Tutorial on variational approximation methods
نویسنده
چکیده
Tutorial topics • A bit of history • Examples of variational methods • A brief intro to graphical models • Variational mean field theory – Accuracy of variational mean field – Structured mean field theory • Variational methods in Bayesian estimation • Convex duality and variational factorization methods – Example: variational inference and the QMR-DT Variational methods • Classical setting: " finding the extremum of an integral involving a function and its derivatives " Example: finding the trajectory of a particle under external field • The key idea here is that the problem of interest is formulated as an optimization problem Variational methods cont'd • Variational methods have a long history in physics, statistics, control theory as well as economics. – calculus of variations (physics) – linear/non-linear moments problems (statistics) – dynamic programming (control theory) • Variational formulations appear naturally also in machine learning contexts: – regularization theory – maximum entropy estimation • Recently variational methods been used and further developed in the context of approximate inference and estimation Examples of variational methods • In classical examples the formulation itself is given but for us this is one of the key problems • We provide here a few examples that highlight 1. how to cast problems as optimization problems 2. how to find an approximate solution when the exact solution is not feasible • The examples we use involve a) finite element methods for solving differential equations b) large deviation methods (Chernoff bound)
منابع مشابه
Variational Inference for Structured NLP Models
Historically, key breakthroughs in structured NLP models, such as chain CRFs or PCFGs, have relied on imposing careful constraints on the locality of features in order to permit efficient dynamic programming for computing expectations or finding the highestscoring structures. However, as modern structured models become more complex and seek to incorporate longer-range features, it is more and m...
متن کاملThe EM algorithm, variational approximations and expectation propagation for mixtures
The material in this chapter is largely tutorial in nature. The main goal is to review two types of deterministic approximation, variational approximations and the expectation propagation approach, which have been developed mainly in the computer science literature, but with some statistical antecedents, to assist approximate Bayesian inference. However, we believe that it is helpful to preface...
متن کاملA Relaxed Extra Gradient Approximation Method of Two Inverse-Strongly Monotone Mappings for a General System of Variational Inequalities, Fixed Point and Equilibrium Problems
متن کامل
Approximation of fixed points for a continuous representation of nonexpansive mappings in Hilbert spaces
This paper introduces an implicit scheme for a continuous representation of nonexpansive mappings on a closed convex subset of a Hilbert space with respect to a sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup. The main result is to prove the strong convergence of the proposed implicit scheme to the unique solutio...
متن کاملTutorial on Variational Autoencoders
In just three years, Variational Autoencoders (VAEs) have emerged as one of the most popular approaches to unsupervised learning of complicated distributions. VAEs are appealing because they are built on top of standard function approximators (neural networks), and can be trained with stochastic gradient descent. VAEs have already shown promise in generating many kinds of complicated data, incl...
متن کامل