David Bindel

Kun Dong, Austin R. Benson, David Bindel

Spectral analysis connects graph structure to the eigenvalues and eigenvectors of associated matrices. Much of spectral graph theory descends directly from spectral geometry, the study of differentiable manifolds through the spectra of associated differential operators. But the translation from spectral geometry to spectral graph theory has largely focused on results involving only a few extreme eigenvalues and their associated eigenvalues. Unlike in geometry, the study of graphs through the overall distribution of eigenvalues - the spectral density - is largely limited to simple random graph models. The interior of the spectrum of real-world graphs remains largely unexplored, difficult to compute and to interpret. In this paper, we delve into the heart of spectral densities of real-world graphs. We borrow tools developed in condensed matter physics, and add novel adaptations to handle the spectral signatures of common graph motifs. The resulting methods are highly efficient, as we illustrate by computing spectral densities for graphs with over a billion edges on a single compute node. Beyond providing visually compelling fingerprints of graphs, we show how the estimation of spectral densities facilitates the computation of many common centrality measures, and use spectral densities to estimate meaningful information about graph structure that cannot be inferred from the extremal eigenpairs alone.

David Bindel, Amanda Hood

Let $T : Ω\rightarrow \bbC^{n \times n}$ be a matrix-valued function that is analytic on some simply-connected domain $Ω\subset \bbC$. A point $λ\in Ω$ is an eigenvalue if the matrix $T(λ)$ is singular. In this paper, we describe new localization results for nonlinear eigenvalue problems that generalize Gershgorin's theorem, pseudospectral inclusion theorems, and the Bauer-Fike theorem. We use our results to analyze three nonlinear eigenvalue problems: an example from delay differential equations, a problem due to Hadeler, and a quantum resonance computation.

Xinran Zhu, Leo Huang, Cameron Ibrahim et al.

The Bayesian transformed Gaussian process (BTG) model, proposed by Kedem and Oliviera, is a fully Bayesian counterpart to the warped Gaussian process (WGP) and marginalizes out a joint prior over input warping and kernel hyperparameters. This fully Bayesian treatment of hyperparameters often provides more accurate regression estimates and superior uncertainty propagation, but is prohibitively expensive. The BTG posterior predictive distribution, itself estimated through high-dimensional integration, must be inverted in order to perform model prediction. To make the Bayesian approach practical and comparable in speed to maximum-likelihood estimation (MLE), we propose principled and fast techniques for computing with BTG. Our framework uses doubly sparse quadrature rules, tight quantile bounds, and rank-one matrix algebra to enable both fast model prediction and model selection. These scalable methods allow us to regress over higher-dimensional datasets and apply BTG with layered transformations that greatly improve its expressibility. We demonstrate that BTG achieves superior empirical performance over MLE-based models.

Amanda Hood, David Bindel

Asymptotic dynamics of ordinary differential equations (ODEs) are commonly understood by looking at eigenvalues of a matrix, and transient dynamics can be bounded above and below by considering the corresponding pseudospectra. While asymptotics for other classes of differential equations have been studied using eigenvalues of a (nonlinear) matrix-valued function, there are no analogous pseudospectral bounds on transient growth. In this paper, we propose extensions of the pseudospectral results for ODEs first to higher order ODEs and then to delay differential equations (DDEs) with constant delay. Results are illustrated with a discretized partial delay differential equation and a model of a semiconductor laser with phase-conjugate feedback.

Gerald Ogbonna, David Bindel, Lindsay C. Anderson

The reliable operation of large-scale electric power networks is increasingly challenging, particularly with the integration of stochastic renewable generation. In this work, we address the problem of minimizing network transients by optimally modifying the underlying network. We formulate the problem in terms of graph Laplacian matrices and show that, under certain assumptions, the problem is convex. We derive a linear matrix inequality whose feasibility guarantees the existence and uniqueness of phase cohesive steady-state angles; this condition can be directly incorporated as a convex constraint in the optimization framework and we provide several geometric interpretations of the optimization problem. The proposed method is validated on the IEEE 30-bus test system, where results demonstrate that our approach effectively identifies critical links on the network. Dynamic simulations show a significant reduction in network transients and overall improvements across several performance metrics. We explore the sparsity-optimality trade-off using a reweighted $\ell_1$ heuristic.

