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.

Active learning is an area of modern machine learning that studies how data points can be sequentially selected for labeling to train a model with as few labeled examples as possible (Settles, 2009). Minimizing the number of labeled examples is critical in any learning scenario where labels are expensive to obtain, such as in healthcare applications where a medical expert must hand-label each training example (Liu, 2004), or where only a limited number of examples can be evaluated, such as in drug discovery (Warmuth et al., 2003). The active selection of data points shares many technical parallels with channel coding with feedback, where a message is encoded into a sequence of symbols transmitted across a noisy channel and each symbol is selected based on the message and past channel outputs. In active learning, the optimal classifier parameters play the role of the "message" while the sequence of examples with noisy labels plays the role of "channel outputs" available through feedback to select the next example for labeling. Both feedback channel coding and active learning seek to minimize the number of encoder actions, leverage a history of noisy observations to select the next most informative action, must account for observation noise, and should operate in a computationally efficient manner. Although there exists a large literature studying the intersection of information theory with machine learning (Xu and Raginsky, 2017) and specifically active learning (Naghshvar et al., 2015), there remain open questions about the best ways to directly leverage techniques in channel coding for active example selection. The main contribution of this work is a formulation of general active learning problems in terms of a feedback coding system, and a demonstration of this approach through the application and analysis of active learning in logistic regression. To motivate this approach, we first examine active learning through the lens of feedback channel coding by identifying communications system components, including a deterministic encoder, noisy channel, channel input constraints, and capacity-achieving distribution. With these components identified, we show how typical structural constraints in active learning problems prevent the direct application of existing feedback coding approaches such as posterior matching (Ma and Coleman, 2011).

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.

Local causal structure learning aims to discover and distinguish direct causes (parents) and direct effects (children) of a variable of interest from data. While emerging successes have been made, existing methods need to search a large space to distinguish direct causes from direct effects of a target variable T. To tackle this issue, we propose a novel Efficient Local Causal Structure learning algorithm, named ELCS. Specifically, we first propose the concept of N-structures, then design an efficient Markov Blanket (MB) discovery subroutine to integrate MB learning with N-structures to learn the MB of T and simultaneously distinguish direct causes from direct effects of T. With the proposed MB subroutine, ELCS starts from the target variable, sequentially finds MBs of variables connected to the target variable and simultaneously constructs local causal structures over MBs until the direct causes and direct effects of the target variable have been distinguished. Using eight Bayesian networks the extensive experiments have validated that ELCS achieves better accuracy and efficiency than the state-of-the-art algorithms.

Markov Decision Process (MDP) is the underlying model for optimal planning for decision-theoretic agents in stochastic environments. Although much research focuses on solving MDP problems both in tabular form or using factored representations, none focused on tensor decomposition methods. Solving MDPs using tensor algebra offers the prospect of leveraging advances in tensor-based computations to further increase solver efficiency. In this paper, we develop an MDP solver for a multidimensional problem using a tensor decomposition method to compress the transition models and optimize the value iteration and policy iteration algorithms. We empirically evaluate our approach against tabular methods and show our approach can compute much larger problems using substantially less memory, opening up new possibilities for tensor-based approaches in stochastic planning

One of the most important problems in transfer learning is the task of domain adaptation, where the goal is to apply an algorithm trained in one or more source domains to a different (but related) target domain. This paper deals with domain adaptation in the presence of covariate shift while there exist invariances across domains. A main limitation of existing causal inference methods for solving this problem is scalability. To overcome this difficulty, we propose SCTL, an algorithm that avoids an exhaustive search and identifies invariant causal features across the source and target domains based on Markov blanket discovery. SCTL does not require to have prior knowledge of the causal structure, the type of interventions, or the intervention targets. There is an intrinsic locality associated with SCTL that makes SCTL practically scalable and robust because local causal discovery increases the power of computational independence tests and makes the task of domain adaptation computationally tractable. We show the scalability and robustness of SCTL for domain adaptation using synthetic and real data sets in low-dimensional and high-dimensional settings.

Naive Bayes is a classification algorithm for binary (two-class) and multiclass classification problems. It is called Naive Bayes or idiot Bayes because the calculations of the probabilities for each class are simplified to make their calculations tractable. Rather than attempting to calculate the probabilities of each attribute value, they are assumed to be conditionally independent given the class value. This is a very strong assumption that is most unlikely in real data, i.e. that the attributes do not interact. Nevertheless, the approach performs surprisingly well on data where this assumption does not hold.

Additive noise models are a class of causal models in which each variable is defined as a function of its causes plus independent noise. In such models, the ordering of variables by marginal variances may be indicative of the causal order. We introduce varsortability as a measure of agreement between the ordering by marginal variance and the causal order. We show how varsortability dominates the performance of continuous structure learning algorithms on synthetic data. On real-world data, varsortability is an implausible and untestable assumption and we find no indication of high varsortability. We aim to raise awareness that varsortability easily occurs in simulated additive noise models. We provide a baseline method that explicitly exploits varsortability and advocate reporting varsortability in benchmarking data.

In this work we propose a batch Bayesian optimization method for combinatorial problems on permutations, which is well suited for expensive cost functions on permutations. We introduce LAW, a new efficient batch acquisition method based on the determinantal point process, using an acquisition weighted kernel. Relying on multiple parallel evaluations, LAW accelerates the search for the optimal permutation. We provide a regret analysis for our method to gain insight in its theoretical properties. We then apply the framework to permutation problems, which have so far received little attention in the Bayesian Optimization literature, despite their practical importance. We call this method LAW2ORDER. We evaluate the method on several standard combinatorial problems involving permutations such as quadratic assignment, flowshop scheduling and the traveling salesman, as well as on a structure learning task.