arXiv subjects

Alexander Terenin

Publications and source records attributed to Alexander Terenin.

36 records · Page 2Linked to original sources

arXiv2023

Geometry-aware Bayesian Optimization in Robotics using Riemannian Matérn Kernels

Bayesian optimization is a data-efficient technique which can be used for control parameter tuning, parametric policy adaptation, and structure design in robotics. Many of these problems require optimization of functions defined on non-Euclidean domains like spheres, rotation groups, or spaces of positive-definite matrices. To do so, one must place a Gaussian process prior, or equivalently define a kernel, on the space of interest. Effective kernels typically reflect the geometry of the spaces they are defined on, but designing them is generally non-trivial. Recent work on the Riemannian Matérn kernels, based on stochastic partial differential equations and spectral theory of the Laplace-Beltrami operator, offers promising avenues towards constructing such geometry-aware kernels. In this paper, we study techniques for implementing these kernels on manifolds of interest in robotics, demonstrate their performance on a set of artificial benchmark functions, and illustrate geometry-aware Bayesian optimization for a variety of robotic applications, covering orientation control, manipulability optimization, and motion planning, while showing its improved performance.

cs.RO↗

arXiv2022

Learning Contact Dynamics using Physically Structured Neural Networks

Learning physically structured representations of dynamical systems that include contact between different objects is an important problem for learning-based approaches in robotics. Black-box neural networks can learn to approximately represent discontinuous dynamics, but they typically require large quantities of data and often suffer from pathological behaviour when forecasting for longer time horizons. In this work, we use connections between deep neural networks and differential equations to design a family of deep network architectures for representing contact dynamics between objects. We show that these networks can learn discontinuous contact events in a data-efficient manner from noisy observations in settings that are traditionally difficult for black-box approaches and recent physics inspired neural networks. Our results indicate that an idealised form of touch feedback -- which is heavily relied upon by biological systems -- is a key component of making this learning problem tractable. Together with the inductive biases introduced through the network architectures, our techniques enable accurate learning of contact dynamics from observations.

cs.LG↗

arXiv2022

Gaussian Processes and Statistical Decision-making in Non-Euclidean Spaces

Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for broadening the applicability of Gaussian processes. This is done in two ways. Firstly, we develop pathwise conditioning techniques for Gaussian processes, which allow one to express posterior random functions as prior random functions plus a dependent update term. We introduce a wide class of efficient approximations built from this viewpoint, which can be randomly sampled once in advance, and evaluated at arbitrary locations without any subsequent stochasticity. This key property improves efficiency and makes it simpler to deploy Gaussian process models in decision-making settings. Secondly, we develop a collection of Gaussian process models over non-Euclidean spaces, including Riemannian manifolds and graphs. We derive fully constructive expressions for the covariance kernels of scalar-valued Gaussian processes on Riemannian manifolds and graphs. Building on these ideas, we describe a formalism for defining vector-valued Gaussian processes on Riemannian manifolds. The introduced techniques allow all of these models to be trained using standard computational methods. In total, these contributions make Gaussian processes easier to work with and allow them to be used within a wider class of domains in an effective and principled manner. This, in turn, makes it possible to potentially apply Gaussian processes to novel decision-making settings.

stat.ML↗

arXiv2021

Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent with geometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners.

stat.ML↗

arXiv2021

Sliced Multi-Marginal Optimal Transport

Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability in the number of measures, samples and dimensionality. In this work, we propose a multi-marginal optimal transport paradigm based on random one-dimensional projections, whose (generalized) distance we term the sliced multi-marginal Wasserstein distance. To construct this distance, we introduce a characterization of the one-dimensional multi-marginal Kantorovich problem and use it to highlight a number of properties of the sliced multi-marginal Wasserstein distance. In particular, we show that (i) the sliced multi-marginal Wasserstein distance is a (generalized) metric that induces the same topology as the standard Wasserstein distance, (ii) it admits a dimension-free sample complexity, (iii) it is tightly connected with the problem of barycentric averaging under the sliced-Wasserstein metric. We conclude by illustrating the sliced multi-marginal Wasserstein on multi-task density estimation and multi-dynamics reinforcement learning problems.

stat.ML↗

arXiv2021

Pathwise Conditioning of Gaussian Processes

As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via sampling. Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at finite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. These methods are prohibitively expensive in cases where we would, ideally, like to draw high-dimensional vectors or even continuous sample paths. In this work, we investigate a different line of reasoning: rather than focusing on distributions, we articulate Gaussian conditionals at the level of random variables. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to efficiently sampling Gaussian process posteriors. Starting from first principles, we derive these methods and analyze the approximation errors they introduce. We, then, ground these results by exploring the practical implications of pathwise conditioning in various applied settings, such as global optimization and reinforcement learning.

