Author name cluster

Richard Turner

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33 papers

1 author row

NeurIPS Conference 2025 Conference Paper

Distributional Training Data Attribution: What do Influence Functions Sample?

Bruno Mlodozeniec
Isaac Reid
Sam Power
David Krueger
Murat Erdogdu
Richard Turner
Roger Grosse

Randomness is an unavoidable part of training deep learning models, yet something that traditional training data attribution algorithms fail to rigorously account for. They ignore the fact that, due to stochasticity in the initialisation and batching, training on the same dataset can yield different models. In this paper, we address this shortcoming through introducing distributional training data attribution (d-TDA), the goal of which is to predict how the distribution of model outputs (over training runs) depends upon the dataset. Intriguingly, we find that influence functions (IFs), a popular data attribution tool, are 'secretly distributional': they emerge from our framework as the limit to unrolled differentiation, without requiring restrictive convexity assumptions. This provides a new perspective on the effectiveness of IFs in deep learning. We demonstrate the practical utility of d-TDA in experiments, including improving data pruning for vision transformers and identifying influential examples with diffusion models.

NeurIPS Conference 2025 Conference Paper

Estimating Interventional Distributions with Uncertain Causal Graphs through Meta-Learning

Anish Dhir
Cristiana Diaconu
Valentinian Lungu
James Requeima
Richard Turner
Mark van der Wilk

In scientific domains---from biology to the social sciences---many questions boil down to \textit{What effect will we observe if we intervene on a particular variable? } If the causal relationships (e. g. ~a causal graph) are known, its possible to estimate the intervention distributions. In the absence of this domain knowledge, the causal structure must be discovered from the available observational data. However, observational data are often compatible with multiple causal graphs, making methods that commit to a single structure prone to overconfidence. A principled way to manage this structural uncertainty is via Bayesian inference, which averages over a posterior distribution on possible causal structures and functional mechanisms. Unfortunately, the number of causal structures grows super-exponentially with the number of nodes in the graph, making computations intractable. We propose to circumvent these challenges by using meta-learning to create an end-to-end model: the Model-Averaged Causal Estimation Transformer Neural Process (MACE-TNP). The model is trained to predict the Bayesian model-averaged interventional posterior distribution, and its end-to-end nature bypasses the need for expensive calculations. Empirically, we demonstrate that MACE-TNP outperforms strong Bayesian baselines. Our work established meta-learning as a flexible and scalable paradigm for approximating complex Bayesian causal inference, that can be scaled to increasingly challenging settings in the future.

NeurIPS Conference 2025 Conference Paper

Purifying Shampoo: Investigating Shampoo's Heuristics by Decomposing its Preconditioner

Runa Eschenhagen
Aaron Defazio
Tsung-Hsien Lee
Richard Turner
Hao-Jun Shi

The recent success of Shampoo in the AlgoPerf contest has sparked renewed interest in Kronecker-factorization-based optimization algorithms for training neural networks. Despite its success, Shampoo relies heavily on several heuristics such as learning rate grafting and stale preconditioning to achieve performance at-scale. These heuristics increase algorithmic complexity, necessitate further hyperparameter tuning, and lack theoretical justification. This paper investigates these heuristics from the angle of Frobenius norm approximation to full-matrix Adam and decouples the preconditioner's eigenvalues and eigenbasis updates. We show that grafting from Adam mitigates the staleness and mis-scaling of the preconditioner's eigenvalues and how correcting the eigenvalues directly eliminates the need for learning rate grafting. To manage the error induced by infrequent eigenbasis computations, we propose an adaptive criterion for determining the eigenbasis computation frequency motivated by terminating a warm-started QR algorithm. This criterion decouples the update frequency of different preconditioner matrices and enables us to investigate the impact of approximation error on convergence. These practical techniques offer a principled angle towards removing Shampoo's heuristics and developing improved Kronecker-factorization-based training algorithms.

