arXiv subjects

Michalis K. Titsias

Publications and source records attributed to Michalis K. Titsias.

At least 37 records · Page 2Linked to original sources

arXiv2021

Information Theoretic Meta Learning with Gaussian Processes

We formulate meta learning using information theoretic concepts; namely, mutual information and the information bottleneck. The idea is to learn a stochastic representation or encoding of the task description, given by a training set, that is highly informative about predicting the validation set. By making use of variational approximations to the mutual information, we derive a general and tractable framework for meta learning. This framework unifies existing gradient-based algorithms and also allows us to derive new algorithms. In particular, we develop a memory-based algorithm that uses Gaussian processes to obtain non-parametric encoding representations. We demonstrate our method on a few-shot regression problem and on four few-shot classification problems, obtaining competitive accuracy when compared to existing baselines.

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arXiv2021

Unbiased Gradient Estimation for Variational Auto-Encoders using Coupled Markov Chains

The variational auto-encoder (VAE) is a deep latent variable model that has two neural networks in an autoencoder-like architecture; one of them parameterizes the model's likelihood. Fitting its parameters via maximum likelihood (ML) is challenging since the computation of the marginal likelihood involves an intractable integral over the latent space; thus the VAE is trained instead by maximizing a variational lower bound. Here, we develop a ML training scheme for VAEs by introducing unbiased estimators of the log-likelihood gradient. We obtain the estimators by augmenting the latent space with a set of importance samples, similarly to the importance weighted auto-encoder (IWAE), and then constructing a Markov chain Monte Carlo coupling procedure on this augmented space. We provide the conditions under which the estimators can be computed in finite time and with finite variance. We show experimentally that VAEs fitted with unbiased estimators exhibit better predictive performance.

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arXiv2020

Sequential Changepoint Detection in Neural Networks with Checkpoints

We introduce a framework for online changepoint detection and simultaneous model learning which is applicable to highly parametrized models, such as deep neural networks. It is based on detecting changepoints across time by sequentially performing generalized likelihood ratio tests that require only evaluations of simple prediction score functions. This procedure makes use of checkpoints, consisting of early versions of the actual model parameters, that allow to detect distributional changes by performing predictions on future data. We define an algorithm that bounds the Type I error in the sequential testing procedure. We demonstrate the efficiency of our method in challenging continual learning applications with unknown task changepoints, and show improved performance compared to online Bayesian changepoint detection.

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arXiv2020

Functional Regularisation for Continual Learning with Gaussian Processes

We introduce a framework for Continual Learning (CL) based on Bayesian inference over the function space rather than the parameters of a deep neural network. This method, referred to as functional regularisation for Continual Learning, avoids forgetting a previous task by constructing and memorising an approximate posterior belief over the underlying task-specific function. To achieve this we rely on a Gaussian process obtained by treating the weights of the last layer of a neural network as random and Gaussian distributed. Then, the training algorithm sequentially encounters tasks and constructs posterior beliefs over the task-specific functions by using inducing point sparse Gaussian process methods. At each step a new task is first learnt and then a summary is constructed consisting of (i) inducing inputs -- a fixed-size subset of the task inputs selected such that it optimally represents the task -- and (ii) a posterior distribution over the function values at these inputs. This summary then regularises learning of future tasks, through Kullback-Leibler regularisation terms. Our method thus unites approaches focused on (pseudo-)rehearsal with those derived from a sequential Bayesian inference perspective in a principled way, leading to strong results on accepted benchmarks.

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arXiv2020

Gradient-based Adaptive Markov Chain Monte Carlo

We introduce a gradient-based learning method to automatically adapt Markov chain Monte Carlo (MCMC) proposal distributions to intractable targets. We define a maximum entropy regularised objective function, referred to as generalised speed measure, which can be robustly optimised over the parameters of the proposal distribution by applying stochastic gradient optimisation. An advantage of our method compared to traditional adaptive MCMC methods is that the adaptation occurs even when candidate state values are rejected. This is a highly desirable property of any adaptation strategy because the adaptation starts in early iterations even if the initial proposal distribution is far from optimum. We apply the framework for learning multivariate random walk Metropolis and Metropolis-adjusted Langevin proposals with full covariance matrices, and provide empirical evidence that our method can outperform other MCMC algorithms, including Hamiltonian Monte Carlo schemes.

