Luis Oala, Manil Maskey, Lilith Bat-Leah et al. · mit
Drawing from discussions at the inaugural DMLR workshop at ICML 2023 and meetings prior, in this report we outline the relevance of community engagement and infrastructure development for the creation of next-generation public datasets that will advance machine learning science. We chart a path forward as a collective effort to sustain the creation and maintenance of these datasets and methods towards positive scientific, societal and business impact.
Diversity is an important criterion for many areas of machine learning (ML), including generative modeling and dataset curation. However, existing metrics for measuring diversity are often domain-specific and limited in flexibility. In this paper, we address the diversity evaluation problem by proposing the Vendi Score, which connects and extends ideas from ecology and quantum statistical mechanics to ML. The Vendi Score is defined as the exponential of the Shannon entropy of the eigenvalues of a similarity matrix. This matrix is induced by a user-defined similarity function applied to the sample to be evaluated for diversity. In taking a similarity function as input, the Vendi Score enables its user to specify any desired form of diversity. Importantly, unlike many existing metrics in ML, the Vendi Score does not require a reference dataset or distribution over samples or labels, it is therefore general and applicable to any generative model, decoding algorithm, and dataset from any domain where similarity can be defined. We showcase the Vendi Score on molecular generative modeling where we found it addresses shortcomings of the current diversity metric of choice in that domain. We also applied the Vendi Score to generative models of images and decoding algorithms of text where we found it confirms known results about diversity in those domains. Furthermore, we used the Vendi Score to measure mode collapse, a known shortcoming of generative adversarial networks (GANs). In particular, the Vendi Score revealed that even GANs that capture all the modes of a labeled dataset can be less diverse than the original dataset. Finally, the interpretability of the Vendi Score allowed us to diagnose several benchmark ML datasets for diversity, opening the door for diversity-informed data augmentation.
Andre Niyongabo Rubungo, Craig Arnold, Barry P. Rand et al. · princeton
The prediction of crystal properties plays a crucial role in the crystal design process. Current methods for predicting crystal properties focus on modeling crystal structures using graph neural networks (GNNs). Although GNNs are powerful, accurately modeling the complex interactions between atoms and molecules within a crystal remains a challenge. Surprisingly, predicting crystal properties from crystal text descriptions is understudied, despite the rich information and expressiveness that text data offer. One of the main reasons is the lack of publicly available data for this task. In this paper, we develop and make public a benchmark dataset (called TextEdge) that contains text descriptions of crystal structures with their properties. We then propose LLM-Prop, a method that leverages the general-purpose learning capabilities of large language models (LLMs) to predict the physical and electronic properties of crystals from their text descriptions. LLM-Prop outperforms the current state-of-the-art GNN-based crystal property predictor by about 4% in predicting band gap, 3% in classifying whether the band gap is direct or indirect, and 66% in predicting unit cell volume. LLM-Prop also outperforms a finetuned MatBERT, a domain-specific pre-trained BERT model, despite having 3 times fewer parameters. Our empirical results may highlight the current inability of GNNs to capture information pertaining to space group symmetry and Wyckoff sites for accurate crystal property prediction.
Measuring diversity accurately is important for many scientific fields, including machine learning (ML), ecology, and chemistry. The Vendi Score was introduced as a generic similarity-based diversity metric that extends the Hill number of order q=1 by leveraging ideas from quantum statistical mechanics. Contrary to many diversity metrics in ecology, the Vendi Score accounts for similarity and does not require knowledge of the prevalence of the categories in the collection to be evaluated for diversity. However, the Vendi Score treats each item in a given collection with a level of sensitivity proportional to the item's prevalence. This is undesirable in settings where there is a significant imbalance in item prevalence. In this paper, we extend the other Hill numbers using similarity to provide flexibility in allocating sensitivity to rare or common items. This leads to a family of diversity metrics -- Vendi scores with different levels of sensitivity -- that can be used in a variety of applications. We study the properties of the scores in a synthetic controlled setting where the ground truth diversity is known. We then test their utility in improving molecular simulations via Vendi Sampling. Finally, we use the Vendi scores to better understand the behavior of image generative models in terms of memorization, duplication, diversity, and sample quality.
