Exact, Fast and Expressive Poisson Point Processes via Squared Neural Families

We introduce squared neural Poisson point processes (SNEPPPs) by parameterising the intensity function by the squared norm of a two layer neural network. When the hidden layer is fixed and the second layer has a single neuron, our approach resembles previous uses of squared Gaussian process or kernel methods, but allowing the hidden layer to be learnt allows for additional flexibility. In many cases of interest, the integrated intensity function admits a closed form and can be computed in quadratic time in the number of hidden neurons. We enumerate a far more extensive number of such cases than has previously been discussed. Our approach is more memory and time efficient than naive implementations of squared or exponentiated kernel methods or Gaussian processes. Maximum likelihood and maximum a posteriori estimates in a reparameterisation of the final layer of the intensity function can be obtained by solving a (strongly) convex optimisation problem using projected gradient descent. We demonstrate SNEPPPs on real, and synthetic benchmarks, and provide a software implementation. https://github.com/RussellTsuchida/snefy

Gaussian Ensemble Belief Propagation for Efficient Inference in High-Dimensional Systems

Efficient inference in high-dimensional models remains a central challenge in machine learning. This paper introduces the Gaussian Ensemble Belief Propagation (GEnBP) algorithm, a fusion of the Ensemble Kalman filter and Gaussian belief propagation (GaBP) methods. GEnBP updates ensembles by passing low-rank local messages in a graphical model structure. This combination inherits favourable qualities from each method. Ensemble techniques allow GEnBP to handle high-dimensional states, parameters and intricate, noisy, black-box generation processes. The use of local messages in a graphical model structure ensures that the approach is suited to distributed computing and can efficiently handle complex dependence structures. GEnBP is particularly advantageous when the ensemble size is considerably smaller than the inference dimension. This scenario often arises in fields such as spatiotemporal modelling, image processing and physical model inversion. GEnBP can be applied to general problem structures, including jointly learning system parameters, observation parameters, and latent state variables.

Squared Neural Families: A New Class of Tractable Density Models

Flexible models for probability distributions are an essential ingredient in many machine learning tasks. We develop and investigate a new class of probability distributions, which we call a Squared Neural Family (SNEFY), formed by squaring the 2-norm of a neural network and normalising it with respect to a base measure. Following the reasoning similar to the well established connections between infinitely wide neural networks and Gaussian processes, we show that SNEFYs admit a closed form normalising constants in many cases of interest, thereby resulting in flexible yet fully tractable density models. SNEFYs strictly generalise classical exponential families, are closed under conditioning, and have tractable marginal distributions. Their utility is illustrated on a variety of density estimation and conditional density estimation tasks. Software available at https://github.com/RussellTsuchida/snefy.

Earth Movers in The Big Data Era: A Review of Optimal Transport in Machine Learning

Optimal Transport (OT) is a mathematical framework that first emerged in the eighteenth century and has led to a plethora of methods for answering many theoretical and applied questions. The last decade is a witness of the remarkable contributions of this classical optimization problem to machine learning. This paper is about where and how optimal transport is used in machine learning with a focus on the question of salable optimal transport. We provide a comprehensive survey of optimal transport while ensuring an accessible presentation as permitted by the nature of the topic and the context. First, we explain optimal transport background and introduce different flavors (i.e. mathematical formulations), properties, and notable applications. We then address the fundamental question of how to scale optimal transport to cope with the current demands of big and high dimensional data. We conduct a systematic analysis of the methods used in the literature for scaling OT and present the findings in a unified taxonomy. We conclude with presenting some open challenges and discussing potential future research directions. A live repository of related OT research papers is maintained in https://github.com/abdelwahed/OT_for_big_data.git.

Deep equilibrium models as estimators for continuous latent variables

Principal Component Analysis (PCA) and its exponential family extensions have three components: observations, latents and parameters of a linear transformation. We consider a generalised setting where the canonical parameters of the exponential family are a nonlinear transformation of the latents. We show explicit relationships between particular neural network architectures and the corresponding statistical models. We find that deep equilibrium models -- a recently introduced class of implicit neural networks -- solve maximum a-posteriori (MAP) estimates for the latents and parameters of the transformation. Our analysis provides a systematic way to relate activation functions, dropout, and layer structure, to statistical assumptions about the observations, thus providing foundational principles for unsupervised DEQs. For hierarchical latents, individual neurons can be interpreted as nodes in a deep graphical model. Our DEQ feature maps are end-to-end differentiable, enabling fine-tuning for downstream tasks.

