Stochastic gradient Langevin dynamics (SGLD) and stochastic gradient Hamiltonian Monte Carlo (SGHMC) are two popular Markov Chain Monte Carlo (MCMC) algorithms for Bayesian inference that can scale to large datasets, allowing to sample from the posterior distribution of a machine learning (ML) model based on the input data and the prior distribution over the model parameters. However, these algorithms do not apply to the decentralized learning setting, when a network of agents are working collaboratively to learn the parameters of an ML model without sharing their individual data due to privacy reasons or communication constraints. We study two algorithms: Decentralized SGLD (DE-SGLD) and Decentralized SGHMC (DE-SGHMC) which are adaptations of SGLD and SGHMC methods that allow scaleable Bayesian inference in the decentralized setting. We show that when the posterior distribution is strongly log-concave, the iterates of these algorithms converge linearly to a neighborhood of the target distribution in the 2-Wasserstein metric. We illustrate the results for decentralized Bayesian linear regression and Bayesian logistic regression problems.

In today's ultra-competitive business universe, Probability and Statistics are the most important fields of study. That is because statistical research presents businesses with the data they need to make informed decisions in every business area, whether it is market research, product development, product launch timing, customer data analysis, sales forecast, or employee performance. But why do you need to master probability and statistics in Python? The course'Mastering Probability and Statistics in Python' is designed carefully to reflect the most in-demand skills that will help you in understanding the concepts and methodology with regards to Python. How is this course different? This course is designed for beginners, although we will go far deep gradually.

Learned neural solvers have successfully been used to solve combinatorial optimization and decision problems. More general counting variants of these problems, however, are still largely solved with hand-crafted solvers. To bridge this gap, we introduce belief propagation neural networks (BPNNs), a class of parameterized operators that operate on factor graphs and generalize Belief Propagation (BP). In its strictest form, a BPNN layer (BPNN-D) is a learned iterative operator that provably maintains many of the desirable properties of BP for any choice of the parameters. Empirically, we show that by training BPNN-D learns to perform the task better than the original BP: it converges 1.7x faster on Ising models while providing tighter bounds. On challenging model counting problems, BPNNs compute estimates 100's of times faster than state-of-the-art handcrafted methods, while returning an estimate of comparable quality.

Bayesian coresets have emerged as a promising approach for scalable Bayesian inference [22, 12, 13, 11]. The key idea is to select a (weighted) subset of the data such that posterior inference using the selected subset closely approximates posterior inference using the full dataset. This creates a tradeoff, where using Bayesian coresets as opposed to the full dataset exchanges approximation accuracy for computational speedups. We study Bayesian coresets as they are easy to implement, effective in practice, and come with useful theoretical guarantees that relate the coreset size with the approximation quality. The main technical challenge in the Bayesian coreset problem lies in handling the combinatorial constraints - we desire to select a few data points out of many as the coreset. The state of the art approaches rely on two ideas: convexification and greedy methods. In convexification [13], the sparsity constraint - i.e., selection of k data samples - is relaxed into a convex l

Many inference problems in structured prediction can be modeled as maximizing a score function on a space of labels, where graphs are a natural representation to decompose the total score into a sum of unary (nodes) and pairwise (edges) scores. Given a generative model with an undirected connected graph $G$ and true vector of binary labels, it has been previously shown that when $G$ has good expansion properties, such as complete graphs or $d$-regular expanders, one can exactly recover the true labels (with high probability and in polynomial time) from a single noisy observation of each edge and node. We analyze the previously studied generative model by Globerson et al. (2015) under a notion of statistical parity. That is, given a fair binary node labeling, we ask the question whether it is possible to recover the fair assignment, with high probability and in polynomial time, from single edge and node observations. We find that, in contrast to the known trade-offs between fairness and model performance, the addition of the fairness constraint improves the probability of exact recovery. We effectively explain this phenomenon and empirically show how graphs with poor expansion properties, such as grids, are now capable to achieve exact recovery with high probability. Finally, as a byproduct of our analysis, we provide a tighter minimum-eigenvalue bound than that of Weyl's inequality.

Influence diagrams (IDs) are well-known formalisms extending Bayesian networks to model decision situations under uncertainty. Although they are convenient as a decision theoretic tool, their knowledge representation ability is limited in capturing other crucial notions such as logical consistency. We complement IDs with the light-weight description logic (DL) EL to overcome such limitations. We consider a setup where DL axioms hold in some contexts, yet the actual context is uncertain. The framework benefits from the convenience of using DL as a domain knowledge representation language and the modelling strength of IDs to deal with decisions over contexts in the presence of contextual uncertainty. We define related reasoning problems and study their computational complexity.

We can use driving data collected over a long period of time to extract rich information about how vehicles behave in different areas of the roads. In this paper, we introduce the concept of directional primitives, which is a representation of prior information of road networks. Specifically, we represent the uncertainty of directions using a mixture of von Mises distributions and associated speeds using gamma distributions. These location-dependent primitives can be combined with motion information of surrounding vehicles to predict their future behavior in the form of probability distributions. Experiments conducted on highways, intersections, and roundabouts in the Carla simulator, as well as real-world urban driving datasets, indicate that primitives lead to better uncertainty-aware motion estimation.

Estimating the score, i.e., the gradient of log density function, from a set of samples generated by an unknown distribution is a fundamental task in inference and learning of probabilistic models that involve flexible yet intractable densities. Kernel estimators based on Stein's methods or score matching have shown promise, however their theoretical properties and relationships have not been fully-understood. We provide a unifying view of these estimators under the framework of regularized nonparametric regression. It allows us to analyse existing estimators and construct new ones with desirable properties by choosing different hypothesis spaces and regularizers. A unified convergence analysis is provided for such estimators. Finally, we propose score estimators based on iterative regularization that enjoy computational benefits from curl-free kernels and fast convergence.

Virtual screening aims to find desirable compounds from chemical library by using computational methods. For this purpose with machine learning, model outputs that can be interpreted as predictive probability will be beneficial, in that a high prediction score corresponds to high probability of correctness. In this work, we present a study on the prediction performance and reliability of graph neural networks trained with the recently proposed Bayesian learning algorithms. Our work shows that Bayesian learning algorithms allow well-calibrated predictions for various GNN architectures and classification tasks. Also, we show the implications of reliable predictions on virtual screening, where Bayesian learning may lead to higher success in finding hit compounds.

We study Sinkhorn EM (sEM), a variant of the expectation maximization (EM) algorithm for mixtures based on entropic optimal transport. sEM differs from the classic EM algorithm in the way responsibilities are computed during the expectation step: rather than assign data points to clusters independently, sEM uses optimal transport to compute responsibilities by incorporating prior information about mixing weights. Like EM, sEM has a natural interpretation as a coordinate ascent procedure, which iteratively constructs and optimizes a lower bound on the log-likelihood. However, we show theoretically and empirically that sEM has better behavior than EM: it possesses better global convergence guarantees and is less prone to getting stuck in bad local optima. We complement these findings with experiments on simulated data as well as in an inference task involving C. elegans neurons and show that sEM learns cell labels significantly better than other approaches.