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arXiv2019

Prescribed Generative Adversarial Networks

Generative adversarial networks (GANs) are a powerful approach to unsupervised learning. They have achieved state-of-the-art performance in the image domain. However, GANs are limited in two ways. They often learn distributions with low support---a phenomenon known as mode collapse---and they do not guarantee the existence of a probability density, which makes evaluating generalization using predictive log-likelihood impossible. In this paper, we develop the prescribed GAN (PresGAN) to address these shortcomings. PresGANs add noise to the output of a density network and optimize an entropy-regularized adversarial loss. The added noise renders tractable approximations of the predictive log-likelihood and stabilizes the training procedure. The entropy regularizer encourages PresGANs to capture all the modes of the data distribution. Fitting PresGANs involves computing the intractable gradients of the entropy regularization term; PresGANs sidestep this intractability using unbiased stochastic estimates. We evaluate PresGANs on several datasets and found they mitigate mode collapse and generate samples with high perceptual quality. We further found that PresGANs reduce the gap in performance in terms of predictive log-likelihood between traditional GANs and variational autoencoders (VAEs).

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arXiv2019

A Contrastive Divergence for Combining Variational Inference and MCMC

We develop a method to combine Markov chain Monte Carlo (MCMC) and variational inference (VI), leveraging the advantages of both inference approaches. Specifically, we improve the variational distribution by running a few MCMC steps. To make inference tractable, we introduce the variational contrastive divergence (VCD), a new divergence that replaces the standard Kullback-Leibler (KL) divergence used in VI. The VCD captures a notion of discrepancy between the initial variational distribution and its improved version (obtained after running the MCMC steps), and it converges asymptotically to the symmetrized KL divergence between the variational distribution and the posterior of interest. The VCD objective can be optimized efficiently with respect to the variational parameters via stochastic optimization. We show experimentally that optimizing the VCD leads to better predictive performance on two latent variable models: logistic matrix factorization and variational autoencoders (VAEs).

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arXiv2019

Unbiased Implicit Variational Inference

We develop unbiased implicit variational inference (UIVI), a method that expands the applicability of variational inference by defining an expressive variational family. UIVI considers an implicit variational distribution obtained in a hierarchical manner using a simple reparameterizable distribution whose variational parameters are defined by arbitrarily flexible deep neural networks. Unlike previous works, UIVI directly optimizes the evidence lower bound (ELBO) rather than an approximation to the ELBO. We demonstrate UIVI on several models, including Bayesian multinomial logistic regression and variational autoencoders, and show that UIVI achieves both tighter ELBO and better predictive performance than existing approaches at a similar computational cost.

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arXiv2018

Bayesian Transfer Reinforcement Learning with Prior Knowledge Rules

We propose a probabilistic framework to directly insert prior knowledge in reinforcement learning (RL) algorithms by defining the behaviour policy as a Bayesian posterior distribution. Such a posterior combines task specific information with prior knowledge, thus allowing to achieve transfer learning across tasks. The resulting method is flexible and it can be easily incorporated to any standard off-policy and on-policy algorithms, such as those based on temporal differences and policy gradients. We develop a specific instance of this Bayesian transfer RL framework by expressing prior knowledge as general deterministic rules that can be useful in a large variety of tasks, such as navigation tasks. Also, we elaborate more on recent probabilistic and entropy-regularised RL by developing a novel temporal learning algorithm and show how to combine it with Bayesian transfer RL. Finally, we demonstrate our method for solving mazes and show that significant speed ups can be obtained.