Efficient Gaussian Process Model on Class-Imbalanced Datasets for Generalized Zero-Shot Learning

Zero-Shot Learning (ZSL) models aim to classify object classes that are not seen during the training process. However, the problem of class imbalance is rarely discussed, despite its presence in several ZSL datasets. In this paper, we propose a Neural Network model that learns a latent feature embedding and a Gaussian Process (GP) regression model that predicts latent feature prototypes of unseen classes. A calibrated classifier is then constructed for ZSL and Generalized ZSL tasks. Our Neural Network model is trained efficiently with a simple training strategy that mitigates the impact of class-imbalanced training data. The model has an average training time of 5 minutes and can achieve state-of-the-art (SOTA) performance on imbalanced ZSL benchmark datasets like AWA2, AWA1 and APY, while having relatively good performance on the SUN and CUB datasets.

Gaussian Process Bandits with Aggregated Feedback

We consider the continuum-armed bandits problem, under a novel setting of recommending the best arms within a fixed budget under aggregated feedback. This is motivated by applications where the precise rewards are impossible or expensive to obtain, while an aggregated reward or feedback, such as the average over a subset, is available. We constrain the set of reward functions by assuming that they are from a Gaussian Process and propose the Gaussian Process Optimistic Optimisation (GPOO) algorithm. We adaptively construct a tree with nodes as subsets of the arm space, where the feedback is the aggregated reward of representatives of a node. We propose a new simple regret notion with respect to aggregated feedback on the recommended arms. We provide theoretical analysis for the proposed algorithm, and recover single point feedback as a special case. We illustrate GPOO and compare it with related algorithms on simulated data.

Avoiding Kernel Fixed Points: Computing with ELU and GELU Infinite Networks

Analysing and computing with Gaussian processes arising from infinitely wide neural networks has recently seen a resurgence in popularity. Despite this, many explicit covariance functions of networks with activation functions used in modern networks remain unknown. Furthermore, while the kernels of deep networks can be computed iteratively, theoretical understanding of deep kernels is lacking, particularly with respect to fixed-point dynamics. Firstly, we derive the covariance functions of MLPs with exponential linear units and Gaussian error linear units and evaluate the performance of the limiting Gaussian processes on some benchmarks. Secondly, and more generally, we introduce a framework for analysing the fixed-point dynamics of iterated kernels corresponding to a broad range of activation functions. We find that unlike some previously studied neural network kernels, these new kernels exhibit non-trivial fixed-point dynamics which are mirrored in finite-width neural networks.

Richer priors for infinitely wide multi-layer perceptrons

It is well-known that the distribution over functions induced through a zero-mean iid prior distribution over the parameters of a multi-layer perceptron (MLP) converges to a Gaussian process (GP), under mild conditions. We extend this result firstly to independent priors with general zero or non-zero means, and secondly to a family of partially exchangeable priors which generalise iid priors. We discuss how the second prior arises naturally when considering an equivalence class of functions in an MLP and through training processes such as stochastic gradient descent. The model resulting from partially exchangeable priors is a GP, with an additional level of inference in the sense that the prior and posterior predictive distributions require marginalisation over hyperparameters. We derive the kernels of the limiting GP in deep MLPs, and show empirically that these kernels avoid certain pathologies present in previously studied priors. We empirically evaluate our claims of convergence by measuring the maximum mean discrepancy between finite width models and limiting models. We compare the performance of our new limiting model to some previously discussed models on synthetic regression problems. We observe increasing ill-conditioning of the marginal likelihood and hyper-posterior as the depth of the model increases, drawing parallels with finite width networks which require notoriously involved optimisation tricks.

Exchangeability and Kernel Invariance in Trained MLPs

In the analysis of machine learning models, it is often convenient to assume that the parameters are IID. This assumption is not satisfied when the parameters are updated through training processes such as SGD. A relaxation of the IID condition is a probabilistic symmetry known as exchangeability. We show the sense in which the weights in MLPs are exchangeable. This yields the result that in certain instances, the layer-wise kernel of fully-connected layers remains approximately constant during training. We identify a sharp change in the macroscopic behavior of networks as the covariance between weights changes from zero.