Darian Nwankwo, David Bindel

Bayesian optimization (BO) methods choose sample points by optimizing an acquisition function derived from a statistical model of the objective. These acquisition functions are chosen to balance sampling regions with predicted good objective values against exploring regions where the objective is uncertain. Standard acquisition functions are myopic, considering only the impact of the next sample, but non-myopic acquisition functions may be more effective. In principle, one could model the sampling by a Markov decision process, and optimally choose the next sample by maximizing an expected reward computed by dynamic programming; however, this is infeasibly expensive. More practical approaches, such as rollout, consider a parametric family of sampling policies. In this paper, we show how to efficiently estimate rollout acquisition functions and their gradients, enabling stochastic gradient-based optimization of sampling policies.

Moontae Lee, Sungjun Cho, Kun Dong et al.

Across many data domains, co-occurrence statistics about the joint appearance of objects are powerfully informative. By transforming unsupervised learning problems into decompositions of co-occurrence statistics, spectral algorithms provide transparent and efficient algorithms for posterior inference such as latent topic analysis and community detection. As object vocabularies grow, however, it becomes rapidly more expensive to store and run inference algorithms on co-occurrence statistics. Rectifying co-occurrence, the key process to uphold model assumptions, becomes increasingly more vital in the presence of rare terms, but current techniques cannot scale to large vocabularies. We propose novel methods that simultaneously compress and rectify co-occurrence statistics, scaling gracefully with the size of vocabulary and the dimension of latent space. We also present new algorithms learning latent variables from the compressed statistics, and verify that our methods perform comparably to previous approaches on both textual and non-textual data.

Dongping Qi, David Bindel, Alexander Vladimirsky

We consider a task of surveillance-evading path-planning in a continuous setting. An Evader strives to escape from a 2D domain while minimizing the risk of detection (and immediate capture). The probability of detection is path-dependent and determined by the spatially inhomogeneous surveillance intensity, which is fixed but a priori unknown and gradually learned in the multi-episodic setting. We introduce a Bayesian reinforcement learning algorithm that relies on a Gaussian Process regression (to model the surveillance intensity function based on the information from prior episodes), numerical methods for Hamilton-Jacobi PDEs (to plan the best continuous trajectories based on the current model), and Confidence Bounds (to balance the exploration vs exploitation). We use numerical experiments and regret metrics to highlight the significant advantages of our approach compared to traditional graph-based algorithms of reinforcement learning.

Misha Padidar, Xinran Zhu, Leo Huang et al.

Gaussian processes with derivative information are useful in many settings where derivative information is available, including numerous Bayesian optimization and regression tasks that arise in the natural sciences. Incorporating derivative observations, however, comes with a dominating $O(N^3D^3)$ computational cost when training on $N$ points in $D$ input dimensions. This is intractable for even moderately sized problems. While recent work has addressed this intractability in the low-$D$ setting, the high-$N$, high-$D$ setting is still unexplored and of great value, particularly as machine learning problems increasingly become high dimensional. In this paper, we introduce methods to achieve fully scalable Gaussian process regression with derivatives using variational inference. Analogous to the use of inducing values to sparsify the labels of a training set, we introduce the concept of inducing directional derivatives to sparsify the partial derivative information of a training set. This enables us to construct a variational posterior that incorporates derivative information but whose size depends neither on the full dataset size $N$ nor the full dimensionality $D$. We demonstrate the full scalability of our approach on a variety of tasks, ranging from a high dimensional stellarator fusion regression task to training graph convolutional neural networks on Pubmed using Bayesian optimization. Surprisingly, we find that our approach can improve regression performance even in settings where only label data is available.

Leo Huang, Andrew Graven, David Bindel

A fundamental problem on graph-structured data is that of quantifying similarity between graphs. Graph kernels are an established technique for such tasks; in particular, those based on random walks and return probabilities have proven to be effective in wide-ranging applications, from bioinformatics to social networks to computer vision. However, random walk kernels generally suffer from slowness and tottering, an effect which causes walks to overemphasize local graph topology, undercutting the importance of global structure. To correct for these issues, we recast return probability graph kernels under the more general framework of density of states -- a framework which uses the lens of spectral analysis to uncover graph motifs and properties hidden within the interior of the spectrum -- and use our interpretation to construct scalable, composite density of states based graph kernels which balance local and global information, leading to higher classification accuracies on a host of benchmark datasets.