Minimizing the inclusive Kullback-Leibler (KL) divergence with stochastic gradient descent (SGD) is challenging since its gradient is defined as an integral over the posterior. Recently, multiple methods have been proposed to run SGD with biased gradient estimates obtained from a Markov chain. This paper provides the first non-asymptotic convergence analysis of these methods by establishing their mixing rate and gradient variance. To do this, we demonstrate that these methods-which we collectively refer to as Markov chain score ascent (MCSA) methods-can be cast as special cases of the Markov chain gradient descent framework. Furthermore, by leveraging this new understanding, we develop a novel MCSA scheme, parallel MCSA (pMCSA), that achieves a tighter bound on the gradient variance. We demonstrate that this improved theoretical result translates to superior empirical performance.
Rafael Espinosa Castañeda, Ashley Dale, Hongchen Wang et al.
Materials science data collection can be expensive, making the reuse and long-term utility of datasets critical important for future discovery campaigns. In practice, researchers prioritize a subset of properties due to research interests. However, ignoring a subset of outcomes in data collection campaigns potentially generate datasets poorly suited for future learning tasks. Here, we present a framework for dataset construction that maximizes informativeness for target properties of interest while preserving performance on untargeted ones. Our approach uses diversity-aware selection to ensure broad coverage of the materials space. In noisy experimental dataset construction, we find that without our diversity-aware framework, prediction performance on untargeted properties can degrade by up to 40% relative to random sampling, whereas applying our framework yields improvements of up to 10% . For targeted properties, performance can degrade with respect to random sampling by up to 12.5% without diversity, while our framework achieves gains of up to 25%. Incorporating diversity into dataset construction not only preserves informativeness for the targeted properties, but also improves materials coverage for potential future objectives. As a result, the constructed datasets remain broadly informative across considered and unconsidered outcomes, ensuring unbiased quality entries and mitigating cold-start limitations in subsequent modeling and discovery campaigns.
Out-of-distribution (OOD) detection is critical for the safe deployment of machine learning systems. Existing post-hoc detectors typically rely on model confidence scores or likelihood estimates in feature space, often under restrictive distributional assumptions. In this work, we introduce a third paradigm and formulate OOD detection from a diversity perspective. We propose the Vendi Novelty Score (VNS), an OOD detector based on the Vendi Scores (VS), a family of similarity-based diversity metrics. VNS quantifies how much a test sample increases the VS of the in-distribution feature set, providing a principled notion of novelty that does not require density modeling. VNS is linear-time, non-parametric, and naturally combines class-conditional (local) and dataset-level (global) novelty signals. Across multiple image classification benchmarks and network architectures, VNS achieves state-of-the-art OOD detection performance. Remarkably, VNS retains this performance when computed using only 1% of the training data, enabling deployment in memory- or access-constrained settings.
Andre Niyongabo Rubungo, Kangming Li, Jason Hattrick-Simpers et al. · princeton
Large language models (LLMs) are increasingly being used in materials science. However, little attention has been given to benchmarking and standardized evaluation for LLM-based materials property prediction, which hinders progress. We present LLM4Mat-Bench, the largest benchmark to date for evaluating the performance of LLMs in predicting the properties of crystalline materials. LLM4Mat-Bench contains about 1.9M crystal structures in total, collected from 10 publicly available materials data sources, and 45 distinct properties. LLM4Mat-Bench features different input modalities: crystal composition, CIF, and crystal text description, with 4.7M, 615.5M, and 3.1B tokens in total for each modality, respectively. We use LLM4Mat-Bench to fine-tune models with different sizes, including LLM-Prop and MatBERT, and provide zero-shot and few-shot prompts to evaluate the property prediction capabilities of LLM-chat-like models, including Llama, Gemma, and Mistral. The results highlight the challenges of general-purpose LLMs in materials science and the need for task-specific predictive models and task-specific instruction-tuned LLMs in materials property prediction.
Experimental design techniques such as active search and Bayesian optimization are widely used in the natural sciences for data collection and discovery. However, existing techniques tend to favor exploitation over exploration of the search space, which causes them to get stuck in local optima. This ``collapse" problem prevents experimental design algorithms from yielding diverse high-quality data. In this paper, we extend the Vendi scores -- a family of interpretable similarity-based diversity metrics -- to account for quality. We then leverage these quality-weighted Vendi scores to tackle experimental design problems across various applications, including drug discovery, materials discovery, and reinforcement learning. We found that quality-weighted Vendi scores allow us to construct policies for experimental design that flexibly balance quality and diversity, and ultimately assemble rich and diverse sets of high-performing data points. Our algorithms led to a 70%-170% increase in the number of effective discoveries compared to baselines.
