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1. Statistical Guarantees for Transformation Based Models with applications to Implicit Variational Inference

   Plummer, Sean ; Zhou, Shuang ; Bhattacharya, Anirban ; Dunson, David ; Pati, Debdeep ( April 2021 , Proceedings of Machine Learning Research)

   null (Ed.)

   Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both nonparametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the L1 sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback–Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference. 

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2. Marginalized Stochastic Natural Gradients for Black-Box Variational Inference

   Ji, Geng ; Sujono, Debora ; Sudderth, Erik B. ( July 2021 , International Conference on Machine Learning)

   Meila, Marina ; Zhang, Tong (Ed.)

   Black-box variational inference algorithms use stochastic sampling to analyze diverse statistical models, like those expressed in probabilistic programming languages, without model-specific derivations. While the popular score-function estimator computes unbiased gradient estimates, its variance is often unacceptably large, especially in models with discrete latent variables. We propose a stochastic natural gradient estimator that is as broadly applicable and unbiased, but improves efficiency by exploiting the curvature of the variational bound, and provably reduces variance by marginalizing discrete latent variables. Our marginalized stochastic natural gradients have intriguing connections to classic coordinate ascent variational inference, but allow parallel updates of variational parameters, and provide superior convergence guarantees relative to naive Monte Carlo approximations. We integrate our method with the probabilistic programming language Pyro and evaluate real-world models of documents, images, networks, and crowd-sourcing. Compared to score-function estimators, we require far fewer Monte Carlo samples and consistently convergence orders of magnitude faster. 

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3. Correlated Equilibria for Approximate Variational Inference in MRFs

   Ortiz, Luis E. ; Wang, Boshen ; Gong, Ze ( January 2020 , Proceedings of Machine Learning Research)

   Jaeger, Manfred ; Nielsen, Thomas Dyhre (Ed.)

   Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of advances in equilibrium computation for probabilistic inference. In particular, we present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. While some previous work explores this direction, we still lack a more precise connection between variational probabilistic inference in MRFs and correlated equilibria. This paper sharpens the connection, which helps us exploit relatively more recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. Our work discusses how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. In addition, inspired by a previously stated game-theoretic view of tree-reweighted message-passing techniques for belief inference as a zero-sum game, we propose a different, general-sum potential game to design approximate fictitious-play techniques. Empirical evaluations on synthetic experiments and on an application to soft de-noising on real-world image datasets illustrate the performance of our proposed approach and shed some light on the conditions under which the resulting belief inference algorithms may be most effective relative to standard state-of-the-art methods. 

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4. OCEAN: Online Task Inference for Compositional Tasks with Context Adaptation

   Ren, Hongyu ; Zhu, Yuke ; Leskovec, Jure ; Anandkumar, Anima ; Garg, Animesh. ( January 2020 , Conference on Uncertainty in Artificial Intelligence (UAI))

   Real-world tasks often exhibit a compositional structure that contains a sequence of simpler sub-tasks. For instance, opening a door requires reaching, grasping, rotating, and pulling the door knob. Such compositional tasks require an agent to reason about the sub-task at hand while orchestrating global behavior accordingly. This can be cast as an online task inference problem, where the current task identity, represented by a context variable, is estimated from the agent’s past experiences with probabilistic inference. Previous approaches have employed simple latent distributions, e.g., Gaussian, to model a single context for the entire task. However, this formulation lacks the expressiveness to capture the composition and transition of the sub-tasks. We propose a variational inference framework OCEAN to perform online task inference for compositional tasks. OCEAN models global and local context variables in a joint latent space, where the global variables represent a mixture of subtasks required for the task, while the local variables capture the transitions between the subtasks. Our framework supports flexible latent distributions based on prior knowledge of the task structure and can be trained in an unsupervised manner. Experimental results show that OCEAN provides more effective task inference with sequential context adaptation and thus leads to a performance boost on complex, multi-stage tasks. 

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5. Excess Risk Bounds for the Bayes Risk using Variational Inference in Latent Gaussian Models

   Sheth, Rishit ; Khardon, Roni ( January 2017 , Advances in neural information processing systems)

   Bayesian models are established as one of the main successful paradigms for complex problems in machine learning. To handle intractable inference, research in this area has developed new approximation methods that are fast and effective. However, theoretical analysis of the performance of such approximations is not well developed. The paper furthers such analysis by providing bounds on the excess risk of variational inference algorithms and related regularized loss minimization algorithms for a large class of latent variable models with Gaussian latent variables. We strengthen previous results for variational algorithms by showing they are competitive with any point-estimate predictor. Unlike previous work, we also provide bounds on the risk of the \emph{Bayesian} predictor and not just the risk of the Gibbs predictor for the same approximate posterior. The bounds are applied in complex models including sparse Gaussian processes and correlated topic models. Theoretical results are complemented by identifying novel approximations to the Bayesian objective that attempt to minimize the risk directly. An empirical evaluation compares the variational and new algorithms shedding further light on their performance. 

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