Kyle Wilson, David Bindel

Rotation Averaging is a non-convex optimization problem that determines orientations of a collection of cameras from their images of a 3D scene. The problem has been studied using a variety of distances and robustifiers. The intrinsic (or geodesic) distance on SO(3) is geometrically meaningful; but while some extrinsic distance-based solvers admit (conditional) guarantees of correctness, no comparable results have been found under the intrinsic metric. In this paper, we study the spatial distribution of local minima. First, we do a novel empirical study to demonstrate sharp transitions in qualitative behavior: as problems become noisier, they transition from a single (easy-to-find) dominant minimum to a cost surface filled with minima. In the second part of this paper we derive a theoretical bound for when this transition occurs. This is an extension of the results of [24], which used local convexity as a proxy to study the difficulty of problem. By recognizing the underlying quotient manifold geometry of the problem we achieve an n-fold improvement over prior work. Incidentally, our analysis also extends the prior $l_2$ work to general $l_p$ costs. Our results suggest using algebraic connectivity as an indicator of problem difficulty.

Eric Hans Lee, David Eriksson, Bolong Cheng et al.

Bayesian optimization (BO) is a class of sample-efficient global optimization methods, where a probabilistic model conditioned on previous observations is used to determine future evaluations via the optimization of an acquisition function. Most acquisition functions are myopic, meaning that they only consider the impact of the next function evaluation. Non-myopic acquisition functions consider the impact of the next $h$ function evaluations and are typically computed through rollout, in which $h$ steps of BO are simulated. These rollout acquisition functions are defined as $h$-dimensional integrals, and are expensive to compute and optimize. We show that a combination of quasi-Monte Carlo, common random numbers, and control variates significantly reduce the computational burden of rollout. We then formulate a policy-search based approach that removes the need to optimize the rollout acquisition function. Finally, we discuss the qualitative behavior of rollout policies in the setting of multi-modal objectives and model error.

David Eriksson, David Bindel, Christine A. Shoemaker

This paper describes Plumbing for Optimization with Asynchronous Parallelism (POAP) and the Python Surrogate Optimization Toolbox (pySOT). POAP is an event-driven framework for building and combining asynchronous optimization strategies, designed for global optimization of expensive functions where concurrent function evaluations are useful. POAP consists of three components: a worker pool capable of function evaluations, strategies to propose evaluations or other actions, and a controller that mediates the interaction between the workers and strategies. pySOT is a collection of synchronous and asynchronous surrogate optimization strategies, implemented in the POAP framework. We support the stochastic RBF method by Regis and Shoemaker along with various extensions of this method, and a general surrogate optimization strategy that covers most Bayesian optimization methods. We have implemented many different surrogate models, experimental designs, acquisition functions, and a large set of test problems. We make an extensive comparison between synchronous and asynchronous parallelism and find that the advantage of asynchronous computation increases as the variance of the evaluation time or number of processors increases. We observe a close to linear speed-up with 4, 8, and 16 processors in both the synchronous and asynchronous setting.

David Eriksson, Kun Dong, Eric Hans Lee et al.

Gaussian processes (GPs) with derivatives are useful in many applications, including Bayesian optimization, implicit surface reconstruction, and terrain reconstruction. Fitting a GP to function values and derivatives at $n$ points in $d$ dimensions requires linear solves and log determinants with an ${n(d+1) \times n(d+1)}$ positive definite matrix -- leading to prohibitive $\mathcal{O}(n^3d^3)$ computations for standard direct methods. We propose iterative solvers using fast $\mathcal{O}(nd)$ matrix-vector multiplications (MVMs), together with pivoted Cholesky preconditioning that cuts the iterations to convergence by several orders of magnitude, allowing for fast kernel learning and prediction. Our approaches, together with dimensionality reduction, enables Bayesian optimization with derivatives to scale to high-dimensional problems and large evaluation budgets.

Jacob R. Gardner, Geoff Pleiss, David Bindel et al.