Optimal trajectory design is computationally expensive for nonlinear and high-dimensional dynamical systems. The challenge arises from the non-convex nature of the optimization problem with multiple local optima, which usually requires a global search. Traditional numerical solvers struggle to find diverse solutions efficiently without appropriate initial guesses. In this paper, we introduce DiffuSolve, a general diffusion model-based solver for non-convex trajectory optimization. An expressive diffusion model is trained on pre-collected locally optimal solutions and efficiently samples initial guesses, which then warm-starts numerical solvers to fine-tune the feasibility and optimality. We also present DiffuSolve+, a novel constrained diffusion model with an additional loss in training that further reduces the problem constraint violations of diffusion samples. Experimental evaluations on three tasks verify the improved robustness, diversity, and a 2$\times$ to 11$\times$ increase in computational efficiency with our proposed method, which generalizes well to trajectory optimization problems of varying challenges.
The evolution of microscopy, beginning with its invention in the late 16th century, has continuously enhanced our ability to explore and understand the microscopic world, enabling increasingly detailed observations of structures and phenomena. In parallel, the rise of data-driven science has underscored the need for sophisticated methods to explore and understand the composition of complex data collections. This paper introduces the Vendiscope, the first algorithmic microscope designed to extend traditional microscopy to computational analysis. The Vendiscope leverages the Vendi scores -- a family of differentiable diversity metrics rooted in ecology and quantum mechanics -- and assigns weights to data points based on their contribution to the overall diversity of the collection. These weights enable high-resolution data analysis at scale. We demonstrate this across biology, materials science, and machine learning (ML). We analyzed the $250$ million protein sequences in the protein universe, discovering that over $200$ million are near-duplicates and that AlphaFold fails on proteins with Gene Ontology (GO) functions that contribute most to diversity. Applying the Vendiscope to the Materials Project database led to similar findings: more than $85\%$ of the crystals with formation energy data are near-duplicates and ML models perform poorly on materials that enhance diversity. Additionally, the Vendiscope can be used to study phenomena such as memorization in generative models. We used the Vendiscope to identify memorized training samples from $13$ different generative models and found that the best-performing ones often memorize the training samples that contribute least to diversity. Our findings demonstrate that the Vendiscope can serve as a powerful tool for data-driven science.
Elizabeth G. Campolongo, Yuan-Tang Chou, Ekaterina Govorkova et al.
Scientific discoveries are often made by finding a pattern or object that was not predicted by the known rules of science. Oftentimes, these anomalous events or objects that do not conform to the norms are an indication that the rules of science governing the data are incomplete, and something new needs to be present to explain these unexpected outliers. The challenge of finding anomalies can be confounding since it requires codifying a complete knowledge of the known scientific behaviors and then projecting these known behaviors on the data to look for deviations. When utilizing machine learning, this presents a particular challenge since we require that the model not only understands scientific data perfectly but also recognizes when the data is inconsistent and out of the scope of its trained behavior. In this paper, we present three datasets aimed at developing machine learning-based anomaly detection for disparate scientific domains covering astrophysics, genomics, and polar science. We present the different datasets along with a scheme to make machine learning challenges around the three datasets findable, accessible, interoperable, and reusable (FAIR). Furthermore, we present an approach that generalizes to future machine learning challenges, enabling the possibility of large, more compute-intensive challenges that can ultimately lead to scientific discovery.
Neural scaling laws are driving the machine learning community toward training ever-larger foundation models across domains, assuring high accuracy and transferable representations for extrapolative tasks. We test this promise in quantum chemistry by scaling model capacity and training data from quantum chemical calculations. As a generalization task, we evaluate the resulting models' predictions of the bond dissociation energy of neutral H$_2$, the simplest possible molecule. We find that, regardless of dataset size or model capacity, models trained only on stable structures fail dramatically to even qualitatively reproduce the H$_2$ energy curve. Only when compressed and stretched geometries are explicitly included in training do the predictions roughly resemble the correct shape. Nonetheless, the largest foundation models trained on the largest and most diverse datasets containing dissociating diatomics exhibit serious failures on simple diatomic molecules. Most strikingly, they cannot reproduce the trivial repulsive energy curve of two bare protons, revealing their failure to learn the basic Coulomb's law involved in electronic structure theory. These results suggest that scaling alone is insufficient for building reliable quantum chemical models.
