In many machine learning applications, well-calibrated posterior predictive distributions are required for a number of closely-related datasets. Given similarity between datasets, it is natural to wish to develop meta-learning algorithms that utilise other datasets to reduce the computational complexity and / or improve predictive performance when deploying models on newly-seen datasets at test time. There have been a number of significant recent developments in meta-learning for predictive distributions, most notably that of the neural process (NP) family (Garnelo et al., 2018a,b; Foong et al., 2020; Gordon et al., 2018, 2019). Despite the utility of these methods on large-scale meta-datasets, they perform poorly in settings where the number of datasets and the total number of datapoints is small. We argue that this is a result of the large number of shared model parameters overfitting to the meta-dataset.

Most of the existing methods for estimating the local intrinsic dimension of a data distribution do not scale well to high-dimensional data. Many of them rely on a non-parametric nearest neighbors approach which suffers from the curse of dimensionality. We attempt to address that challenge by proposing a novel approach to the problem: Local Intrinsic Dimension estimation using approximate Likelihood (LIDL). Our method relies on an arbitrary density estimation method as its subroutine and hence tries to sidestep the dimensionality challenge by making use of the recent progress in parametric neural methods for likelihood estimation. We carefully investigate the empirical properties of the proposed method, compare them with our theoretical predictions, and show that LIDL yields competitive results on the standard benchmarks for this problem and that it scales to thousands of dimensions. What is more, we anticipate this approach to improve further with the continuing advances in the density estimation literature.

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.

We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this'Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models. Keywords: Approximate Bayesian inference, optimisation, variational inference, expectation propagation, Gaussian processes.

Michael Jordan, Jason Lee, and Yun Yang just arXived a paper with their proposal on handling large datasets through distributed computing, thus contributing to the currently very active research topic of approximate solutions in large Bayesian models. The core of the proposal is summarised by the screenshot above, where the approximate likelihood replaces the exact likelihood with a first order Taylor expansion. The first term is the likelihood computed for a given subsample (or a given thread) at a ratio of one to N and the difference of the gradients is only computed once at a good enough guess. While the paper also considers M-estimators and non-Bayesian settings, the Bayesian part thus consists in running a regular MCMC when the log-target is approximated by the above. I first thought this proposal amounted to a Gaussian approximation à la Simon Wood or to an INLA approach but this is not the case: the first term of the approximate likelihood is exact and hence can be of any form, while the scalar product is linear in?, providing a sort of first order approximation, albeit frozen at the chosen starting value.

Probabilistic models can be defined by an energy function, where the probability of each state is proportional to the exponential of the state's negative energy. This paper considers a generalization of energy-based models in which the probability of a state is proportional to an arbitrary positive, strictly decreasing, and twice differentiable function of the state's energy. The precise shape of the nonlinear map from energies to unnormalized probabilities has to be learned from data together with the parameters of the energy function. As a case study we show that the above generalization of a fully visible Boltzmann machine yields an accurate model of neural activity of retinal ganglion cells. We attribute this success to the model's ability to easily capture distributions whose probabilities span a large dynamic range, a possible consequence of latent variables that globally couple the system. Similar features have recently been observed in many datasets, suggesting that our new method has wide applicability.