Vinayak Rao

Imon Banerjee, Harsha Honnappa, Vinayak Rao

In this work, we study a natural nonparametric estimator of the transition probability matrices of a finite controlled Markov chain. We consider an offline setting with a fixed dataset, collected using a so-called logging policy. We develop sample complexity bounds for the estimator and establish conditions for minimaxity. Our statistical bounds depend on the logging policy through its mixing properties. We show that achieving a particular statistical risk bound involves a subtle and interesting trade-off between the strength of the mixing properties and the number of samples. We demonstrate the validity of our results under various examples, such as ergodic Markov chains, weakly ergodic inhomogeneous Markov chains, and controlled Markov chains with non-stationary Markov, episodic, and greedy controls. Lastly, we use these sample complexity bounds to establish concomitant ones for offline evaluation of stationary Markov control policies.

Yangze Zhou, Vinayak Rao, Bruno Ribeiro

Node classification is a central task in relational learning, with the current state-of-the-art hinging on two key principles: (i) predictions are permutation-invariant to the ordering of a node's neighbors, and (ii) predictions are a function of the node's $r$-hop neighborhood topology and attributes, $r \geq 2$. Both graph neural networks and collective inference methods (e.g., belief propagation) rely on information from up to $r$-hops away. In this work, we study if the use of more powerful permutation-invariant functions can sometimes avoid the need for classifiers to collect information beyond $1$-hop. Towards this, we introduce a new architecture, the Set Twister, which generalizes DeepSets (Zaheer et al., 2017), a simple and widely-used permutation-invariant representation. Set Twister theoretically increases expressiveness of DeepSets, allowing it to capture higher-order dependencies, while keeping its simplicity and low computational cost. Empirically, we see accuracy improvements of Set Twister over DeepSets as well as a variety of graph neural networks and collective inference schemes in several tasks, while showcasing its implementation simplicity and computational efficiency.

Mohamed A. Zahran, Leonardo Teixeira, Vinayak Rao et al.

This work proposes an unsupervised learning framework for trajectory (sequence) outlier detection that combines ranking tests with user sequence models. The overall framework identifies sequence outliers at a desired false positive rate (FPR), in an otherwise parameter-free manner. We evaluate our methodology on a collection of real and simulated datasets based on user actions at the websites last.fm and msnbc.com, where we know ground truth, and demonstrate improved accuracy over existing approaches. We also apply our approach to a large real-world dataset of Pinterest and Facebook users, where we find that users tend to re-share Pinterest posts of Facebook friends significantly more than other types of users, pointing to a potential influence of Facebook friendship on sharing behavior on Pinterest.

Jordan Awan, Vinayak Rao

Differential privacy (DP) offers strong theoretical privacy guarantees, but implementations of DP mechanisms may be vulnerable to side-channel attacks, such as timing attacks. When sampling methods such as MCMC or rejection sampling are used to implement a mechanism, the runtime can leak private information. We characterize the additional privacy cost due to the runtime of a rejection sampler in terms of both $(ε,δ)$-DP as well as $f$-DP. We also show that unless the acceptance probability is constant across databases, the runtime of a rejection sampler does not satisfy $ε$-DP for any $ε$. We show that there is a similar breakdown in privacy with adaptive rejection samplers. We propose three modifications to the rejection sampling algorithm, with varying assumptions, to protect against timing attacks by making the runtime independent of the data. The modification with the weakest assumptions is an approximate sampler, introducing a small increase in the privacy cost, whereas the other modifications give perfect samplers. We also use our techniques to develop an adaptive rejection sampler for log-Hölder densities, which also has data-independent runtime. We give several examples of DP mechanisms that fit the assumptions of our methods and can thus be implemented using our samplers.

Guilherme Gomes, Vinayak Rao, Jennifer Neville

Clustering and community detection with multiple graphs have typically focused on aligned graphs, where there is a mapping between nodes across the graphs (e.g., multi-view, multi-layer, temporal graphs). However, there are numerous application areas with multiple graphs that are only partially aligned, or even unaligned. These graphs are often drawn from the same population, with communities of potentially different sizes that exhibit similar structure. In this paper, we develop a joint stochastic blockmodel (Joint SBM) to estimate shared communities across sets of heterogeneous non-aligned graphs. We derive an efficient spectral clustering approach to learn the parameters of the joint SBM. We evaluate the model on both synthetic and real-world datasets and show that the joint model is able to exploit cross-graph information to better estimate the communities compared to learning separate SBMs on each individual graph.

Ryan L. Murphy, Balasubramaniam Srinivasan, Vinayak Rao et al.

This work generalizes graph neural networks (GNNs) beyond those based on the Weisfeiler-Lehman (WL) algorithm, graph Laplacians, and diffusions. Our approach, denoted Relational Pooling (RP), draws from the theory of finite partial exchangeability to provide a framework with maximal representation power for graphs. RP can work with existing graph representation models and, somewhat counterintuitively, can make them even more powerful than the original WL isomorphism test. Additionally, RP allows architectures like Recurrent Neural Networks and Convolutional Neural Networks to be used in a theoretically sound approach for graph classification. We demonstrate improved performance of RP-based graph representations over state-of-the-art methods on a number of tasks.

Ryan L. Murphy, Balasubramaniam Srinivasan, Vinayak Rao et al.