Alternators have recently been introduced as a framework for modeling time-dependent data. They often outperform other popular frameworks, such as state-space models and diffusion models, on challenging time-series tasks. This paper introduces a new Alternator model, called Alternator++, which enhances the flexibility of traditional Alternators by explicitly modeling the noise terms used to sample the latent and observed trajectories, drawing on the idea of noise models from the diffusion modeling literature. Alternator++ optimizes the sum of the Alternator loss and a noise-matching loss. The latter forces the noise trajectories generated by the two noise models to approximate the noise trajectories that produce the observed and latent trajectories. We demonstrate the effectiveness of Alternator++ in tasks such as density estimation, time series imputation, and forecasting, showing that it outperforms several strong baselines, including Mambas, ScoreGrad, and Dyffusion.
From pedestrians to Kuramoto oscillators, interactions between agents govern how dynamical systems evolve in space and time. Discovering how these agents relate to each other has the potential to improve our understanding of the often complex dynamics that underlie these systems. Recent works learn to categorize relationships between agents based on observations of their physical behavior. These approaches model relationship categories as outcomes of a categorical distribution which is limiting and contrary to real-world systems, where relationship categories often intermingle and interact. In this work, we introduce a level of abstraction between the observable behavior of agents and the latent categories that determine their behavior. To do this, we learn a mapping from agent observations to agent preferences for a set of latent categories. The learned preferences and inter-agent proximity are integrated in a nonlinear opinion dynamics model, which allows us to naturally identify mutually exclusive categories, predict an agent's evolution in time, and control an agent's behavior. Through extensive experiments, we demonstrate the utility of our model for learning interpretable categories, and the efficacy of our model for long-horizon trajectory prediction.
Variational auto-encoders (VAEs) are a powerful approach to unsupervised learning. They enable scalable approximate posterior inference in latent-variable models using variational inference (VI). A VAE posits a variational family parameterized by a deep neural network called an encoder that takes data as input. This encoder is shared across all the observations, which amortizes the cost of inference. However the encoder of a VAE has the undesirable property that it maps a given observation and a semantics-preserving transformation of it to different latent representations. This "inconsistency" of the encoder lowers the quality of the learned representations, especially for downstream tasks, and also negatively affects generalization. In this paper, we propose a regularization method to enforce consistency in VAEs. The idea is to minimize the Kullback-Leibler (KL) divergence between the variational distribution when conditioning on the observation and the variational distribution when conditioning on a random semantic-preserving transformation of this observation. This regularization is applicable to any VAE. In our experiments we apply it to four different VAE variants on several benchmark datasets and found it always improves the quality of the learned representations but also leads to better generalization. In particular, when applied to the Nouveau Variational Auto-Encoder (NVAE), our regularization method yields state-of-the-art performance on MNIST and CIFAR-10. We also applied our method to 3D data and found it learns representations of superior quality as measured by accuracy on a downstream classification task.
Probabilistic graphical modeling (PGM) provides a framework for formulating an interpretable generative process of data and expressing uncertainty about unknowns, but it lacks flexibility. Deep learning (DL) is an alternative framework for learning from data that has achieved great empirical success in recent years. DL offers great flexibility, but it lacks the interpretability and calibration of PGM. This thesis develops deep probabilistic graphical modeling (DPGM.) DPGM consists in leveraging DL to make PGM more flexible. DPGM brings about new methods for learning from data that exhibit the advantages of both PGM and DL. We use DL within PGM to build flexible models endowed with an interpretable latent structure. One model class we develop extends exponential family PCA using neural networks to improve predictive performance while enforcing the interpretability of the latent factors. Another model class we introduce enables accounting for long-term dependencies when modeling sequential data, which is a challenge when using purely DL or PGM approaches. Finally, DPGM successfully solves several outstanding problems of probabilistic topic models, a widely used family of models in PGM. DPGM also brings about new algorithms for learning with complex data. We develop reweighted expectation maximization, an algorithm that unifies several existing maximum likelihood-based algorithms for learning models parameterized by neural networks. This unifying view is made possible using expectation maximization, a canonical inference algorithm in PGM. We also develop entropy-regularized adversarial learning, a learning paradigm that deviates from the traditional maximum likelihood approach used in PGM. From the DL perspective, entropy-regularized adversarial learning provides a solution to the long-standing mode collapse problem of generative adversarial networks, a widely used DL approach.
Adji B. Dieng, Francisco J. R. Ruiz, David M. Blei et al.
