally, we indicate how to extend the model to signals which contain multiple modulators and carriers.

#### Two problems with variational expectation maximisation for time-series models

Richard E. Turner, Maneesh Sahani, 2011. (In Bayesian Time series models). Edited by D. Barber, T. Cemgil, S. Chiappa. Cambridge University Press.

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Variational methods are a key component of the approximate inference and learning toolbox. These methods fill an important middle ground, retaining distributional information about uncertainty in latent variables, unlike maximum a posteriori methods (MAP), and yet generally requiring less computational time than Monte Carlo Markov Chain methods. In particular the variational Expectation Maximisation (vEM) and variational Bayes algorithms, both involving variational optimisation of a free-energy, are widely used in time-series modelling. Here, we investigate the success of vEM in simple probabilistic time-series models. First we consider the inference step of vEM, and show that a consequence of the well-known compactness property of variational inference is a failure to propagate uncertainty in time, thus limiting the usefulness of the retained distributional information. In particular, the uncertainty may appear to be smallest precisely when the approximation is poorest. Second, we consider parameter learning and analytically reveal systematic biases in the parameters found by vEM. Surprisingly, simpler variational approximations (such a mean-field) can lead to less bias than more complicated structured approximations.

#### Demodulation as Probabilistic Inference

Richard E. Turner, Maneesh Sahani, 2011. (Transactions on Audio, Speech and Language Processing).

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Demodulation is an ill-posed problem whenever both carrier and envelope signals are broadband and unknown. Here, we approach this problem using the methods of probabilistic inference. The new approach, called Probabilistic Amplitude Demodulation (PAD), is computationally challenging but improves on existing methods in a number of ways. By contrast to previous approaches to demodulation, it satisfies five key desiderata: PAD has soft constraints because it is probabilistic; PAD is able to automatically adjust to the signal because it learns parameters; PAD is user-steerable because the solution can be shaped by user-specific prior information; PAD is robust to broad-band noise because this is modelled explicitly; and PAD’s solution is self-consistent, empirically satisfying a Carrier Identity property. Furthermore, the probabilistic view naturally encompasses noise and uncertainty, allowing PAD to cope with missing data and return error bars on carrier and envelope estimates. Finally, we show that when PAD is applied to a bandpass-filtered signal, the stop-band energy of the inferred carrier is minimal, making PAD well-suited to sub-band demodulation.

#### Probabilistic amplitude and frequency demodulation

Richard E. Turner, Maneesh Sahani, 2011. (In Advances in Neural Information Processing Systems 24). The MIT Press.

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A number of recent scientific and engineering problems require signals to be decomposed into a product of a slowly varying positive envelope and a quickly varying carrier whose instantaneous frequency also varies slowly over time. Although signal processing provides algorithms for so-called amplitude- and frequency-demodulation (AFD), there are well known problems with all of the existing methods. Motivated by the fact that AFD is ill-posed, we approach the problem using probabilistic inference. The new approach, called probabilistic amplitude and frequency demodulation (PAFD), models instantaneous frequency using an auto-regressive generalization of the von Mises distribution, and the envelopes using Gaussian auto-regressive dynamics with a positivity constraint. A novel form of expectation propagation is used for inference. We demonstrate that although PAFD is computationally demanding, it outperforms previous approaches on synthetic and real signals in clean, noisy and missing data settings.

#### Decomposing signals into a sum of amplitude and frequency modulated sinusoids using probabilistic inference

Richard E. Turner, Maneesh Sahani, march 2012. (In Acoustics, Speech and Signal Processing (ICASSP), 2012 IEEE International Conference on). **DOI**: 10.1109/ICASSP.2012.6288343. **ISSN**: 1520-6149.

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There are many methods for decomposing signals into a sum of amplitude and frequency modulated sinusoids. In this paper we take a new estimation based approach. Identifying the problem as ill-posed, we show how to regularize the solution by imposing soft constraints on the amplitude and phase variables of the sinusoids. Estimation proceeds using a version of Kalman smoothing. We evaluate the method on synthetic and natural, clean and noisy signals, showing that it outperforms previous decompositions, but at a higher computational cost.

#### A Maximum-Likelihood Interpretation for Slow Feature Analysis

Richard E. Turner, Maneesh Sahani, 2007. (Neural Computation). Cambridge, MA, USA. MIT Press. **DOI**: http://dx.doi.org/10.1162/neco.2007.19.4.1022. **ISSN**: 0899-7667.

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The brain extracts useful features from a maelstrom of sensory information, and a fundamental goal of theoretical neuroscience is to work out how it does so. One proposed feature extraction strategy is motivated by the observation that the meaning of sensory data, such as the identity of a moving visual object, is often more persistent than the activation of any single sensory receptor. This notion is embodied in the slow feature analysis (SFA) algorithm, which uses “slowness” as an heuristic by which to extract semantic information from multi-dimensional time-series. Here, we develop a probabilistic interpretation of this algorithm showing that inference and learning in the limiting case of a suitable probabilistic model yield exactly the results of SFA. Similar equivalences have proved useful in interpreting and extending comparable algorithms such as independent component analysis. For SFA, we use the equivalent probabilistic model as a conceptual spring-board, with which to motivate several novel extensions to the algorithm.