We consider a simple and overarching representation for permutation-invariant functions of sequences (or multiset functions). Our approach, which we call Janossy pooling, expresses a permutation-invariant function as the average of a permutation-sensitive function applied to all reorderings of the input sequence. This allows us to leverage the rich and mature literature on permutation-sensitive functions to construct novel and flexible permutation-invariant functions. If carried out naively, Janossy pooling can be computationally prohibitive. To allow computational tractability, we consider three kinds of approximations: canonical orderings of sequences, functions with $k$-order interactions, and stochastic optimization algorithms with random permutations. Our framework unifies a variety of existing work in the literature, and suggests possible modeling and algorithmic extensions. We explore a few in our experiments, which demonstrate improved performance over current state-of-the-art methods.

Putu Ayu Sudyanti, Vinayak Rao

This paper addresses challenges in flexibly modeling multimodal data that lie on constrained spaces. Such data are commonly found in spatial applications, such as climatology and criminology, where measurements are restricted to a geographical area. Other settings include domains where unsuitable recordings are discarded, such as flow-cytometry measurements. A simple approach to modeling such data is through the use of mixture models, especially nonparametric mixture models. Mixture models, while flexible and theoretically well-understood, are unsuitable for settings involving complicated constraints, leading to difficulties in specifying the component distributions and in evaluating normalization constants. Bayesian inference over the parameters of these models results in posterior distributions that are doubly-intractable. We address this problem via an algorithm based on rejection sampling and data augmentation. We view samples from a truncated distribution as outcomes of a rejection sampling scheme, where proposals are made from a simple mixture model and are rejected if they violate the constraints. Our scheme proceeds by imputing the rejected samples given mixture parameters and then resampling parameters given all samples. We study two modeling approaches: mixtures of truncated Gaussians and truncated mixtures of Gaussians, along with their associated Markov chain Monte Carlo sampling algorithms. We also discuss variations of the models, as well as approximations to improve mixing, reduce computational cost, and lower variance. We present results on simulated data and apply our algorithms to two problems; one involving flow-cytometry data, and the other, crime recorded in the city of Chicago.

Guilherme Gomes, Vinayak Rao, Jennifer Neville

In this work, we consider hypothesis testing and anomaly detection on datasets where each observation is a weighted network. Examples of such data include brain connectivity networks from fMRI flow data, or word co-occurrence counts for populations of individuals. Current approaches to hypothesis testing for weighted networks typically requires thresholding the edge-weights, to transform the data to binary networks. This results in a loss of information, and outcomes are sensitivity to choice of threshold levels. Our work avoids this, and we consider weighted-graph observations in two situations, 1) where each graph belongs to one of two populations, and 2) where entities belong to one of two populations, with each entity possessing multiple graphs (indexed e.g. by time). Specifically, we propose a hierarchical Bayesian hypothesis testing framework that models each population with a mixture of latent space models for weighted networks, and then tests populations of networks for differences in distribution over components. Our framework is capable of population-level, entity-specific, as well as edge-specific hypothesis testing. We apply it to synthetic data and three real-world datasets: two social media datasets involving word co-occurrences from discussions on Twitter of the political unrest in Brazil, and on Instagram concerning Attention Deficit Hyperactivity Disorder (ADHD) and depression drugs, and one medical dataset involving fMRI brain-scans of human subjects. The results show that our proposed method has lower Type I error and higher statistical power compared to alternatives that need to threshold the edge weights. Moreover, they show our proposed method is better suited to deal with highly heterogeneous datasets.

Aditi Iyer, Bingjing Tang, Vinayak Rao et al.

We propose a novel two-phase approach to functional network estimation of multi-subject functional Magnetic Resonance Imaging (fMRI) data, which applies model-based image segmentation to determine a group-representative connectivity map. In our approach, we first improve clustering-based Independent Component Analysis (ICA) to generate maps of components occurring consistently across subjects, and then estimate the group-representative map through MAP-MRF (Maximum a priori - Markov random field) labeling. For the latter, we provide a novel and efficient variational Bayes algorithm. We study the performance of the proposed method using synthesized data following a theoretical model, and demonstrate its viability in blind extraction of group-representative functional networks using simulated fMRI data. We anticipate the proposed method will be applied in identifying common neuronal characteristics in a population, and could be further extended to real-world clinical diagnosis.

Xin Yuan, Vinayak Rao, Shaobo Han et al.

A new shrinkage-based construction is developed for a compressible vector $\boldsymbol{x}\in\mathbb{R}^n$, for cases in which the components of $\xv$ are naturally associated with a tree structure. Important examples are when $\xv$ corresponds to the coefficients of a wavelet or block-DCT representation of data. The method we consider in detail, and for which numerical results are presented, is based on increments of a gamma process. However, we demonstrate that the general framework is appropriate for many other types of shrinkage priors, all within the Lévy process family, with the gamma process a special case. Bayesian inference is carried out by approximating the posterior with samples from an MCMC algorithm, as well as by constructing a heuristic variational approximation to the posterior. We also consider expectation-maximization (EM) for a MAP (point) solution. State-of-the-art results are manifested for compressive sensing and denoising applications, the latter with spiky (non-Gaussian) noise.

Vinayak Rao, Yee Whye Teh

Markov jump processes and continuous time Bayesian networks are important classes of continuous time dynamical systems. In this paper, we tackle the problem of inferring unobserved paths in these models by introducing a fast auxiliary variable Gibbs sampler. Our approach is based on the idea of uniformization, and sets up a Markov chain over paths by sampling a finite set of virtual jump times and then running a standard hidden Markov model forward filtering-backward sampling algorithm over states at the set of extant and virtual jump times. We demonstrate significant computational benefits over a state-of-the-art Gibbs sampler on a number of continuous time Bayesian networks.