Autoregressive models use chain rule to define a joint probability distribution as a product of conditionals. These conditionals need to be normalized, imposing constraints on the functional families that can be used. To increase flexibility, we propose autoregressive conditional score models (AR-CSM) where we parameterize the joint distribution in terms of the derivatives of univariate log-conditionals (scores), which need not be normalized. To train AR-CSM, we introduce a new divergence between distributions named Composite Score Matching (CSM). For AR-CSM models, this divergence between data and model distributions can be computed and optimized efficiently, requiring no expensive sampling or adversarial training.

We study the off-policy evaluation (OPE) problem in reinforcement learning with linear function approximation, which aims to estimate the value function of a target policy based on the offline data collected by a behavior policy. We propose to incorporate the variance information of the value function to improve the sample efficiency of OPE. More specifically, for time-inhomogeneous episodic linear Markov decision processes (MDPs), we propose an algorithm, \texttt{VA-OPE}, which uses the estimated variance of the value function to reweight the Bellman residual in Fitted Q-Iteration. We show that our algorithm achieves a tighter error bound than the best-known result. We also provide a fine-grained characterization of the distribution shift between the behavior policy and the target policy.

We give methods for Bayesian inference of directed acyclic graphs, DAGs, and the induced causal effects from passively observed complete data. Our methods build on a recent Markov chain Monte Carlo scheme for learning Bayesian networks, which enables efficient approximate sampling from the graph posterior, provided that each node is assigned a small number K of candidate parents. We present algorithmic techniques to significantly reduce the space and time requirements, which make the use of substantially larger values of K feasible. Furthermore, we investigate the problem of selecting the candidate parents per node so as to maximize the covered posterior mass. Finally, we combine our sampling method with a novel Bayesian approach for estimating causal effects in linear Gaussian DAG models.

Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and Fisher information matrix, to incorporate geometric information into SVGD updates. We achieve this by presenting a generalization of SVGD that replaces the scalar-valued kernels in vanilla SVGD with more general matrix-valued kernels. This yields a significant extension of SVGD, and more importantly, allows us to flexibly incorporate various preconditioning matricesto accelerate the exploration in the probability landscape. Empirical results show that our method outperforms vanilla SVGD and a variety of baseline approaches over a range of real-world Bayesian inference tasks.

Flow-based generative models parameterize probability distributions through an invertible transformation and can be trained by maximum likelihood. Invertible residual networks provide a flexible family of transformations where only Lipschitz conditions rather than strict architectural constraints are needed for enforcing invertibility. However, prior work trained invertible residual networks for density estimation by relying on biased log-density estimates whose bias increased with the network's expressiveness. We give a tractable unbiased estimate of the log density, and reduce the memory required during training by a factor of ten. Furthermore, we improve invertible residual blocks by proposing the use of activation functions that avoid gradient saturation and generalizing the Lipschitz condition to induced mixed norms.

Bayesian decision theory outlines a rigorous framework for making optimal decisions based on maximizing expected utility over a model posterior. However, practitioners often do not have access to the full posterior and resort to approximate inference strategies. In such cases, taking the eventual decision-making task into account while performing the inference allows for calibrating the posterior approximation to maximize the utility. We present an automatic pipeline that co-opts continuous utilities into variational inference algorithms to account for decision-making. We provide practical strategies for approximating and maximizing the gain, and empirically demonstrate consistent improvement when calibrating approximations for specific utilities.

Variational Autoencoders (VAEs) are powerful probabilistic models to learn representations of complex data distributions. One important limitation of VAEs is the strong prior assumption that latent representations learned by the model follow a simple uni-modal Gaussian distribution. Further, the variational training procedure poses considerable practical challenges. Recently proposed regularized autoencoders offer a deterministic autoencoding framework, that simplifies the original VAE objective and is significantly easier to train. Since these models only provide weak control over the learned latent distribution, they require an ex-post density estimation step to generate samples comparable to those of VAEs.

We show that the matrix perspective function, which is jointly convex in the Cartesian product of a standard Euclidean vector space and a conformal space of symmetric matrices, has a proximity operator in an almost closed form. The only implicit part is to solve a semismooth, univariate root finding problem. We uncover the connection between our problem of study and the matrix nearness problem. Through this connection, we propose a quadratically convergent Newton algorithm for the root finding problem.Experiments verify that the evaluation of the proximity operator requires at most 8 Newton steps, taking less than 5s for 2000 by 2000 matrices on a standard laptop. Using this routine as a building block, we demonstrate the usefulness of the studied proximity operator in constrained maximum likelihood estimation of Gaussian mean and covariance, peudolikelihood-based graphical model selection, and a matrix variant of the scaled lasso problem.

Applying probabilistic models to reinforcement learning (RL) enables the uses of powerful optimisation tools such as variational inference in RL. However, existing inference frameworks and their algorithms pose significant challenges for learning optimal policies, e.g., the lack of mode capturing behaviour in pseudo-likelihood methods, difficulties learning deterministic policies in maximum entropy RL based approaches, and a lack of analysis when function approximators are used. We propose VIREL, a theoretically grounded probabilistic inference framework for RL that utilises a parametrised action-value function to summarise future dynamics of the underlying MDP, generalising existing approaches. VIREL also benefits from a mode-seeking form of KL divergence, the ability to learn deterministic optimal polices naturally from inference, and the ability to optimise value functions and policies in separate, iterative steps. In applying variational expectation-maximisation to VIREL, we thus show that the actor-critic algorithm can be reduced to expectation-maximisation, with policy improvement equivalent to an E-step and policy evaluation to an M-step.

Value function approximation has demonstrated phenomenal empirical success in reinforcement learning (RL). Nevertheless, despite a handful of recent progress on developing theory for RL with linear function approximation, the understanding of \emph{general} function approximation schemes largely remains missing. In this paper, we establish the first provably efficient RL algorithm with general value function approximation. We show that if the value functions admit an approximation with a function class \mathcal{F}, our algorithm achieves a regret bound of \widetilde{O}(\mathrm{poly}(dH)\sqrt{T}) where d is a complexity measure of \mathcal{F} that depends on the eluder dimension [Russo and Van Roy, 2013] and log-covering numbers, H is the planning horizon, and T is the number interactions with the environment. Moreover, our algorithm is model-free and provides a framework to justify the effectiveness of algorithms used in practice.