Abstract:The performance of the Monte Carlo sampling methods relies on the crucial choice of a proposal density. The notion of optimality is fundamental to design suitable adaptive procedures of the proposal density within Monte Carlo schemes. This work is an exhaustive review around the concept of optimality in importance sampling. Several frameworks are described and analyzed, such as the marginal likelihood approximation for model selection, the use of multiple proposal densities, a sequence of tempered posteriors, and noisy scenarios including the applications to approximate Bayesian computation (ABC) and reinforcement learning, to name a few. Some theoretical and empirical comparisons are also provided.
Abstract:Flexible and scalable decentralized learning solutions are fundamentally important in the application of multi-agent systems. While several recent approaches introduce (ensembles of) kernel machines in the distributed setting, Bayesian solutions are much more limited. We introduce a fully decentralized, asymptotically exact solution to computing the random feature approximation of Gaussian processes. We further address the choice of hyperparameters by introducing an ensembling scheme for Bayesian multiple kernel learning based on online Bayesian model averaging. The resulting algorithm is tested against Bayesian and frequentist methods on simulated and real-world datasets.
Abstract:In science and engineering, we often work with models designed for accurate prediction of variables of interest. Recognizing that these models are approximations of reality, it becomes desirable to apply multiple models to the same data and integrate their outcomes. In this paper, we operate within the Bayesian paradigm, relying on Gaussian processes as our models. These models generate predictive probability density functions (pdfs), and the objective is to integrate them systematically, employing both linear and log-linear pooling. We introduce novel approaches for log-linear pooling, determining input-dependent weights for the predictive pdfs of the Gaussian processes. The aggregation of the pdfs is realized through Monte Carlo sampling, drawing samples of weights from their posterior. The performance of these methods, as well as those based on linear pooling, is demonstrated using a synthetic dataset.
Abstract:Reinforcement Learning has drawn huge interest as a tool for solving optimal control problems. Solving a given problem (task or environment) involves converging towards an optimal policy. However, there might exist multiple optimal policies that can dramatically differ in their behaviour; for example, some may be faster than the others but at the expense of greater risk. We consider and study a distribution of optimal policies. We design a curiosity-augmented Metropolis algorithm (CAMEO), such that we can sample optimal policies, and such that these policies effectively adopt diverse behaviours, since this implies greater coverage of the different possible optimal policies. In experimental simulations we show that CAMEO indeed obtains policies that all solve classic control problems, and even in the challenging case of environments that provide sparse rewards. We further show that the different policies we sample present different risk profiles, corresponding to interesting practical applications in interpretability, and represents a first step towards learning the distribution of optimal policies itself.
Abstract:In this work, we analyze the noisy importance sampling (IS), i.e., IS working with noisy evaluations of the target density. We present the general framework and derive optimal proposal densities for noisy IS estimators. The optimal proposals incorporate the information of the variance of the noisy realizations, proposing points in regions where the noise power is higher. We also compare the use of the optimal proposals with previous optimality approaches considered in a noisy IS framework.
Abstract:This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of the most relevant methodologies are compared through theoretical comparisons and numerical experiments.