Generative adversarial networks (GANs) are a powerful approach to unsupervised learning. They have achieved state-of-the-art performance in the image domain. However, GANs are limited in two ways. They often learn distributions with low support---a phenomenon known as mode collapse---and they do not guarantee the existence of a probability density, which makes evaluating generalization using predictive log-likelihood impossible. In this paper, we develop the prescribed GAN (PresGAN) to address these shortcomings. PresGANs add noise to the output of a density network and optimize an entropy-regularized adversarial loss. The added noise renders tractable approximations of the predictive log-likelihood and stabilizes the training procedure. The entropy regularizer encourages PresGANs to capture all the modes of the data distribution. Fitting PresGANs involves computing the intractable gradients of the entropy regularization term; PresGANs sidestep this intractability using unbiased stochastic estimates. We evaluate PresGANs on several datasets and found they mitigate mode collapse and generate samples with high perceptual quality. We further found that PresGANs reduce the gap in performance in terms of predictive log-likelihood between traditional GANs and variational autoencoders (VAEs).
Adji B. Dieng, Francisco J. R. Ruiz, David M. Blei
Topic modeling analyzes documents to learn meaningful patterns of words. For documents collected in sequence, dynamic topic models capture how these patterns vary over time. We develop the dynamic embedded topic model (D-ETM), a generative model of documents that combines dynamic latent Dirichlet allocation (D-LDA) and word embeddings. The D-ETM models each word with a categorical distribution parameterized by the inner product between the word embedding and a per-time-step embedding representation of its assigned topic. The D-ETM learns smooth topic trajectories by defining a random walk prior over the embedding representations of the topics. We fit the D-ETM using structured amortized variational inference with a recurrent neural network. On three different corpora---a collection of United Nations debates, a set of ACL abstracts, and a dataset of Science Magazine articles---we found that the D-ETM outperforms D-LDA on a document completion task. We further found that the D-ETM learns more diverse and coherent topics than D-LDA while requiring significantly less time to fit.
Adji B. Dieng, Francisco J. R. Ruiz, David M. Blei
Topic modeling analyzes documents to learn meaningful patterns of words. However, existing topic models fail to learn interpretable topics when working with large and heavy-tailed vocabularies. To this end, we develop the Embedded Topic Model (ETM), a generative model of documents that marries traditional topic models with word embeddings. In particular, it models each word with a categorical distribution whose natural parameter is the inner product between a word embedding and an embedding of its assigned topic. To fit the ETM, we develop an efficient amortized variational inference algorithm. The ETM discovers interpretable topics even with large vocabularies that include rare words and stop words. It outperforms existing document models, such as latent Dirichlet allocation (LDA), in terms of both topic quality and predictive performance.
Training deep generative models with maximum likelihood remains a challenge. The typical workaround is to use variational inference (VI) and maximize a lower bound to the log marginal likelihood of the data. Variational auto-encoders (VAEs) adopt this approach. They further amortize the cost of inference by using a recognition network to parameterize the variational family. Amortized VI scales approximate posterior inference in deep generative models to large datasets. However it introduces an amortization gap and leads to approximate posteriors of reduced expressivity due to the problem known as posterior collapse. In this paper, we consider expectation maximization (EM) as a paradigm for fitting deep generative models. Unlike VI, EM directly maximizes the log marginal likelihood of the data. We rediscover the importance weighted auto-encoder (IWAE) as an instance of EM and propose a new EM-based algorithm for fitting deep generative models called reweighted expectation maximization (REM). REM learns better generative models than the IWAE by decoupling the learning dynamics of the generative model and the recognition network using a separate expressive proposal found by moment matching. We compared REM to the VAE and the IWAE on several density estimation benchmarks and found it leads to significantly better performance as measured by log-likelihood.
Variational autoencoders learn distributions of high-dimensional data. They model data with a deep latent-variable model and then fit the model by maximizing a lower bound of the log marginal likelihood. VAEs can capture complex distributions, but they can also suffer from an issue known as "latent variable collapse," especially if the likelihood model is powerful. Specifically, the lower bound involves an approximate posterior of the latent variables; this posterior "collapses" when it is set equal to the prior, i.e., when the approximate posterior is independent of the data. While VAEs learn good generative models, latent variable collapse prevents them from learning useful representations. In this paper, we propose a simple new way to avoid latent variable collapse by including skip connections in our generative model; these connections enforce strong links between the latent variables and the likelihood function. We study generative skip models both theoretically and empirically. Theoretically, we prove that skip models increase the mutual information between the observations and the inferred latent variables. Empirically, we study images (MNIST and Omniglot) and text (Yahoo). Compared to existing VAE architectures, we show that generative skip models maintain similar predictive performance but lead to less collapse and provide more meaningful representations of the data.
