Channel-Driven Monte Carlo Sampling for Bayesian Distributed Learning in Wireless Data Centers

Conventional frequentist learning, as assumed by existing federated learning protocols, is limited in its ability to quantify uncertainty, incorporate prior knowledge, guide active learning, and enable continual learning. Bayesian learning provides a principled approach to address all these limitations, at the cost of an increase in computational complexity. This paper studies distributed Bayesian learning in a wireless data center setting encompassing a central server and multiple distributed workers. Prior work on wireless distributed learning has focused exclusively on frequentist learning, and has introduced the idea of leveraging uncoded transmission to enable "over-the-air" computing. Unlike frequentist learning, Bayesian learning aims at evaluating approximations or samples from a global posterior distribution in the model parameter space. This work investigates for the first time the design of distributed one-shot, or "embarrassingly parallel", Bayesian learning protocols in wireless data centers via consensus Monte Carlo (CMC). Uncoded transmission is introduced not only as a way to implement "over-the-air" computing, but also as a mechanism to deploy channel-driven MC sampling: Rather than treating channel noise as a nuisance to be mitigated, channel-driven sampling utilizes channel noise as an integral part of the MC sampling process. A simple wireless CMC scheme is first proposed that is asymptotically optimal under Gaussian local posteriors. Then, for arbitrary local posteriors, a variational optimization strategy is introduced. Simulation results demonstrate that, if properly accounted for, channel noise can indeed contribute to MC sampling and does not necessarily decrease the accuracy level.Comment: Under Revisio

Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach

Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain Monte Carlo, remain the gold standard for approximate Bayesian inference because they have a robust finite-sample theory and reliable convergence diagnostics. However, alternative methods, which are more scalable or apply to problems where Markov Chain Monte Carlo cannot be used, lack the same finite-data approximation theory and tools for evaluating their accuracy. In this work, we develop a flexible new approach to bounding the error of mean and uncertainty estimates of scalable inference algorithms. Our strategy is to control the estimation errors in terms of Wasserstein distance, then bound the Wasserstein distance via a generalized notion of Fisher distance. Unlike computing the Wasserstein distance, which requires access to the normalized posterior distribution, the Fisher distance is tractable to compute because it requires access only to the gradient of the log posterior density. We demonstrate the usefulness of our Fisher distance approach by deriving bounds on the Wasserstein error of the Laplace approximation and Hilbert coresets. We anticipate that our approach will be applicable to many other approximate inference methods such as the integrated Laplace approximation, variational inference, and approximate Bayesian computationComment: 22 pages, 2 figure

PASS-GLM: polynomial approximate sufficient statistics for scalable Bayesian GLM inference

Generalized linear models (GLMs) -- such as logistic regression, Poisson regression, and robust regression -- provide interpretable models for diverse data types. Probabilistic approaches, particularly Bayesian ones, allow coherent estimates of uncertainty, incorporation of prior information, and sharing of power across experiments via hierarchical models. In practice, however, the approximate Bayesian methods necessary for inference have either failed to scale to large data sets or failed to provide theoretical guarantees on the quality of inference. We propose a new approach based on constructing polynomial approximate sufficient statistics for GLMs (PASS-GLM). We demonstrate that our method admits a simple algorithm as well as trivial streaming and distributed extensions that do not compound error across computations. We provide theoretical guarantees on the quality of point (MAP) estimates, the approximate posterior, and posterior mean and uncertainty estimates. We validate our approach empirically in the case of logistic regression using a quadratic approximation and show competitive performance with stochastic gradient descent, MCMC, and the Laplace approximation in terms of speed and multiple measures of accuracy -- including on an advertising data set with 40 million data points and 20,000 covariates.Comment: In Proceedings of the 31st Annual Conference on Neural Information Processing Systems (NIPS 2017). v3: corrected typos in Appendix

Global consensus Monte Carlo

To conduct Bayesian inference with large data sets, it is often convenient or necessary to distribute the data across multiple machines. We consider a likelihood function expressed as a product of terms, each associated with a subset of the data. Inspired by global variable consensus optimisation, we introduce an instrumental hierarchical model associating auxiliary statistical parameters with each term, which are conditionally independent given the top-level parameters. One of these top-level parameters controls the unconditional strength of association between the auxiliary parameters. This model leads to a distributed MCMC algorithm on an extended state space yielding approximations of posterior expectations. A trade-off between computational tractability and fidelity to the original model can be controlled by changing the association strength in the instrumental model. We further propose the use of a SMC sampler with a sequence of association strengths, allowing both the automatic determination of appropriate strengths and for a bias correction technique to be applied. In contrast to similar distributed Monte Carlo algorithms, this approach requires few distributional assumptions. The performance of the algorithms is illustrated with a number of simulated examples