NeurIPS Conference 2023 Conference Paper

Geometric Neural Diffusion Processes

Emile Mathieu
Vincent Dutordoir
Michael Hutchinson
Valentin De Bortoli
Yee Whye Teh
Richard Turner

Denoising diffusion models have proven to be a flexible and effective paradigm for generative modelling. Their recent extension to infinite dimensional Euclidean spaces has allowed for the modelling of stochastic processes. However, many problems in the natural sciences incorporate symmetries and involve data living in non-Euclidean spaces. In this work, we extend the framework of diffusion models to incorporate a series of geometric priors in infinite-dimension modelling. We do so by a) constructing a noising process which admits, as limiting distribution, a geometric Gaussian process that transforms under the symmetry group of interest, and b) approximating the score with a neural network that is equivariant w. r. t. this group. We show that with these conditions, the generative functional model admits the same symmetry. We demonstrate scalability and capacity of the model, using a novel Langevin-based conditional sampler, to fit complex scalar and vector fields, with Euclidean and spherical codomain, on synthetic and real-world weather data.

NeurIPS Conference 2023 Conference Paper

Kronecker-Factored Approximate Curvature for Modern Neural Network Architectures

Runa Eschenhagen
Alexander Immer
Richard Turner
Frank Schneider
Philipp Hennig

The core components of many modern neural network architectures, such as transformers, convolutional, or graph neural networks, can be expressed as linear layers with *weight-sharing*. Kronecker-Factored Approximate Curvature (K-FAC), a second-order optimisation method, has shown promise to speed up neural network training and thereby reduce computational costs. However, there is currently no framework to apply it to generic architectures, specifically ones with linear weight-sharing layers. In this work, we identify two different settings of linear weight-sharing layers which motivate two flavours of K-FAC -- *expand* and *reduce*. We show that they are exact for deep linear networks with weight-sharing in their respective setting. Notably, K-FAC-reduce is generally faster than K-FAC-expand, which we leverage to speed up automatic hyperparameter selection via optimising the marginal likelihood for a Wide ResNet. Finally, we observe little difference between these two K-FAC variations when using them to train both a graph neural network and a vision transformer. However, both variations are able to reach a fixed validation metric target in $50$-$75$\% of the number of steps of a first-order reference run, which translates into a comparable improvement in wall-clock time. This highlights the potential of applying K-FAC to modern neural network architectures.

NeurIPS Conference 2023 Conference Paper

PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers

Phillip Lippe
Bas Veeling
Paris Perdikaris
Richard Turner
Johannes Brandstetter

Time-dependent partial differential equations (PDEs) are ubiquitous in science and engineering. Recently, mostly due to the high computational cost of traditional solution techniques, deep neural network based surrogates have gained increased interest. The practical utility of such neural PDE solvers relies on their ability to provide accurate, stable predictions over long time horizons, which is a notoriously hard problem. In this work, we present a large-scale analysis of common temporal rollout strategies, identifying the neglect of non-dominant spatial frequency information, often associated with high frequencies in PDE solutions, as the primary pitfall limiting stable, accurate rollout performance. Based on these insights, we draw inspiration from recent advances in diffusion models to introduce PDE-Refiner; a novel model class that enables more accurate modeling of all frequency components via a multistep refinement process. We validate PDE-Refiner on challenging benchmarks of complex fluid dynamics, demonstrating stable and accurate rollouts that consistently outperform state-of-the-art models, including neural, numerical, and hybrid neural-numerical architectures. We further demonstrate that PDE-Refiner greatly enhances data efficiency, since the denoising objective implicitly induces a novel form of spectral data augmentation. Finally, PDE-Refiner's connection to diffusion models enables an accurate and efficient assessment of the model's predictive uncertainty, allowing us to estimate when the surrogate becomes inaccurate.

NeurIPS Conference 2022 Conference Paper

Contextual Squeeze-and-Excitation for Efficient Few-Shot Image Classification

Massimiliano Patacchiola
John Bronskill
Aliaksandra Shysheya
Katja Hofmann
Sebastian Nowozin
Richard Turner

Recent years have seen a growth in user-centric applications that require effective knowledge transfer across tasks in the low-data regime. An example is personalization, where a pretrained system is adapted by learning on small amounts of labeled data belonging to a specific user. This setting requires high accuracy under low computational complexity, therefore the Pareto frontier of accuracy vs. adaptation cost plays a crucial role. In this paper we push this Pareto frontier in the few-shot image classification setting with a key contribution: a new adaptive block called Contextual Squeeze-and-Excitation (CaSE) that adjusts a pretrained neural network on a new task to significantly improve performance with a single forward pass of the user data (context). We use meta-trained CaSE blocks to conditionally adapt the body of a network and a fine-tuning routine to adapt a linear head, defining a method called UpperCaSE. UpperCaSE achieves a new state-of-the-art accuracy relative to meta-learners on the 26 datasets of VTAB+MD and on a challenging real-world personalization benchmark (ORBIT), narrowing the gap with leading fine-tuning methods with the benefit of orders of magnitude lower adaptation cost.