Adji B. Dieng, Rajesh Ranganath, Jaan Altosaar et al.
Recurrent neural networks (RNNs) are powerful models of sequential data. They have been successfully used in domains such as text and speech. However, RNNs are susceptible to overfitting; regularization is important. In this paper we develop Noisin, a new method for regularizing RNNs. Noisin injects random noise into the hidden states of the RNN and then maximizes the corresponding marginal likelihood of the data. We show how Noisin applies to any RNN and we study many different types of noise. Noisin is unbiased--it preserves the underlying RNN on average. We characterize how Noisin regularizes its RNN both theoretically and empirically. On language modeling benchmarks, Noisin improves over dropout by as much as 12.2% on the Penn Treebank and 9.4% on the Wikitext-2 dataset. We also compared the state-of-the-art language model of Yang et al. 2017, both with and without Noisin. On the Penn Treebank, the method with Noisin more quickly reaches state-of-the-art performance.
Francisco J. R. Ruiz, Michalis K. Titsias, Adji B. Dieng et al.
Categorical distributions are ubiquitous in machine learning, e.g., in classification, language models, and recommendation systems. However, when the number of possible outcomes is very large, using categorical distributions becomes computationally expensive, as the complexity scales linearly with the number of outcomes. To address this problem, we propose augment and reduce (A&R), a method to alleviate the computational complexity. A&R uses two ideas: latent variable augmentation and stochastic variational inference. It maximizes a lower bound on the marginal likelihood of the data. Unlike existing methods which are specific to softmax, A&R is more general and is amenable to other categorical models, such as multinomial probit. On several large-scale classification problems, we show that A&R provides a tighter bound on the marginal likelihood and has better predictive performance than existing approaches.
In this paper, we propose TopicRNN, a recurrent neural network (RNN)-based language model designed to directly capture the global semantic meaning relating words in a document via latent topics. Because of their sequential nature, RNNs are good at capturing the local structure of a word sequence - both semantic and syntactic - but might face difficulty remembering long-range dependencies. Intuitively, these long-range dependencies are of semantic nature. In contrast, latent topic models are able to capture the global underlying semantic structure of a document but do not account for word ordering. The proposed TopicRNN model integrates the merits of RNNs and latent topic models: it captures local (syntactic) dependencies using an RNN and global (semantic) dependencies using latent topics. Unlike previous work on contextual RNN language modeling, our model is learned end-to-end. Empirical results on word prediction show that TopicRNN outperforms existing contextual RNN baselines. In addition, TopicRNN can be used as an unsupervised feature extractor for documents. We do this for sentiment analysis on the IMDB movie review dataset and report an error rate of $6.28\%$. This is comparable to the state-of-the-art $5.91\%$ resulting from a semi-supervised approach. Finally, TopicRNN also yields sensible topics, making it a useful alternative to document models such as latent Dirichlet allocation.
Adji B. Dieng, Dustin Tran, Rajesh Ranganath et al.
Variational inference (VI) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$. Closeness is usually measured via a divergence $D(q || p)$ from $q$ to $p$. While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose CHIVI, a black-box variational inference algorithm that minimizes $D_χ(p || q)$, the $χ$-divergence from $p$ to $q$. CHIVI minimizes an upper bound of the model evidence, which we term the $χ$ upper bound (CUBO). Minimizing the CUBO leads to improved posterior uncertainty, and it can also be used with the classical VI lower bound (ELBO) to provide a sandwich estimate of the model evidence. We study CHIVI on three models: probit regression, Gaussian process classification, and a Cox process model of basketball plays. When compared to expectation propagation and classical VI, CHIVI produces better error rates and more accurate estimates of posterior variance.
Probabilistic modeling is a powerful approach for analyzing empirical information. We describe Edward, a library for probabilistic modeling. Edward's design reflects an iterative process pioneered by George Box: build a model of a phenomenon, make inferences about the model given data, and criticize the model's fit to the data. Edward supports a broad class of probabilistic models, efficient algorithms for inference, and many techniques for model criticism. The library builds on top of TensorFlow to support distributed training and hardware such as GPUs. Edward enables the development of complex probabilistic models and their algorithms at a massive scale.