The recently proposed Thermodynamic Variational Objective (TVO) leverages thermodynamic integration to provide a family of variational inference objectives, which both tighten and generalize the ubiquitous Evidence Lower Bound (ELBO). However, the tightness of TVO bounds was not previously known, an expensive grid search was used to choose a "schedule" of intermediate distributions, and model learning suffered with ostensibly tighter bounds. In this work, we propose an exponential family interpretation of the geometric mixture curve underlying the TVO and various path sampling methods, which allows us to characterize the gap in TVO likelihood bounds as a sum of KL divergences. We propose to choose intermediate distributions using equal spacing in the moment parameters of our exponential family, which matches grid search performance and allows the schedule to adaptively update over the course of training. Finally, we derive a doubly reparameterized gradient estimator which improves model learning and allows the TVO to benefit from more refined bounds. To further contextualize our contributions, we provide a unified framework for understanding thermodynamic integration and the TVO using Taylor series remainders.
Rutgers, The State University of New Jersey（罗格斯大学，新泽西州立大学）
Bayesian models provide a framework for probabilistic modelling of complex datasets. However, many of such models are computationally demanding especially in the presence of large datasets. On the other hand, in sensor network applications, statistical (Bayesian) parameter estimation usually needs distributed algorithms, in which both data and computation are distributed across the nodes of the network. In this paper we propose a general framework for distributed Bayesian learning using Bregman Alternating Direction Method of Multipliers (B-ADMM). We demonstrate the utility of our framework, w…
We study quasi-Newton methods from the viewpoint of information geometry induced associated with Bregman divergences. Fletcher has studied a variational problem which derives the approximate Hessian update formula of the quasi-Newton methods. We point out that the variational problem is identical to optimization of the Kullback-Leibler divergence, which is a discrepancy measure between two probability distributions. The Kullback-Leibler divergence for the multinomial normal distribution corresponds to the objective function Fletcher has considered. We introduce the Bregman divergence as an ext…
University of Pittsburgh（匹兹堡大学）；Aalborg University（奥尔堡大学）
Graph-based methods provide a powerful tool set for many non-parametric frameworks in Machine Learning. In general, the memory and computational complexity of these methods is quadratic in the number of examples in the data which makes them quickly infeasible for moderate to large scale datasets. A significant effort to find more efficient solutions to the problem has been made in the literature. One of the state-of-the-art methods that has been recently introduced is the Variational Dual-Tree (VDT) framework. Despite some of its unique features, VDT is currently restricted only to Euclidean s…
ENS（巴黎高等师范学院）；Univ. PSL（巴黎文理研究大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；Inria（法国国家信息与自动化研究所）；Criteo AI Lab（Criteo人工智能实验室）
In this paper, we analyze the local convergence rate of optimistic mirror descent methods in stochastic variational inequalities, a class of optimization problems with important applications to learning theory and machine learning. Our analysis reveals an intricate relation between the algorithm's rate of convergence and the local geometry induced by the method's underlying Bregman function. We quantify this relation by means of the Legendre exponent, a notion that we introduce to measure the growth rate of the Bregman divergence relative to the ambient norm near a solution. We show that this …
MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）
The family of f-divergences is ubiquitously applied to generative modeling in order to adapt the distribution of the model to that of the data. Well-definedness of f-divergences, however, requires the distributions of the data and model to overlap completely in every time step of training. As a result, as soon as the support of distributions of data and model contain non-overlapping portions, gradient based training of the corresponding model becomes hopeless. Recent advances in generative modeling are full of remedies for handling this support mismatch problem: key ideas include either modify…
Greedy algorithms have long been a workhorse for learning graphical models, and more broadly for learning statistical models with sparse structure. In the context of learning directed acyclic graphs, greedy algorithms are popular despite their worst-case exponential runtime. In practice, however, they are very efficient. We provide new insight into this phenomenon by studying a general greedy score-based algorithm for learning DAGs. Unlike edge-greedy algorithms such as the popular GES and hill-climbing algorithms, our approach is vertex-greedy and requires at most a polynomial number of score…
This document describes concisely the ubiquitous class of exponential family distributions met in statistics. The first part recalls definitions and summarizes main properties and duality with Bregman divergences (all proofs are skipped). The second part lists decompositions and related formula of common exponential family distributions. We recall the Fisher-Rao-Riemannian geometries and the dual affine connection information geometries of statistical manifolds. It is intended to maintain and update this document and catalog by adding new distribution items.
University of Alberta（阿尔伯塔大学）；Imperial College London（伦敦帝国学院）
Recently, much work has been done on extending the scope of online learning and incremental stochastic optimization algorithms. In this paper we contribute to this effort in two ways: First, based on a new regret decomposition and a generalization of Bregman divergences, we provide a self-contained, modular analysis of the two workhorses of online learning: (general) adaptive versions of Mirror Descent (MD) and the Follow-the-Regularized-Leader (FTRL) algorithms. The analysis is done with extra care so as not to introduce assumptions not needed in the proofs and allows to combine, in a straigh…
Data Science Institute（数据科学研究所）；Imperial College（帝国理工学院）；Institute of Security Science and Technology（安全科学与技术研究所）
Two main families of reinforcement learning algorithms, Q-learning and policy gradients, have recently been proven to be equivalent when using a softmax relaxation on one part, and an entropic regularization on the other. We relate this result to the well-known convex duality of Shannon entropy and the softmax function. Such a result is also known as the Donsker-Varadhan formula. This provides a short proof of the equivalence. We then interpret this duality further, and use ideas of convex analysis to prove a new policy inequality relative to soft Q-learning.