NeurIPS Conference 2021 Conference Paper

Collapsed Variational Bounds for Bayesian Neural Networks

Marcin Tomczak
Siddharth Swaroop
Andrew Foong
Richard Turner

Recent interest in learning large variational Bayesian Neural Networks (BNNs) has been partly hampered by poor predictive performance caused by underfitting, and their performance is known to be very sensitive to the prior over weights. Current practice often fixes the prior parameters to standard values or tunes them using heuristics or cross-validation. In this paper, we treat prior parameters in a distributional way by extending the model and collapsing the variational bound with respect to their posteriors. This leads to novel and tighter Evidence Lower Bounds (ELBOs) for performing variational inference (VI) in BNNs. Our experiments show that the new bounds significantly improve the performance of Gaussian mean-field VI applied to BNNs on a variety of data sets, demonstrating that mean-field VI works well even in deep models. We also find that the tighter ELBOs can be good optimization targets for learning the hyperparameters of hierarchical priors.

NeurIPS Conference 2021 Conference Paper

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

Andrew Foong
Wessel Bruinsma
David Burt
Richard Turner

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 find 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.

NeurIPS Conference 2021 Conference Paper

Memory Efficient Meta-Learning with Large Images

John Bronskill
Daniela Massiceti
Massimiliano Patacchiola
Katja Hofmann
Sebastian Nowozin
Richard Turner

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.

NeurIPS Conference 2020 Conference Paper

Continual Deep Learning by Functional Regularisation of Memorable Past

Pingbo Pan
Siddharth Swaroop
Alexander Immer
Runa Eschenhagen
Richard Turner
Mohammad Emtiyaz Khan

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

NeurIPS Conference 2020 Conference Paper

Efficient Low Rank Gaussian Variational Inference for Neural Networks

Marcin Tomczak
Siddharth Swaroop
Richard Turner

Bayesian neural networks are enjoying a renaissance driven in part by recent advances in variational inference (VI). The most common form of VI employs a fully factorized or mean-field distribution, but this is known to suffer from several pathologies, especially as we expect posterior distributions with highly correlated parameters. Current algorithms that capture these correlations with a Gaussian approximating family are difficult to scale to large models due to computational costs and high variance of gradient updates. By using a new form of the reparametrization trick, we derive a computationally efficient algorithm for performing VI with a Gaussian family with a low-rank plus diagonal covariance structure. We scale to deep feed-forward and convolutional architectures. We find that adding low-rank terms to parametrized diagonal covariance does not improve predictive performance except on small networks, but low-rank terms added to a constant diagonal covariance improves performance on small and large-scale network architectures.

NeurIPS Conference 2020 Conference Paper

Meta-Learning Stationary Stochastic Process Prediction with Convolutional Neural Processes

Andrew Foong
Wessel Bruinsma
Jonathan Gordon
Yann Dubois
James Requeima
Richard Turner

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.

NeurIPS Conference 2020 Conference Paper

On the Expressiveness of Approximate Inference in Bayesian Neural Networks

Andrew Foong
David Burt
Yingzhen Li
Richard Turner

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood. We study the quality of common variational methods in approximating the Bayesian predictive distribution. For single-hidden layer ReLU BNNs, we prove a fundamental limitation in function-space of two of the most commonly used distributions defined in weight-space: mean-field Gaussian and Monte Carlo dropout. We find there are simple cases where neither method can have substantially increased uncertainty in between well-separated regions of low uncertainty. We provide strong empirical evidence that exact inference does not have this pathology, hence it is due to the approximation and not the model. In contrast, for deep networks, we prove a universality result showing that there exist approximate posteriors in the above classes which provide flexible uncertainty estimates. However, we find empirically that pathologies of a similar form as in the single-hidden layer case can persist when performing variational inference in deeper networks. Our results motivate careful consideration of the implications of approximate inference methods in BNNs.

