Jekyll2022-09-23T11:31:14+00:00https://avt.im/feed.xmlAlexander TereninAlexander TereninPathwise Conditioning and Non-Euclidean Gaussian Processes2022-07-14T00:00:00+00:002022-07-14T00:00:00+00:00https://avt.im/talks/2022/07/14/Pathwise-Conditioning<p>In Gaussian processes, conditioning and computation of posterior distributions is usually done in a distributional fashion by working with finite-dimensional marginals. However, there is another way to think about conditioning: using actual random functions rather than their probability distributions. This perspective is particularly helpful in decision-theoretic settings such as Bayesian optimization, where it enables efficient computation of a wider class of acquisition functions than otherwise possible. In this talk, we describe these recent advances, and discuss their broader implications to Bayesian modeling.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninIn Gaussian processes, conditioning and computation of posterior distributions is usually done in a distributional fashion by working with finite-dimensional marginals. However, there is another way to think about conditioning: using actual random functions rather than their probability distributions. This perspective is particularly helpful in decision-theoretic settings such as Bayesian optimization, where it enables efficient computation of a wider class of acquisition functions than otherwise possible. In this talk, we describe these recent advances, and discuss their broader implications to Bayesian modeling.Pathwise Conditioning and Non-Euclidean Gaussian Processes2022-07-13T00:00:00+00:002022-07-13T00:00:00+00:00https://avt.im/talks/2022/07/13/Pathwise-Conditioning<p>In Gaussian processes, conditioning and computation of posterior distributions is usually done in a distributional fashion by working with finite-dimensional marginals. However, there is another way to think about conditioning: using actual random functions rather than their probability distributions. This perspective is particularly helpful in decision-theoretic settings such as Bayesian optimization, where it enables efficient computation of a wider class of acquisition functions than otherwise possible. In this talk, we describe these recent advances, and discuss their broader implications to Bayesian modeling.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninIn Gaussian processes, conditioning and computation of posterior distributions is usually done in a distributional fashion by working with finite-dimensional marginals. However, there is another way to think about conditioning: using actual random functions rather than their probability distributions. This perspective is particularly helpful in decision-theoretic settings such as Bayesian optimization, where it enables efficient computation of a wider class of acquisition functions than otherwise possible. In this talk, we describe these recent advances, and discuss their broader implications to Bayesian modeling.Pathwise Conditioning and Non-Euclidean Gaussian Processes2022-07-01T00:00:00+00:002022-07-01T00:00:00+00:00https://avt.im/talks/2022/07/01/Pathwise-Conditioning<p>In Gaussian processes, conditioning and computation of posterior distributions is usually done in a distributional fashion by working with finite-dimensional marginals. However, there is another way to think about conditioning: using actual random functions rather than their probability distributions. This perspective is particularly helpful in decision-theoretic settings such as Bayesian optimization, where it enables efficient computation of a wider class of acquisition functions than otherwise possible. In this talk, we describe these recent advances, and discuss their broader implications to Bayesian modeling.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninIn Gaussian processes, conditioning and computation of posterior distributions is usually done in a distributional fashion by working with finite-dimensional marginals. However, there is another way to think about conditioning: using actual random functions rather than their probability distributions. This perspective is particularly helpful in decision-theoretic settings such as Bayesian optimization, where it enables efficient computation of a wider class of acquisition functions than otherwise possible. In this talk, we describe these recent advances, and discuss their broader implications to Bayesian modeling.Machines Regret Their Actions Too: A Brief Tutorial on Multi-armed Bandits2022-06-10T00:00:00+00:002022-06-10T00:00:00+00:00https://avt.im/talks/2022/06/10/Bandits-Tutorial<p>Multi-armed bandits are a class of sequential decision problems which include uncertainty. One of their defining characteristics is the presence of explore-exploit tradeoffs, which require one to balance taking advantage of information that is known with trying different options in order to learn more information in order to make optimal decisions. In this tutorial, we introduce the problem setting and basic techniques of analysis. We conclude by discussing how explore-exploit tradeoffs appear in more general settings, and how the ideas presented can aid in understanding of areas like reinforcement learning.</p>Alexander TereninMulti-armed bandits are a class of sequential decision problems which include uncertainty. One of their defining characteristics is the presence of explore-exploit tradeoffs, which require one to balance taking advantage of information that is known with trying different options in order to learn more information in order to make optimal decisions. In this tutorial, we introduce the problem setting and basic techniques of analysis. We conclude by discussing how explore-exploit tradeoffs appear in more general settings, and how the ideas presented can aid in understanding of areas like reinforcement learning.Non-Euclidean Matérn Gaussian Processes2022-06-08T00:00:00+00:002022-06-08T00:00:00+00:00https://avt.im/talks/2022/06/08/Non-Euclidean-Matern-GP<p>In recent years, the machine learning community has become increasingly interested in learning in settings where data lives in non-Euclidean spaces, for instance in applications to physics and engineering, or other settings where it is important that symmetries are enforced.