Despite advances in scalable models, the inference tools used for Gaussian processes (GPs) have yet to fully capitalize on developments in computing hardware. We present an efficient and general approach to GP inference based on Blackbox Matrix-Matrix multiplication (BBMM). BBMM inference uses a modified batched version of the conjugate gradients algorithm to derive all terms for training and inference in a single call. BBMM reduces the asymptotic complexity of exact GP inference from $O(n^3)$ to $O(n^2)$. Adapting this algorithm to scalable approximations and complex GP models simply requires a routine for efficient matrix-matrix multiplication with the kernel and its derivative. In addition, BBMM uses a specialized preconditioner to substantially speed up convergence. In experiments we show that BBMM effectively uses GPU hardware to dramatically accelerate both exact GP inference and scalable approximations. Additionally, we provide GPyTorch, a software platform for scalable GP inference via BBMM, built on PyTorch.

Moontae Lee, David Bindel, David Mimno

Spectral topic modeling algorithms operate on matrices/tensors of word co-occurrence statistics to learn topic-specific word distributions. This approach removes the dependence on the original documents and produces substantial gains in efficiency and provable topic inference, but at a cost: the model can no longer provide information about the topic composition of individual documents. Recently Thresholded Linear Inverse (TLI) is proposed to map the observed words of each document back to its topic composition. However, its linear characteristics limit the inference quality without considering the important prior information over topics. In this paper, we evaluate Simple Probabilistic Inverse (SPI) method and novel Prior-aware Dual Decomposition (PADD) that is capable of learning document-specific topic compositions in parallel. Experiments show that PADD successfully leverages topic correlations as a prior, notably outperforming TLI and learning quality topic compositions comparable to Gibbs sampling on various data.

Kun Dong, David Eriksson, Hannes Nickisch et al.

For applications as varied as Bayesian neural networks, determinantal point processes, elliptical graphical models, and kernel learning for Gaussian processes (GPs), one must compute a log determinant of an $n \times n$ positive definite matrix, and its derivatives - leading to prohibitive $\mathcal{O}(n^3)$ computations. We propose novel $\mathcal{O}(n)$ approaches to estimating these quantities from only fast matrix vector multiplications (MVMs). These stochastic approximations are based on Chebyshev, Lanczos, and surrogate models, and converge quickly even for kernel matrices that have challenging spectra. We leverage these approximations to develop a scalable Gaussian process approach to kernel learning. We find that Lanczos is generally superior to Chebyshev for kernel learning, and that a surrogate approach can be highly efficient and accurate with popular kernels.

Moontae Lee, David Bindel, David Mimno

Spectral inference provides fast algorithms and provable optimality for latent topic analysis. But for real data these algorithms require additional ad-hoc heuristics, and even then often produce unusable results. We explain this poor performance by casting the problem of topic inference in the framework of Joint Stochastic Matrix Factorization (JSMF) and showing that previous methods violate the theoretical conditions necessary for a good solution to exist. We then propose a novel rectification method that learns high quality topics and their interactions even on small, noisy data. This method achieves results comparable to probabilistic techniques in several domains while maintaining scalability and provable optimality.

Colin Ponce, David Bindel

In this paper, we present a Fingerprint Linear Estimation Routine (FLiER) to identify topology changes in power networks using readings from sparsely-deployed phasor measurement units (PMUs). When a power line, load, or generator trips in a network, or when a substation is reconfigured, the event leaves a unique "voltage fingerprint" of bus voltage changes that we can identify using only the portion of the network directly observed by the PMUs. The naive brute-force approach to identify a failed line from such voltage fingerprints, though simple and accurate, is slow. We derive an approximate algorithm based on a local linearization and a novel filtering approach that is faster and only slightly less accurate. We present experimental results using the IEEE 57-bus, IEEE 118-bus, and Polish 1999-2000 winter peak networks.

David Bindel, Maciej Zworski

We consider one dimensional scattering and show how the presence of a mild positive barrier separating the interaction region from infinity implies that the bound and antibound states are symmetric modulo exponentially small errors in 1/h. This simple result was inspired by a numerical experiment and we describe the numerical scheme for an efficient computation of resonances in one dimension.