NeurIPS Conference 2020 Conference Paper

VAEM: a Deep Generative Model for Heterogeneous Mixed Type Data

Chao Ma
Sebastian Tschiatschek
Richard Turner
José Miguel Hernández-Lobato
Cheng Zhang

Deep generative models often perform poorly in real-world applications due to the heterogeneity of natural data sets. Heterogeneity arises from data containing different types of features (categorical, ordinal, continuous, etc. ) and features of the same type having different marginal distributions. We propose an extension of variational autoencoders (VAEs) called VAEM to handle such heterogeneous data. VAEM is a deep generative model that is trained in a two stage manner, such that the first stage provides a more uniform representation of the data to the second stage, thereby sidestepping the problems caused by heterogeneous data. We provide extensions of VAEM to handle partially observed data, and demonstrate its performance in data generation, missing data prediction and sequential feature selection tasks. Our results show that VAEM broadens the range of real-world applications where deep generative models can be successfully deployed.

NeurIPS Conference 2019 Conference Paper

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

James Requeima
Jonathan Gordon
John Bronskill
Sebastian Nowozin
Richard Turner

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.

NeurIPS Conference 2019 Conference Paper

Icebreaker: Element-wise Efficient Information Acquisition with a Bayesian Deep Latent Gaussian Model

Wenbo Gong
Sebastian Tschiatschek
Sebastian Nowozin
Richard Turner
José Miguel Hernández-Lobato
Cheng Zhang

In this paper, we address the ice-start problem, i. e. , the challenge of deploying machine learning models when only a little or no training data is initially available, and acquiring each feature element of data is associated with costs. This setting is representative of the real-world machine learning applications. For instance, in the health care domain, obtaining every single measurement comes with a cost. We propose Icebreaker, a principled framework for elementwise training data acquisition. Icebreaker introduces a full Bayesian Deep Latent Gaussian Model (BELGAM) with a novel inference method, which combines recent advances in amortized inference and stochastic gradient MCMC to enable fast and accurate posterior inference. By utilizing BELGAM’s ability to fully quantify model uncertainty, we also propose two information acquisition functions for imputation and active prediction problems. We demonstrate that BELGAM performs significantly better than previous variational autoencoder (VAE) based models, when the data set size is small, using both machine learning benchmarks and real world recommender systems and health-care applications. Moreover, Icebreaker not only demonstrates improved performance compared to baselines, but it is also capable of achieving better test performance with less training data available.

NeurIPS Conference 2019 Conference Paper

Practical Deep Learning with Bayesian Principles

Kazuki Osawa
Siddharth Swaroop
Mohammad Emtiyaz Khan
Anirudh Jain
Runa Eschenhagen
Richard Turner
Rio Yokota

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.

NeurIPS Conference 2018 Conference Paper

Geometrically Coupled Monte Carlo Sampling

Mark Rowland
Krzysztof Choromanski
François Chalus
Aldo Pacchiano
Tamas Sarlos
Richard Turner
Adrian Weller

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 QMC, by studying discrepancy. We explore our findings empirically, and observe benefits of our sampling schemes for reinforcement learning and generative modelling.

NeurIPS Conference 2018 Conference Paper

Infinite-Horizon Gaussian Processes

Arno Solin
James Hensman
Richard Turner

Gaussian processes provide a flexible framework for forecasting, removing noise, and interpreting long temporal datasets. State space modelling (Kalman filtering) enables these non-parametric models to be deployed on long datasets by reducing the complexity to linear in the number of data points. The complexity is still cubic in the state dimension m which is an impediment to practical application. In certain special cases (Gaussian likelihood, regular spacing) the GP posterior will reach a steady posterior state when the data are very long. We leverage this and formulate an inference scheme for GPs with general likelihoods, where inference is based on single-sweep EP (assumed density filtering). The infinite-horizon model tackles the cubic cost in the state dimensionality and reduces the cost in the state dimension m to O(m^2) per data point. The model is extended to online-learning of hyperparameters. We show examples for large finite-length modelling problems, and present how the method runs in real-time on a smartphone on a continuous data stream updated at 100 Hz.