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arXiv2018

Fully Scalable Gaussian Processes using Subspace Inducing Inputs

We introduce fully scalable Gaussian processes, an implementation scheme that tackles the problem of treating a high number of training instances together with high dimensional input data. Our key idea is a representation trick over the inducing variables called subspace inducing inputs. This is combined with certain matrix-preconditioning based parametrizations of the variational distributions that lead to simplified and numerically stable variational lower bounds. Our illustrative applications are based on challenging extreme multi-label classification problems with the extra burden of the very large number of class labels. We demonstrate the usefulness of our approach by presenting predictive performances together with low computational times in datasets with extremely large number of instances and input dimensions.

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arXiv2018

Augment and Reduce: Stochastic Inference for Large Categorical Distributions

Categorical distributions are ubiquitous in machine learning, e.g., in classification, language models, and recommendation systems. However, when the number of possible outcomes is very large, using categorical distributions becomes computationally expensive, as the complexity scales linearly with the number of outcomes. To address this problem, we propose augment and reduce (A&R), a method to alleviate the computational complexity. A&R uses two ideas: latent variable augmentation and stochastic variational inference. It maximizes a lower bound on the marginal likelihood of the data. Unlike existing methods which are specific to softmax, A&R is more general and is amenable to other categorical models, such as multinomial probit. On several large-scale classification problems, we show that A&R provides a tighter bound on the marginal likelihood and has better predictive performance than existing approaches.

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arXiv2018

Auxiliary gradient-based sampling algorithms

We introduce a new family of MCMC samplers that combine auxiliary variables, Gibbs sampling and Taylor expansions of the target density. Our approach permits the marginalisation over the auxiliary variables yielding marginal samplers, or the augmentation of the auxiliary variables, yielding auxiliary samplers. The well-known Metropolis-adjusted Langevin algorithm (MALA) and preconditioned Crank-Nicolson Langevin (pCNL) algorithm are shown to be special cases. We prove that marginal samplers are superior in terms of asymptotic variance and demonstrate cases where they are slower in computing time compared to auxiliary samplers. In the context of latent Gaussian models we propose new auxiliary and marginal samplers whose implementation requires a single tuning parameter, which can be found automatically during the transient phase. Extensive experimentation shows that the increase in efficiency (measured as effective sample size per unit of computing time) relative to (optimised implementations of) pCNL, elliptical slice sampling and MALA ranges from 10-fold in binary classification problems to 25-fold in log-Gaussian Cox processes to 100-fold in Gaussian process regression, and it is on par with Riemann manifold Hamiltonian Monte Carlo in an example where the latter has the same complexity as the aforementioned algorithms. We explain this remarkable improvement in terms of the way alternative samplers try to approximate the eigenvalues of the target. We introduce a novel MCMC sampling scheme for hyperparameter learning that builds upon the auxiliary samplers. The MATLAB code for reproducing the experiments in the article is publicly available and a Supplement to this article contains additional experiments and implementation details.

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arXiv2017

Learning Model Reparametrizations: Implicit Variational Inference by Fitting MCMC distributions

We introduce a new algorithm for approximate inference that combines reparametrization, Markov chain Monte Carlo and variational methods. We construct a very flexible implicit variational distribution synthesized by an arbitrary Markov chain Monte Carlo operation and a deterministic transformation that can be optimized using the reparametrization trick. Unlike current methods for implicit variational inference, our method avoids the computation of log density ratios and therefore it is easily applicable to arbitrary continuous and differentiable models. We demonstrate the proposed algorithm for fitting banana-shaped distributions and for training variational autoencoders.