stat.ML↗

arXiv2021

Aligning Time Series on Incomparable Spaces

Dynamic time warping (DTW) is a useful method for aligning, comparing and combining time series, but it requires them to live in comparable spaces. In this work, we consider a setting in which time series live on different spaces without a sensible ground metric, causing DTW to become ill-defined. To alleviate this, we propose Gromov dynamic time warping (GDTW), a distance between time series on potentially incomparable spaces that avoids the comparability requirement by instead considering intra-relational geometry. We demonstrate its effectiveness at aligning, combining and comparing time series living on incomparable spaces. We further propose a smoothed version of GDTW as a differentiable loss and assess its properties in a variety of settings, including barycentric averaging, generative modeling and imitation learning.

cs.LG↗

arXiv2020

Pólya Urn Latent Dirichlet Allocation: a doubly sparse massively parallel sampler

Latent Dirichlet Allocation (LDA) is a topic model widely used in natural language processing and machine learning. Most approaches to training the model rely on iterative algorithms, which makes it difficult to run LDA on big corpora that are best analyzed in parallel and distributed computational environments. Indeed, current approaches to parallel inference either don't converge to the correct posterior or require storage of large dense matrices in memory. We present a novel sampler that overcomes both problems, and we show that this sampler is faster, both empirically and theoretically, than previous Gibbs samplers for LDA. We do so by employing a novel Pólya-urn-based approximation in the sparse partially collapsed sampler for LDA. We prove that the approximation error vanishes with data size, making our algorithm asymptotically exact, a property of importance for large-scale topic models. In addition, we show, via an explicit example, that - contrary to popular belief in the topic modeling literature - partially collapsed samplers can be more efficient than fully collapsed samplers. We conclude by comparing the performance of our algorithm with that of other approaches on well-known corpora.

stat.ML↗

arXiv2020

Sparse Parallel Training of Hierarchical Dirichlet Process Topic Models

To scale non-parametric extensions of probabilistic topic models such as Latent Dirichlet allocation to larger data sets, practitioners rely increasingly on parallel and distributed systems. In this work, we study data-parallel training for the hierarchical Dirichlet process (HDP) topic model. Based upon a representation of certain conditional distributions within an HDP, we propose a doubly sparse data-parallel sampler for the HDP topic model. This sampler utilizes all available sources of sparsity found in natural language - an important way to make computation efficient. We benchmark our method on a well-known corpus (PubMed) with 8m documents and 768m tokens, using a single multi-core machine in under four days.

stat.ML↗

arXiv2020

Efficiently Sampling Functions from Gaussian Process Posteriors

Gaussian processes are the gold standard for many real-world modeling problems, especially in cases where a model's success hinges upon its ability to faithfully represent predictive uncertainty. These problems typically exist as parts of larger frameworks, wherein quantities of interest are ultimately defined by integrating over posterior distributions. These quantities are frequently intractable, motivating the use of Monte Carlo methods. Despite substantial progress in scaling up Gaussian processes to large training sets, methods for accurately generating draws from their posterior distributions still scale cubically in the number of test locations. We identify a decomposition of Gaussian processes that naturally lends itself to scalable sampling by separating out the prior from the data. Building off of this factorization, we propose an easy-to-use and general-purpose approach for fast posterior sampling, which seamlessly pairs with sparse approximations to afford scalability both during training and at test time. In a series of experiments designed to test competing sampling schemes' statistical properties and practical ramifications, we demonstrate how decoupled sample paths accurately represent Gaussian process posteriors at a fraction of the usual cost.

stat.ML↗

arXiv2020

Variational Integrator Networks for Physically Structured Embeddings

Learning workable representations of dynamical systems is becoming an increasingly important problem in a number of application areas. By leveraging recent work connecting deep neural networks to systems of differential equations, we propose \emph{variational integrator networks}, a class of neural network architectures designed to preserve the geometric structure of physical systems. This class of network architectures facilitates accurate long-term prediction, interpretability, and data-efficient learning, while still remaining highly flexible and capable of modeling complex behavior. We demonstrate that they can accurately learn dynamical systems from both noisy observations in phase space and from image pixels within which the unknown dynamics are embedded.

stat.ML↗

arXiv2020

Asynchronous Gibbs Sampling

Gibbs sampling is a Markov Chain Monte Carlo (MCMC) method often used in Bayesian learning. MCMC methods can be difficult to deploy on parallel and distributed systems due to their inherently sequential nature. We study asynchronous Gibbs sampling, which achieves parallelism by simply ignoring sequential requirements. This method has been shown to produce good empirical results for some hierarchical models, and is popular in the topic modeling community, but was also shown to diverge for other targets. We introduce a theoretical framework for analyzing asynchronous Gibbs sampling and other extensions of MCMC that do not possess the Markov property. We prove that asynchronous Gibbs can be modified so that it converges under appropriate regularity conditions -- we call this the exact asynchronous Gibbs algorithm. We study asynchronous Gibbs on a set of examples by comparing the exact and approximate algorithms, including two where it works well, and one where it fails dramatically. We conclude with a set of heuristics to describe settings where the algorithm can be effectively used.