JMLR Journal 2018 Journal Article

Invariant Models for Causal Transfer Learning

Mateo Rojas-Carulla
Bernhard Schölkopf
Richard Turner
Jonas Peters

Methods of transfer learning try to combine knowledge from several related tasks (or domains) to improve performance on a test task. Inspired by causal methodology, we relax the usual covariate shift assumption and assume that it holds true for a subset of predictor variables: the conditional distribution of the target variable given this subset of predictors is invariant over all tasks. We show how this assumption can be motivated from ideas in the field of causality. We focus on the problem of Domain Generalization, in which no examples from the test task are observed. We prove that in an adversarial setting using this subset for prediction is optimal in Domain Generalization; we further provide examples, in which the tasks are sufficiently diverse and the estimator therefore outperforms pooling the data, even on average. If examples from the test task are available, we also provide a method to transfer knowledge from the training tasks and exploit all available features for prediction. However, we provide no guarantees for this method. We introduce a practical method which allows for automatic inference of the above subset and provide corresponding code. We present results on synthetic data sets and a gene deletion data set. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

NeurIPS Conference 2017 Conference Paper

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

Shixiang (Shane) Gu
Timothy Lillicrap
Richard Turner
Zoubin Ghahramani
Bernhard Schölkopf
Sergey Levine

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.

NeurIPS Conference 2017 Conference Paper

Streaming Sparse Gaussian Process Approximations

Thang Bui
Cuong Nguyen
Richard Turner

Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates 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 methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is assessed using synthetic and real-world datasets.

AAAI Conference 2017 Conference Paper

The Multivariate Generalised von Mises Distribution: Inference and Applications

Alexandre Navarro
Jes Frellsen
Richard Turner

Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First we introduce a new multivariate distribution over circular variables, called the multivariate Generalised von Mises (mGvM) distribution. This distribution can be constructed by restricting and renormalising a general multivariate Gaussian distribution to the unit hyper-torus. Previously proposed multivariate circular distributions are shown to be special cases of this construction. Second, we introduce a new probabilistic model for circular regression inspired by Gaussian Processes, and a method for probabilistic Principal Component Analysis with circular hidden variables. These models can leverage standard modelling tools (e. g. kernel functions and automatic relevance determination). Third, we show that the posterior distribution in these models is a mGvM distribution which enables development of an efﬁcient variational free-energy scheme for performing approximate inference and approximate maximum-likelihood learning.

NeurIPS Conference 2016 Conference Paper

Rényi Divergence Variational Inference

Yingzhen Li
Richard Turner

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.

NeurIPS Conference 2015 Conference Paper

Learning Stationary Time Series using Gaussian Processes with Nonparametric Kernels

Felipe Tobar
Thang Bui
Richard Turner

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 nonparametric-window 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 free-energy 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.

NeurIPS Conference 2015 Conference Paper

Neural Adaptive Sequential Monte Carlo

Shixiang (Shane) Gu
Zoubin Ghahramani
Richard Turner

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 parameterized proposal distribution and it supports online and batch variants. We use the new framework to adapt powerful proposal distributions with rich parameterizations 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 latent variable recurrent neural network (LV-RNN) 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.

NeurIPS Conference 2015 Conference Paper

Stochastic Expectation Propagation

Yingzhen Li
José Miguel Hernández-Lobato
Richard Turner

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.

NeurIPS Conference 2014 Conference Paper

Tree-structured Gaussian Process Approximations

Thang Bui
Richard Turner

Gaussian process regression can be accelerated by constructing a small pseudo-dataset to summarise 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) minimisation. 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.

NeurIPS Conference 2011 Conference Paper

Probabilistic amplitude and frequency demodulation

Richard Turner
Maneesh Sahani

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.

NeurIPS Conference 2009 Conference Paper

Occlusive Components Analysis

Jörg Lücke
Richard Turner
Maneesh Sahani
Marc Henniges

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-cause component extraction methods.

NeurIPS Conference 2007 Conference Paper

Modeling Natural Sounds with Modulation Cascade Processes

Richard Turner
Maneesh Sahani

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 (~0. 1s); glottal pulses (~0. 01s); and formants (<0. 001s). The auditory system uses information from each of these time-scales to solve complicated tasks such as auditory scene analysis. 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.

NeurIPS Conference 2007 Conference Paper

On Sparsity and Overcompleteness in Image Models

Pietro Berkes
Richard Turner
Maneesh Sahani

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 overcomplete.