Mixed Variational Flows for Discrete Variables

Variational flows allow practitioners to learn complex continuous distributions, but approximating discrete distributions remains a challenge. Current methodologies typically embed the discrete target in a continuous space - usually via continuous relaxation or dequantization - and then apply a continuous flow. These approaches involve a surrogate target that may not capture the original discrete target, might have biased or unstable gradients, and can create a difficult optimization problem. In this work, we develop a variational flow family for discrete distributions without any continuous embedding. First, we develop a measure-preserving and discrete (MAD) invertible map that leaves the discrete target invariant, and then create a mixed variational flow (MAD Mix) based on that map. We also develop an extension to MAD Mix that handles joint discrete and continuous models. Our experiments suggest that MAD Mix produces more reliable approximations than continuous-embedding flows while being significantly faster to train.

Embracing the chaos: analysis and diagnosis of numerical instability in variational flows

In this paper, we investigate the impact of numerical instability on the reliability of sampling, density evaluation, and evidence lower bound (ELBO) estimation in variational flows. We first empirically demonstrate that common flows can exhibit a catastrophic accumulation of error: the numerical flow map deviates significantly from the exact map -- which affects sampling -- and the numerical inverse flow map does not accurately recover the initial input -- which affects density and ELBO computations. Surprisingly though, we find that results produced by flows are often accurate enough for applications despite the presence of serious numerical instability. In this work, we treat variational flows as dynamical systems, and leverage shadowing theory to elucidate this behavior via theoretical guarantees on the error of sampling, density evaluation, and ELBO estimation. Finally, we develop and empirically test a diagnostic procedure that can be used to validate results produced by numerically unstable flows in practice.

Machine Learning and the Future of Bayesian Computation

Bayesian models are a powerful tool for studying complex data, allowing the analyst to encode rich hierarchical dependencies and leverage prior information. Most importantly, they facilitate a complete characterization of uncertainty through the posterior distribution. Practical posterior computation is commonly performed via MCMC, which can be computationally infeasible for high dimensional models with many observations. In this article we discuss the potential to improve posterior computation using ideas from machine learning. Concrete future directions are explored in vignettes on normalizing flows, Bayesian coresets, distributed Bayesian inference, and variational inference.

Conditional Permutation Invariant Flows

We present a novel, conditional generative probabilistic model of set-valued data with a tractable log density. This model is a continuous normalizing flow governed by permutation equivariant dynamics. These dynamics are driven by a learnable per-set-element term and pairwise interactions, both parametrized by deep neural networks. We illustrate the utility of this model via applications including (1) complex traffic scene generation conditioned on visually specified map information, and (2) object bounding box generation conditioned directly on images. We train our model by maximizing the expected likelihood of labeled conditional data under our flow, with the aid of a penalty that ensures the dynamics are smooth and hence efficiently solvable. Our method significantly outperforms non-permutation invariant baselines in terms of log likelihood and domain-specific metrics (offroad, collision, and combined infractions), yielding realistic samples that are difficult to distinguish from real data.

Ergodic variational flows

This work presents a new class of variational family -- ergodic variational flows -- that not only enables tractable i.i.d. sampling and density evaluation, but also comes with MCMC-like convergence guarantees. Ergodic variational flows consist of a mixture of repeated applications of a measure-preserving and ergodic map to an initial reference distribution. We provide mild conditions under which the variational distribution converges weakly and in total variation to the target as the number of steps in the flow increases; this convergence holds regardless of the value of variational parameters, although different parameter values may result in faster or slower convergence. Further, we develop a particular instantiation of the general family using Hamiltonian dynamics combined with deterministic momentum refreshment. Simulated and real data experiments provide an empirical verification of the convergence theory and demonstrate that samples produced by the method are of comparable quality to a state-of-the-art MCMC method.

Fast Bayesian Coresets via Subsampling and Quasi-Newton Refinement

Bayesian coresets approximate a posterior distribution by building a small weighted subset of the data points. Any inference procedure that is too computationally expensive to be run on the full posterior can instead be run inexpensively on the coreset, with results that approximate those on the full data. However, current approaches are limited by either a significant run-time or the need for the user to specify a low-cost approximation to the full posterior. We propose a Bayesian coreset construction algorithm that first selects a uniformly random subset of data, and then optimizes the weights using a novel quasi-Newton method. Our algorithm is simple to implement, does not require the user to specify a low-cost posterior approximation, and is the first to come with a general high-probability bound on the KL divergence of the output coreset posterior. Experiments demonstrate that the method provides orders of magnitude improvement in construction time against the state-of-the-art black-box method. Moreover, it provides significant improvements in coreset quality against alternatives with comparable construction times, with far less storage cost and user input required.