stat.CO↗

arXiv2020

Cox's Theorem and the Jaynesian Interpretation of Probability

There are multiple proposed interpretations of probability theory: one such interpretation is true-false logic under uncertainty. Cox's Theorem is a representation theorem that states, under a certain set of axioms describing the meaning of uncertainty, that every true-false logic under uncertainty is isomorphic to conditional probability theory. This result was used by Jaynes to develop a philosophical framework in which statistical inference under uncertainty should be conducted through the use of probability, via Bayes' Rule. Unfortunately, most existing correct proofs of Cox's Theorem require restrictive assumptions: for instance, many do not apply even to the simple example of rolling a pair of fair dice. We offer a new axiomatization by replacing various technical conditions with an axiom stating that our theory must be consistent with respect to repeated events. We discuss the implications of our results, both for the philosophy of probability and for the philosophy of statistics.

math.ST↗

arXiv2018

A Piecewise Deterministic Markov Process via $(r,θ)$ swaps in hyperspherical coordinates

Recently, a class of stochastic processes known as piecewise deterministic Markov processes has been used to define continuous-time Markov chain Monte Carlo algorithms with a number of attractive properties, including compatibility with stochastic gradients like those typically found in optimization and variational inference, and high efficiency on certain big data problems. Not many processes in this class that are capable of targeting arbitrary invariant distributions are currently known, and within one subclass all previously known processes utilize linear transition functions. In this work, we derive a process whose transition function is nonlinear through solving its Fokker-Planck equation in hyperspherical coordinates. We explore its behavior on Gaussian targets, as well as a Bayesian logistic regression model with synthetic data. We discuss implications to both the theory of piecewise deterministic Markov processes, and to Bayesian statisticians as well as physicists seeking to use them for simulation-based computation.

stat.CO↗

arXiv2018

Techniques for proving Asynchronous Convergence results for Markov Chain Monte Carlo methods

Markov Chain Monte Carlo (MCMC) methods such as Gibbs sampling are finding widespread use in applied statistics and machine learning. These often lead to difficult computational problems, which are increasingly being solved on parallel and distributed systems such as compute clusters. Recent work has proposed running iterative algorithms such as gradient descent and MCMC in parallel asynchronously for increased performance, with good empirical results in certain problems. Unfortunately, for MCMC this parallelization technique requires new convergence theory, as it has been explicitly demonstrated to lead to divergence on some examples. Recent theory on Asynchronous Gibbs sampling describes why these algorithms can fail, and provides a way to alter them to make them converge. In this article, we describe how to apply this theory in a generic setting, to understand the asynchronous behavior of any MCMC algorithm, including those implemented using parameter servers, and those not based on Gibbs sampling.

stat.ML↗

arXiv2018

GPU-accelerated Gibbs sampling: a case study of the Horseshoe Probit model

Gibbs sampling is a widely used Markov chain Monte Carlo (MCMC) method for numerically approximating integrals of interest in Bayesian statistics and other mathematical sciences. Many implementations of MCMC methods do not extend easily to parallel computing environments, as their inherently sequential nature incurs a large synchronization cost. In the case study illustrated by this paper, we show how to do Gibbs sampling in a fully data-parallel manner on a graphics processing unit, for a large class of exchangeable models that admit latent variable representations. Our approach takes a systems perspective, with emphasis placed on efficient use of compute hardware. We demonstrate our method on a Horseshoe Probit regression model and find that our implementation scales effectively to thousands of predictors and millions of data points simultaneously.

stat.CO↗

arXiv2017

Comment: A brief survey of the current state of play for Bayesian computation in data science at Big-Data scale

We wish to contribute to the discussion of "Comparing Consensus Monte Carlo Strategies for Distributed Bayesian Computation" by offering our views on the current best methods for Bayesian computation, both at big-data scale and with smaller data sets, as summarized in Table 1. This table is certainly an over-simplification of a highly complicated area of research in constant (present and likely future) flux, but we believe that constructing summaries of this type is worthwhile despite their drawbacks, if only to facilitate further discussion.

stat.CO↗

arXiv2017

A Noninformative Prior on a Space of Distribution Functions

In a given problem, the Bayesian statistical paradigm requires the specification of a prior distribution that quantifies relevant information about the unknowns of main interest external to the data. In cases where little such information is available, the problem under study may possess an invariance under a transformation group that encodes a lack of information, leading to a unique prior---this idea was explored at length by E.T. Jaynes. Previous successful examples have included location-scale invariance under linear transformation, multiplicative invariance of the rate at which events in a counting process are observed, and the derivation of the Haldane prior for a Bernoulli success probability. In this paper we show that this method can be extended, by generalizing Jaynes, in two ways: (1) to yield families of approximately invariant priors, and (2) to the infinite-dimensional setting, yielding families of priors on spaces of distribution functions. Our results can be used to describe conditions under which a particular Dirichlet Process posterior arises from an optimal Bayesian analysis, in the sense that invariances in the prior and likelihood lead to one and only one posterior distribution.

math.ST↗