Matthew J. Johnson

Roy Frostig, Matthew J. Johnson, Dougal Maclaurin et al.

We decompose reverse-mode automatic differentiation into (forward-mode) linearization followed by transposition. Doing so isolates the essential difference between forward- and reverse-mode AD, and simplifies their joint implementation. In particular, once forward-mode AD rules are defined for every primitive operation in a source language, only linear primitives require an additional transposition rule in order to arrive at a complete reverse-mode AD implementation. This is how reverse-mode AD is written in JAX and Dex.

Matthew D. Hoffman, Matthew J. Johnson, Dustin Tran

Deriving conditional and marginal distributions using conjugacy relationships can be time consuming and error prone. In this paper, we propose a strategy for automating such derivations. Unlike previous systems which focus on relationships between pairs of random variables, our system (which we call Autoconj) operates directly on Python functions that compute log-joint distribution functions. Autoconj provides support for conjugacy-exploiting algorithms in any Python embedded PPL. This paves the way for accelerating development of novel inference algorithms and structure-exploiting modeling strategies.

Sharad Vikram, Matthew D. Hoffman, Matthew J. Johnson

In variational autoencoders, the prior on the latent codes $z$ is often treated as an afterthought, but the prior shapes the kind of latent representation that the model learns. If the goal is to learn a representation that is interpretable and useful, then the prior should reflect the ways in which the high-level factors that describe the data vary. The "default" prior is an isotropic normal, but if the natural factors of variation in the dataset exhibit discrete structure or are not independent, then the isotropic-normal prior will actually encourage learning representations that mask this structure. To alleviate this problem, we propose using a flexible Bayesian nonparametric hierarchical clustering prior based on the time-marginalized coalescent (TMC). To scale learning to large datasets, we develop a new inducing-point approximation and inference algorithm. We then apply the method without supervision to several datasets and examine the interpretability and practical performance of the inferred hierarchies and learned latent space.

Marvin Zhang, Sharad Vikram, Laura Smith et al.

Model-based reinforcement learning (RL) has proven to be a data efficient approach for learning control tasks but is difficult to utilize in domains with complex observations such as images. In this paper, we present a method for learning representations that are suitable for iterative model-based policy improvement, even when the underlying dynamical system has complex dynamics and image observations, in that these representations are optimized for inferring simple dynamics and cost models given data from the current policy. This enables a model-based RL method based on the linear-quadratic regulator (LQR) to be used for systems with image observations. We evaluate our approach on a range of robotics tasks, including manipulation with a real-world robotic arm directly from images. We find that our method produces substantially better final performance than other model-based RL methods while being significantly more efficient than model-free RL.

Ryan P. Adams, Jeffrey Pennington, Matthew J. Johnson et al.

Many important problems are characterized by the eigenvalues of a large matrix. For example, the difficulty of many optimization problems, such as those arising from the fitting of large models in statistics and machine learning, can be investigated via the spectrum of the Hessian of the empirical loss function. Network data can be understood via the eigenstructure of a graph Laplacian matrix using spectral graph theory. Quantum simulations and other many-body problems are often characterized via the eigenvalues of the solution space, as are various dynamic systems. However, naive eigenvalue estimation is computationally expensive even when the matrix can be represented; in many of these situations the matrix is so large as to only be available implicitly via products with vectors. Even worse, one may only have noisy estimates of such matrix vector products. In this work, we combine several different techniques for randomized estimation and show that it is possible to construct unbiased estimators to answer a broad class of questions about the spectra of such implicit matrices, even in the presence of noise. We validate these methods on large-scale problems in which graph theory and random matrix theory provide ground truth.

Ardavan Saeedi, Matthew D. Hoffman, Stephen J. DiVerdi et al.

Professional-grade software applications are powerful but complicated$-$expert users can achieve impressive results, but novices often struggle to complete even basic tasks. Photo editing is a prime example: after loading a photo, the user is confronted with an array of cryptic sliders like "clarity", "temp", and "highlights". An automatically generated suggestion could help, but there is no single "correct" edit for a given image$-$different experts may make very different aesthetic decisions when faced with the same image, and a single expert may make different choices depending on the intended use of the image (or on a whim). We therefore want a system that can propose multiple diverse, high-quality edits while also learning from and adapting to a user's aesthetic preferences. In this work, we develop a statistical model that meets these objectives. Our model builds on recent advances in neural network generative modeling and scalable inference, and uses hierarchical structure to learn editing patterns across many diverse users. Empirically, we find that our model outperforms other approaches on this challenging multimodal prediction task.

Scott W. Linderman, Andrew C. Miller, Ryan P. Adams et al.

Many natural systems, such as neurons firing in the brain or basketball teams traversing a court, give rise to time series data with complex, nonlinear dynamics. We can gain insight into these systems by decomposing the data into segments that are each explained by simpler dynamic units. Building on switching linear dynamical systems (SLDS), we present a new model class that not only discovers these dynamical units, but also explains how their switching behavior depends on observations or continuous latent states. These "recurrent" switching linear dynamical systems provide further insight by discovering the conditions under which each unit is deployed, something that traditional SLDS models fail to do. We leverage recent algorithmic advances in approximate inference to make Bayesian inference in these models easy, fast, and scalable.