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arXiv2017

Bayesian Boolean Matrix Factorisation

Boolean matrix factorisation aims to decompose a binary data matrix into an approximate Boolean product of two low rank, binary matrices: one containing meaningful patterns, the other quantifying how the observations can be expressed as a combination of these patterns. We introduce the OrMachine, a probabilistic generative model for Boolean matrix factorisation and derive a Metropolised Gibbs sampler that facilitates efficient parallel posterior inference. On real world and simulated data, our method outperforms all currently existing approaches for Boolean matrix factorisation and completion. This is the first method to provide full posterior inference for Boolean Matrix factorisation which is relevant in applications, e.g. for controlling false positive rates in collaborative filtering and, crucially, improves the interpretability of the inferred patterns. The proposed algorithm scales to large datasets as we demonstrate by analysing single cell gene expression data in 1.3 million mouse brain cells across 11 thousand genes on commodity hardware.

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arXiv2016

One-vs-Each Approximation to Softmax for Scalable Estimation of Probabilities

The softmax representation of probabilities for categorical variables plays a prominent role in modern machine learning with numerous applications in areas such as large scale classification, neural language modeling and recommendation systems. However, softmax estimation is very expensive for large scale inference because of the high cost associated with computing the normalizing constant. Here, we introduce an efficient approximation to softmax probabilities which takes the form of a rigorous lower bound on the exact probability. This bound is expressed as a product over pairwise probabilities and it leads to scalable estimation based on stochastic optimization. It allows us to perform doubly stochastic estimation by subsampling both training instances and class labels. We show that the new bound has interesting theoretical properties and we demonstrate its use in classification problems.

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arXiv2016

The Generalized Reparameterization Gradient

The reparameterization gradient has become a widely used method to obtain Monte Carlo gradients to optimize the variational objective. However, this technique does not easily apply to commonly used distributions such as beta or gamma without further approximations, and most practical applications of the reparameterization gradient fit Gaussian distributions. In this paper, we introduce the generalized reparameterization gradient, a method that extends the reparameterization gradient to a wider class of variational distributions. Generalized reparameterizations use invertible transformations of the latent variables which lead to transformed distributions that weakly depend on the variational parameters. This results in new Monte Carlo gradients that combine reparameterization gradients and score function gradients. We demonstrate our approach on variational inference for two complex probabilistic models. The generalized reparameterization is effective: even a single sample from the variational distribution is enough to obtain a low-variance gradient.

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arXiv2016

Overdispersed Black-Box Variational Inference

We introduce overdispersed black-box variational inference, a method to reduce the variance of the Monte Carlo estimator of the gradient in black-box variational inference. Instead of taking samples from the variational distribution, we use importance sampling to take samples from an overdispersed distribution in the same exponential family as the variational approximation. Our approach is general since it can be readily applied to any exponential family distribution, which is the typical choice for the variational approximation. We run experiments on two non-conjugate probabilistic models to show that our method effectively reduces the variance, and the overhead introduced by the computation of the proposal parameters and the importance weights is negligible. We find that our overdispersed importance sampling scheme provides lower variance than black-box variational inference, even when the latter uses twice the number of samples. This results in faster convergence of the black-box inference procedure.

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arXiv2015

Inference for determinantal point processes without spectral knowledge

Determinantal point processes (DPPs) are point process models that naturally encode diversity between the points of a given realization, through a positive definite kernel $K$. DPPs possess desirable properties, such as exact sampling or analyticity of the moments, but learning the parameters of kernel $K$ through likelihood-based inference is not straightforward. First, the kernel that appears in the likelihood is not $K$, but another kernel $L$ related to $K$ through an often intractable spectral decomposition. This issue is typically bypassed in machine learning by directly parametrizing the kernel $L$, at the price of some interpretability of the model parameters. We follow this approach here. Second, the likelihood has an intractable normalizing constant, which takes the form of a large determinant in the case of a DPP over a finite set of objects, and the form of a Fredholm determinant in the case of a DPP over a continuous domain. Our main contribution is to derive bounds on the likelihood of a DPP, both for finite and continuous domains. Unlike previous work, our bounds are cheap to evaluate since they do not rely on approximating the spectrum of a large matrix or an operator. Through usual arguments, these bounds thus yield cheap variational inference and moderately expensive exact Markov chain Monte Carlo inference methods for DPPs.

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