In this talk, we will develop a class of Gaussian process models defined on Riemannian manifolds and graphs, and show how to effectively perform all computations needed to train these models using standard automatic-differentiation-based methods.
This gives an effective framework to deploy data-efficient interactive decision-making systems such as Bayesian optimization to settings with symmetries and invariances.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninIn recent years, the machine learning community has become increasingly interested in learning in settings where data lives in non-Euclidean spaces, for instance in applications to physics and engineering, or other settings where it is important that symmetries are enforced. In this talk, we will develop a class of Gaussian process models defined on Riemannian manifolds and graphs, and show how to effectively perform all computations needed to train these models using standard automatic-differentiation-based methods. This gives an effective framework to deploy data-efficient interactive decision-making systems such as Bayesian optimization to settings with symmetries and invariances.Non-Euclidean Matérn Gaussian Processes2022-05-27T00:00:00+00:002022-05-27T00:00:00+00:00https://avt.im/talks/2022/05/27/Non-Euclidean-Matern-GP<p>In recent years, the machine learning community has become increasingly interested in learning in settings where data lives in non-Euclidean spaces, for instance in applications to physics and engineering, or other settings where it is important that symmetries are enforced.
In this talk, we will develop a class of Gaussian process models defined on Riemannian manifolds and graphs, and show how to effectively perform all computations needed to train these models using standard automatic-differentiation-based methods.
This gives an effective framework to deploy data-efficient interactive decision-making systems such as Bayesian optimization to settings with symmetries and invariances.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninIn recent years, the machine learning community has become increasingly interested in learning in settings where data lives in non-Euclidean spaces, for instance in applications to physics and engineering, or other settings where it is important that symmetries are enforced. In this talk, we will develop a class of Gaussian process models defined on Riemannian manifolds and graphs, and show how to effectively perform all computations needed to train these models using standard automatic-differentiation-based methods. This gives an effective framework to deploy data-efficient interactive decision-making systems such as Bayesian optimization to settings with symmetries and invariances.Physically Structured Neural Networks for Smooth and Contact Dynamics2022-05-25T00:00:00+00:002022-05-25T00:00:00+00:00https://avt.im/talks/2022/05/25/Physically-Structured-Networks<p>A neural network’s architecture encodes key information and inductive biases that are used to guide its predictions. In this talk, we discuss recent work which leverages the perspective of neural ordinary differential equations to design network architectures that encode the structures of classical mechanics. We examine the cases of both smooth dynamics and non-smooth contact dynamics. The architectures obtained are easy to understand, show excellent performance and data-efficiency on simple benchmark tasks, and are a promising emerging tool for use in robot learning and related areas.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninA neural network’s architecture encodes key information and inductive biases that are used to guide its predictions. In this talk, we discuss recent work which leverages the perspective of neural ordinary differential equations to design network architectures that encode the structures of classical mechanics. We examine the cases of both smooth dynamics and non-smooth contact dynamics. The architectures obtained are easy to understand, show excellent performance and data-efficiency on simple benchmark tasks, and are a promising emerging tool for use in robot learning and related areas.Non-Euclidean Matérn Gaussian Processes2022-05-06T00:00:00+00:002022-05-06T00:00:00+00:00https://avt.im/talks/2022/05/06/Non-Euclidean-Matern-GP<p>In recent years, the machine learning community has become increasingly interested in learning in settings where data lives in non-Euclidean spaces, for instance in applications to physics and engineering, or other settings where it is important that symmetries are enforced.
In this talk, we will develop a class of Gaussian process models defined on Riemannian manifolds and graphs, and show how to effectively perform all computations needed to train these models using standard automatic-differentiation-based methods.
This gives an effective framework to deploy data-efficient interactive decision-making systems such as Bayesian optimization to settings with symmetries and invariances.</p>
<p>Alexander Terenin is a Postdoctoral Research Associate at the University of Cambridge. He is interested in statistical machine learning, particularly in settings where the data is not fixed, but is gathered interactively by the learning machine. This leads naturally to Gaussian processes and data-efficient interactive decision-making systems such as Bayesian optimization, to areas such as multi-armed bandits and reinforcement learning, and to techniques for incorporating inductive biases and prior information such as symmetries into machine learning models.</p>Alexander TereninIn recent years, the machine learning community has become increasingly interested in learning in settings where data lives in non-Euclidean spaces, for instance in applications to physics and engineering, or other settings where it is important that symmetries are enforced. In this talk, we will develop a class of Gaussian process models defined on Riemannian manifolds and graphs, and show how to effectively perform all computations needed to train these models using standard automatic-differentiation-based methods. This gives an effective framework to deploy data-efficient interactive decision-making systems such as Bayesian optimization to settings with symmetries and invariances.Matérn Gaussian Processes on Riemannian Manifolds2022-03-25T00:00:00+00:002022-03-25T00:00:00+00:00https://avt.im/talks/2022/03/25/Riemannian-Matern-GP<p>Gaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Matérn class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace–Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Matérn to the widely-used squared exponential Gaussian process. By allowing Riemannian Matérn Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.</p>Alexander TereninGaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Matérn class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace–