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Register a new userStein Variational Adaptive Importance Sampling
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Jun Han Mr
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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We propose a novel adaptive importance sampling algorithm which incorporates Stein variational gradient decent algorithm (SVGD) with importance sampling (IS). Our algorithm leverages the nonparametric transforms in SVGD to iteratively decrease the KL divergence between our importance proposal and the target distribution. The advantages of this algorithm are twofold: first, our algorithm turns SVGD into a standard IS algorithm, allowing us to use standard diagnostic and analytic tools of IS to evaluate and interpret the results; second, we do not restrict the choice of our importance proposal to predefined distribution families like traditional (adaptive) IS methods. Empirical experiments demonstrate that our algorithm performs well on evaluating partition functions of restricted Boltzmann machines and testing likelihood of variational auto-encoders.
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As part of Probabilistic Risk Assessment studies, it is necessary to study the fragility of mechanical and civil engineered structures when subjected to seismic loads. This risk can be measured with fragility curves, which express the probability of failure of the structure conditionally to a seismic intensity measure. The estimation of fragility curves relies on time-consuming numerical simulations, so that careful experimental design is required in order to gain the maximum information on the structures fragility with a limited number of code evaluations. We propose and implement an active learning methodology based on adaptive importance sampling in order to reduce the variance of the training loss. The efficiency of the proposed method in terms of bias, standard deviation and prediction interval coverage are theoretically and numerically characterized.
Variational Inference (VI) is a popular alternative to asymptotically exact sampling in Bayesian inference. Its main workhorse is optimization over a reverse Kullback-Leibler divergence (RKL), which typically underestimates the tail of the posterior leading to miscalibration and potential degeneracy. Importance sampling (IS), on the other hand, is often used to fine-tune and de-bias the estimates of approximate Bayesian inference procedures. The quality of IS crucially depends on the choice of the proposal distribution. Ideally, the proposal distribution has heavier tails than the target, which is rarely achievable by minimizing the RKL. We thus propose a novel combination of optimization and sampling techniques for approximate Bayesian inference by constructing an IS proposal distribution through the minimization of a forward KL (FKL) divergence. This approach guarantees asymptotic consistency and a fast convergence towards both the optimal IS estimator and the optimal variational approximation. We empirically demonstrate on real data that our method is competitive with variational boosting and MCMC.
Stein variational gradient decent (SVGD) has been shown to be a powerful approximate inference algorithm for complex distributions. However, the standard SVGD requires calculating the gradient of the target density and cannot be applied when the gradient is unavailable. In this work, we develop a gradient-free variant of SVGD (GF-SVGD), which replaces the true gradient with a surrogate gradient, and corrects the induced bias by re-weighting the gradients in a proper form. We show that our GF-SVGD can be viewed as the standard SVGD with a special choice of kernel, and hence directly inherits the theoretical properties of SVGD. We shed insights on the empirical choice of the surrogate gradient and propose an annealed GF-SVGD that leverages the idea of simulated annealing to improve the performance on high dimensional complex distributions. Empirical studies show that our method consistently outperforms a number of recent advanced gradient-free MCMC methods.
The Adaptive Multiple Importance Sampling (AMIS) algorithm is aimed at an optimal recycling of past simulations in an iterated importance sampling scheme. The difference with earlier adaptive importance sampling implementations like Population Monte Carlo is that the importance weights of all simulated values, past as well as present, are recomputed at each iteration, following the technique of the deterministic multiple mixture estimator of Owen and Zhou (2000). Although the convergence properties of the algorithm cannot be fully investigated, we demonstrate through a challenging banana shape target distribution and a population genetics example that the improvement brought by this technique is substantial.
Monte Carlo methods represent the de facto standard for approximating complicated integrals involving multidimensional target distributions. In order to generate random realizations from the target distribution, Monte Carlo techniques use simpler proposal probability densities to draw candidate samples. The performance of any such method is strictly related to the specification of the proposal distribution, such that unfortunate choices easily wreak havoc on the resulting estimators. In this work, we introduce a layered (i.e., hierarchical) procedure to generate samples employed within a Monte Carlo scheme. This approach ensures that an appropriate equivalent proposal density is always obtained automatically (thus eliminating the risk of a catastrophic performance), although at the expense of a moderate increase in the complexity. Furthermore, we provide a general unified importance sampling (IS) framework, where multiple proposal densities are employed and several IS schemes are introduced by applying the so-called deterministic mixture approach. Finally, given these schemes, we also propose a novel class of adaptive importance samplers using a population of proposals, where the adaptation is driven by independent parallel or interacting Markov Chain Monte Carlo (MCMC) chains. The resulting algorithms efficiently combine the benefits of both IS and MCMC methods.
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