Bayesian inference via sparse Hamiltonian flows

A Bayesian coreset is a small, weighted subset of data that replaces the full dataset during Bayesian inference, with the goal of reducing computational cost. Although past work has shown empirically that there often exists a coreset with low inferential error, efficiently constructing such a coreset remains a challenge. Current methods tend to be slow, require a secondary inference step after coreset construction, and do not provide bounds on the data marginal evidence. In this work, we introduce a new method -- sparse Hamiltonian flows -- that addresses all three of these challenges. The method involves first subsampling the data uniformly, and then optimizing a Hamiltonian flow parametrized by coreset weights and including periodic momentum quasi-refreshment steps. Theoretical results show that the method enables an exponential compression of the dataset in a representative model, and that the quasi-refreshment steps reduce the KL divergence to the target. Real and synthetic experiments demonstrate that sparse Hamiltonian flows provide accurate posterior approximations with significantly reduced runtime compared with competing dynamical-system-based inference methods.

The computational asymptotics of Gaussian variational inference

Variational inference is a popular alternative to Markov chain Monte Carlo methods that constructs a Bayesian posterior approximation by minimizing a discrepancy to the true posterior within a pre-specified family. This converts Bayesian inference into an optimization problem, enabling the use of simple and scalable stochastic optimization algorithms. However, a key limitation of variational inference is that the optimal approximation is typically not tractable to compute; even in simple settings the problem is nonconvex. Thus, recently developed statistical guarantees -- which all involve the (data) asymptotic properties of the optimal variational distribution -- are not reliably obtained in practice. In this work, we provide two major contributions: a theoretical analysis of the asymptotic convexity properties of variational inference in the popular setting with a Gaussian family; and consistent stochastic variational inference (CSVI), an algorithm that exploits these properties to find the optimal approximation in the asymptotic regime. CSVI consists of a tractable initialization procedure that finds the local basin of the optimal solution, and a scaled gradient descent algorithm that stays locally confined to that basin. Experiments on nonconvex synthetic and real-data examples show that compared with standard stochastic gradient descent, CSVI improves the likelihood of obtaining the globally optimal posterior approximation.

Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture

We present a Physics-Informed Neural Network (PINN) to simulate the thermochemical evolution of a composite material on a tool undergoing cure in an autoclave. In particular, we solve the governing coupled system of differential equations -- including conductive heat transfer and resin cure kinetics -- by optimizing the parameters of a deep neural network using a physics-based loss function. To account for the vastly different behaviour of thermal conduction and resin cure, we design a PINN consisting of two disconnected subnetworks, and develop a sequential training algorithm that mitigates instability present in traditional training methods. Further, we incorporate explicit discontinuities into the DNN at the composite-tool interface and enforce known physical behaviour directly in the loss function to improve the solution near the interface. Finally, we train the PINN with a technique that automatically adapts the weights on the loss terms corresponding to PDE, boundary, interface, and initial conditions. The performance of the proposed PINN is demonstrated in multiple scenarios with different material thicknesses and thermal boundary conditions.

Finite mixture models are typically inconsistent for the number of components

Scientists and engineers are often interested in learning the number of subpopulations (or components) present in a data set. Practitioners commonly use a Dirichlet process mixture model (DPMM) for this purpose; in particular, they count the number of clusters---i.e. components containing at least one data point---in the DPMM posterior. But Miller and Harrison (2013) warn that the DPMM cluster-count posterior is severely inconsistent for the number of latent components when the data are truly generated from a finite mixture; that is, the cluster-count posterior probability on the true generating number of components goes to zero in the limit of infinite data. A potential alternative is to use a finite mixture model (FMM) with a prior on the number of components. Past work has shown the resulting FMM component-count posterior is consistent. But existing results crucially depend on the assumption that the component likelihoods are perfectly specified. In practice, this assumption is unrealistic, and empirical evidence (Miller and Dunson, 2019) suggests that the FMM posterior on the number of components is sensitive to the likelihood choice. In this paper, we add rigor to data-analysis folk wisdom by proving that under even the slightest model misspecification, the FMM posterior on the number of components is ultraseverely inconsistent: for any finite $k \in \mathbb{N}$, the posterior probability that the number of components is $k$ converges to 0 in the limit of infinite data. We illustrate practical consequences of our theory on simulated and real data sets.