Continual Learning (CL) algorithms have recently received a lot of attention as they attempt to overcome the need to train with an i.i.d. sample from some unknown target data distribution. Building on prior work, we study principled ways to tackle the CL problem by adopting a Bayesian perspective and focus on continually learning a task-specific posterior distribution via a shared meta-model, a task-conditioned hypernetwork. This approach, which we term Posterior-replay CL, is in sharp contrast to most Bayesian CL approaches that focus on the recursive update of a single posterior distribution. The benefits of our approach are (1) an increased flexibility to model solutions in weight space and therewith less susceptibility to task dissimilarity, (2) access to principled task-specific predictive uncertainty estimates, that can be used to infer task identity during test time and to detect task boundaries during training, and (3) the ability to revisit and update task-specific posteriors in a principled manner without requiring access to past data. The proposed framework is versatile, which we demonstrate using simple posterior approximations (such as Gaussians) as well as powerful, implicit distributions modelled via a neural network. We illustrate the conceptual advance of our framework on low-dimensional problems and show performance gains on computer vision benchmarks.

Bayesian optimization (BO) is a powerful paradigm for efficient optimization of black-box objective functions. High-dimensional BO presents a particular challenge, in part because the curse of dimensionality makes it difficult to define as well as do inference over a suitable class of surrogate models. We argue that Gaussian process surrogate models defined on sparse axis-aligned subspaces offer an attractive compromise between flexibility and parsimony. We demonstrate that our approach, which relies on Hamiltonian Monte Carlo for inference, can rapidly identify sparse subspaces relevant to modeling the unknown objective function, enabling sample-efficient high-dimensional BO. In an extensive suite of experiments comparing to existing methods for high-dimensional BO we demonstrate that our algorithm, Sparse Axis-Aligned Subspace BO (SAASBO), achieves excellent performance on several synthetic and real-world problems without the need to set problem-specific hyperparameters.

We develop variational Laplace for Bayesian neural networks (BNNs) which exploits a local approximation of the curvature of the likelihood to estimate the ELBO without the need for stochastic sampling of the neural-network weights. Variational Laplace performs better on image classification tasks than MAP inference and far better than standard variational inference with stochastic sampling despite using the same mean-field Gaussian approximate posterior. The Variational Laplace objective is simple to evaluate, as it is (in essence) the log-likelihood, plus weight-decay, plus a squared-gradient regularizer. Finally, we emphasise care needed in benchmarking standard VI as there is a risk of stopping before the variance parameters have converged. We show that early-stopping can be avoided by increasing the learning rate for the variance parameters.

We will be discussing the Gaussian Mixture Model. A basic prerequisite to this blog is that one must know about the Gaussian distribution. The Gaussian is also called the normal distribution by statistics people. However, the GMM is called the GMM because in this scenario the G stands for Gaussian and it's not called a normal mixture model. Let's first start with a basic intuition of what a Gaussian mixture is by considering only a single Gaussian we can begin with a typical example.

Missing time-series data is a prevalent practical problem. Imputation methods in time-series data often are applied to the full panel data with the purpose of training a model for a downstream out-of-sample task. For example, in finance, imputation of missing returns may be applied prior to training a portfolio optimization model. Unfortunately, this practice may result in a look-ahead-bias in the future performance on the downstream task. There is an inherent trade-off between the look-ahead-bias of using the full data set for imputation and the larger variance in the imputation from using only the training data. By connecting layers of information revealed in time, we propose a Bayesian posterior consensus distribution which optimally controls the variance and look-ahead-bias trade-off in the imputation. We demonstrate the benefit of our methodology both in synthetic and real financial data.

We introduce a principled approach to detecting out-of-distribution (OOD) data by exploiting a connection to data curation. In data curation, we exclude ambiguous or difficult-to-classify input points from the dataset, and these excluded points are by definition OOD. We can therefore obtain the likelihood for OOD points by using a principled generative model of data-curation initially developed to explain the cold-posterior effect in Bayesian neural networks (Aitchison 2020). This model gives higher OOD probabilities when predictive uncertainty is higher and can be trained using maximum-likelihood jointly over the in-distribution and OOD points. This approach gives superior performance to past methods that did not provide a probability for OOD points, and therefore could not be trained using maximum-likelihood.

Purpose: We propose a general framework for quantifying predictive uncertainties of dose-related quantities and leveraging this information in a dose mimicking problem in the context of automated radiation therapy treatment planning. Methods: A three-step pipeline, comprising feature extraction, dose statistic prediction and dose mimicking, is employed. In particular, the features are produced by a convolutional variational autoencoder and used as inputs in a previously developed nonparametric Bayesian statistical method, estimating the multivariate predictive distribution of a collection of predefined dose statistics. Specially developed objective functions are then used to construct a dose mimicking problem based on the produced distributions, creating deliverable treatment plans. Results: The numerical experiments are performed using a dataset of 94 retrospective treatment plans of prostate cancer patients. We show that the features extracted by the variational autoencoder captures geometric information of substantial relevance to the dose statistic prediction problem, that the estimated predictive distributions are reasonable and outperforms a benchmark method, and that the deliverable plans agree well with their clinical counterparts. Conclusions: We demonstrate that prediction of dose-related quantities may be extended to include uncertainty estimation and that such probabilistic information may be leveraged in a dose mimicking problem. The treatment plans produced by the proposed pipeline resemble their original counterparts well, illustrating the merits of a holistic approach to automated planning based on probabilistic modeling.

This work considers variational Bayesian inference as an inexpensive and scalable alternative to a fully Bayesian approach in the context of sparsity-promoting priors. In particular, the priors considered arise from scale mixtures of Normal distributions with a generalized inverse Gaussian mixing distribution. This includes the variational Bayesian LASSO as an inexpensive and scalable alternative to the Bayesian LASSO introduced in [56]. It also includes priors which more strongly promote sparsity. For linear models the method requires only the iterative solution of deterministic least squares problems. Furthermore, for $n\rightarrow \infty$ data points and p unknown covariates the method can be implemented exactly online with a cost of O(p$^3$) in computation and O(p$^2$) in memory. For large p an approximation is able to achieve promising results for a cost of O(p) in both computation and memory. Strategies for hyper-parameter tuning are also considered. The method is implemented for real and simulated data. It is shown that the performance in terms of variable selection and uncertainty quantification of the variational Bayesian LASSO can be comparable to the Bayesian LASSO for problems which are tractable with that method, and for a fraction of the cost. The present method comfortably handles n = p = 131,073 on a laptop in minutes, and n = 10$^5$, p = 10$^6$ overnight.

Deep kernel learning and related techniques promise to combine the representational power of neural networks with the reliable uncertainty estimates of Gaussian processes. One crucial aspect of these models is an expectation that, because they are treated as Gaussian process models optimized using the marginal likelihood, they are protected from overfitting. However, we identify pathological behavior, including overfitting, on a simple toy example. We explore this pathology, explaining its origins and considering how it applies to real datasets. Through careful experimentation on UCI datasets, CIFAR-10, and the UTKFace dataset, we find that the overfitting from overparameterized deep kernel learning, in which the model is "somewhat Bayesian", can in certain scenarios be worse than that from not being Bayesian at all. However, we find that a fully Bayesian treatment of deep kernel learning can rectify this overfitting and obtain the desired performance improvements over standard neural networks and Gaussian processes.

Quantum machine learning is an emerging field at the intersection of machine learning and quantum computing. Classical cross entropy plays a central role in machine learning. We define its quantum generalization, the quantum cross entropy, and investigate its relations with the quantum fidelity and the maximum likelihood principle. We also discuss its physical implications on quantum measurements.