\textit{Adaptive importance sampling} (AIS) uses past samples to update the \textit{sampling policy} $q_t$ at each stage $t$. Each stage $t$ is formed with two steps : (i) to explore the space with $n_t$ points according to $q_t$ and (ii) to exploit the current amount of information to update the sampling policy. The very fundamental question raised in this paper concerns the behavior of empirical sums based on AIS. Without making any assumption on the \textit{allocation policy} $n_t$, the theory developed involves no restriction on the split of computational resources between the explore (i) and the exploit (ii) step. It is shown that AIS is asymptotically optimal : the asymptotic behavior of AIS is the same as some ``oracle'' strategy that knows the targeted sampling policy from the beginning. From a practical perspective, weighted AIS is introduced, a new method that allows to forget poor samples from early stages.

Probability estimation is one of the fundamental tasks in statistics and machine learning. However, standard methods for probability estimation on discrete objects do not handle object structure in a satisfactory manner. In this paper, we derive a general Bayesian network formulation for probability estimation on leaf-labeled trees that enables flexible approximations which can generalize beyond observations. We show that efficient algorithms for learning Bayesian networks can be easily extended to probability estimation on this challenging structured space. Experiments on both synthetic and real data show that our methods greatly outperform the current practice of using the empirical distribution, as well as a previous effort for probability estimation on trees.

Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space the time-discretization error can dominate when we are near the boundary of the space. We demonstrate that because of this, current SGMCMC methods for the simplex struggle with sparse simplex spaces; when many of the components are close to zero. Unfortunately, many popular large-scale Bayesian models, such as network or topic models, require inference on sparse simplex spaces. To avoid the biases caused by this discretization error, we propose the stochastic Cox-Ingersoll-Ross process (SCIR), which removes all discretization error and we prove that samples from the SCIR process are asymptotically unbiased. We discuss how this idea can be extended to target other constrained spaces. Use of the SCIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.

Collecting the large datasets needed to train deep neural networks can be very difficult, particularly for the many applications for which sharing and pooling data is complicated by practical, ethical, or legal concerns. However, it may be the case that derivative datasets or predictive models developed within individual sites can be shared and combined with fewer restrictions. Training on distributed data and combining the resulting networks is often viewed as continual learning, but these methods require networks to be trained sequentially. In this paper, we introduce distributed weight consolidation (DWC), a continual learning method to consolidate the weights of separate neural networks, each trained on an independent dataset. We evaluated DWC with a brain segmentation case study, where we consolidated dilated convolutional neural networks trained on independent structural magnetic resonance imaging (sMRI) datasets from different sites. We found that DWC led to increased performance on test sets from the different sites, while maintaining generalization performance for a very large and completely independent multi-site dataset, compared to an ensemble baseline.

Conditional Density Estimation (CDE) models deal with estimating conditional distributions. The conditions imposed on the distribution are the inputs of the model. CDE is a challenging task as there is a fundamental trade-off between model complexity, representational capacity and overfitting. In this work, we propose to extend the model's input with latent variables and use Gaussian processes (GP) to map this augmented input onto samples from the conditional distribution. Our Bayesian approach allows for the modeling of small datasets, but we also provide the machinery for it to be applied to big data using stochastic variational inference. Our approach can be used to model densities even in sparse data regions, and allows for sharing learned structure between conditions. We illustrate the effectiveness and wide-reaching applicability of our model on a variety of real-world problems, such as spatio-temporal density estimation of taxi drop-offs, non-Gaussian noise modeling, and few-shot learning on omniglot images.

Learning from pairwise measurements naturally arises from many applications, such as rank aggregation, ordinal embedding, and crowdsourcing. However, most existing models and algorithms are susceptible to potential model misspecification. In this paper, we study a semiparametric model where the pairwise measurements follow a natural exponential family distribution with an unknown base measure. Such a semiparametric model includes various popular parametric models, such as the Bradley-Terry-Luce model and the paired cardinal model, as special cases. To estimate this semiparametric model without specifying the base measure, we propose a data augmentation technique to create virtual examples, which enables us to define a contrastive estimator. In particular, we prove that such a contrastive estimator is invariant to model misspecification within the natural exponential family, and moreover, attains the optimal statistical rate of convergence up to a logarithmic factor. We provide numerical experiments to corroborate our theory.

We address the problem of Bayesian structure learning for domains with hundreds of variables by employing non-parametric bootstrap, recursively. We propose a method that covers both model averaging and model selection in the same framework. The proposed method deals with the main weakness of constraint-based learning---sensitivity to errors in the independence tests---by a novel way of combining bootstrap with constraint-based learning. Essentially, we provide an algorithm for learning a tree, in which each node represents a scored CPDAG for a subset of variables and the level of the node corresponds to the maximal order of conditional independencies that are encoded in the graph. As higher order independencies are tested in deeper recursive calls, they benefit from more bootstrap samples, and therefore are more resistant to the curse-of-dimensionality. Moreover, the re-use of stable low order independencies allows greater computational efficiency. We also provide an algorithm for sampling CPDAGs efficiently from their posterior given the learned tree. That is, not from the full posterior, but from a reduced space of CPDAGs encoded in the learned tree. We empirically demonstrate that the proposed algorithm scales well to hundreds of variables, and learns better MAP models and more reliable causal relationships between variables, than other state-of-the-art-methods.

The broad set of deep generative models (DGMs) has achieved remarkable advances. However, it is often difficult to incorporate rich structured domain knowledge with the end-to-end DGMs. Posterior regularization (PR) offers a principled framework to impose structured constraints on probabilistic models, but has limited applicability to the diverse DGMs that can lack a Bayesian formulation or even explicit density evaluation. PR also requires constraints to be fully specified {\it a priori}, which is impractical or suboptimal for complex knowledge with learnable uncertain parts. In this paper, we establish mathematical correspondence between PR and reinforcement learning (RL), and, based on the connection, expand PR to learn constraints as the extrinsic reward in RL. The resulting algorithm is model-agnostic to apply to any DGMs, and is flexible to adapt arbitrary constraints with the model jointly. Experiments on human image generation and templated sentence generation show models with learned knowledge constraints by our algorithm greatly improve over base generative models.

Bayesian optimization usually assumes that a Bayesian prior is given. However, the strong theoretical guarantees in Bayesian optimization are often regrettably compromised in practice because of unknown parameters in the prior. In this paper, we adopt a variant of empirical Bayes and show that, by estimating the Gaussian process prior from offline data sampled from the same prior and constructing unbiased estimators of the posterior, variants of both GP-UCB and \emph{probability of improvement} achieve a near-zero regret bound, which decreases to a constant proportional to the observational noise as the number of offline data and the number of online evaluations increase. Empirically, we have verified our approach on challenging simulated robotic problems featuring task and motion planning.

Variational inference plays a vital role in learning graphical models, especially on large-scale datasets. Much of its success depends on a proper choice of auxiliary distribution class for posterior approximation. However, how to pursue an auxiliary distribution class that achieves both good approximation ability and computation efficiency remains a core challenge. In this paper, we proposed coupled variational Bayes which exploits the primal-dual view of the ELBO with the variational distribution class generated by an optimization procedure, which is termed optimization embedding. This flexible function class couples the variational distribution with the original parameters in the graphical models, allowing end-to-end learning of the graphical models by back-propagation through the variational distribution. Theoretically, we establish an interesting connection to gradient flow and demonstrate the extreme flexibility of this implicit distribution family in the limit sense. Empirically, we demonstrate the effectiveness of the proposed method on multiple graphical models with either continuous or discrete latent variables comparing to state-of-the-art methods.