Matthew J. Johnson, David Duvenaud, Alexander B. Wiltschko et al.

We propose a general modeling and inference framework that composes probabilistic graphical models with deep learning methods and combines their respective strengths. Our model family augments graphical structure in latent variables with neural network observation models. For inference, we extend variational autoencoders to use graphical model approximating distributions with recognition networks that output conjugate potentials. All components of these models are learned simultaneously with a single objective, giving a scalable algorithm that leverages stochastic variational inference, natural gradients, graphical model message passing, and the reparameterization trick. We illustrate this framework with several example models and an application to mouse behavioral phenotyping.

Scott W. Linderman, Matthew J. Johnson, Ryan P. Adams

Many practical modeling problems involve discrete data that are best represented as draws from multinomial or categorical distributions. For example, nucleotides in a DNA sequence, children's names in a given state and year, and text documents are all commonly modeled with multinomial distributions. In all of these cases, we expect some form of dependency between the draws: the nucleotide at one position in the DNA strand may depend on the preceding nucleotides, children's names are highly correlated from year to year, and topics in text may be correlated and dynamic. These dependencies are not naturally captured by the typical Dirichlet-multinomial formulation. Here, we leverage a logistic stick-breaking representation and recent innovations in Pólya-gamma augmentation to reformulate the multinomial distribution in terms of latent variables with jointly Gaussian likelihoods, enabling us to take advantage of a host of Bayesian inference techniques for Gaussian models with minimal overhead.

Jonathan H. Huggins, Ardavan Saeedi, Matthew J. Johnson

In this note we provide detailed derivations of two versions of small-variance asymptotics for hierarchical Dirichlet process (HDP) mixture models and the HDP hidden Markov model (HDP-HMM, a.k.a. the infinite HMM). We include derivations for the probabilities of certain CRP and CRF partitions, which are of more general interest.

Scott W. Linderman, Matthew J. Johnson, Matthew A. Wilson et al.

Rodent hippocampal population codes represent important spatial information about the environment during navigation. Several computational methods have been developed to uncover the neural representation of spatial topology embedded in rodent hippocampal ensemble spike activity. Here we extend our previous work and propose a nonparametric Bayesian approach to infer rat hippocampal population codes during spatial navigation. To tackle the model selection problem, we leverage a nonparametric Bayesian model. Specifically, to analyze rat hippocampal ensemble spiking activity, we apply a hierarchical Dirichlet process-hidden Markov model (HDP-HMM) using two Bayesian inference methods, one based on Markov chain Monte Carlo (MCMC) and the other based on variational Bayes (VB). We demonstrate the effectiveness of our Bayesian approaches on recordings from a freely-behaving rat navigating in an open field environment. We find that MCMC-based inference with Hamiltonian Monte Carlo (HMC) hyperparameter sampling is flexible and efficient, and outperforms VB and MCMC approaches with hyperparameters set by empirical Bayes.

Matthew J. Johnson, Alan Willsky

There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the traditional HMM. However, in many settings the HDP-HMM's strict Markovian constraints are undesirable, particularly if we wish to learn or encode non-geometric state durations. We can extend the HDP-HMM to capture such structure by drawing upon explicit-duration semi-Markovianity, which has been developed in the parametric setting to allow construction of highly interpretable models that admit natural prior information on state durations. In this paper we introduce the explicitduration HDP-HSMM and develop posterior sampling algorithms for efficient inference in both the direct-assignment and weak-limit approximation settings. We demonstrate the utility of the model and our inference methods on synthetic data as well as experiments on a speaker diarization problem and an example of learning the patterns in Morse code.

Matthew J. Johnson, Alan S. Willsky

There is much interest in the Hierarchical Dirichlet Process Hidden Markov Model (HDP-HMM) as a natural Bayesian nonparametric extension of the ubiquitous Hidden Markov Model for learning from sequential and time-series data. However, in many settings the HDP-HMM's strict Markovian constraints are undesirable, particularly if we wish to learn or encode non-geometric state durations. We can extend the HDP-HMM to capture such structure by drawing upon explicit-duration semi-Markovianity, which has been developed mainly in the parametric frequentist setting, to allow construction of highly interpretable models that admit natural prior information on state durations. In this paper we introduce the explicit-duration Hierarchical Dirichlet Process Hidden semi-Markov Model (HDP-HSMM) and develop sampling algorithms for efficient posterior inference. The methods we introduce also provide new methods for sampling inference in the finite Bayesian HSMM. Our modular Gibbs sampling methods can be embedded in samplers for larger hierarchical Bayesian models, adding semi-Markov chain modeling as another tool in the Bayesian inference toolbox. We demonstrate the utility of the HDP-HSMM and our inference methods on both synthetic and real experiments.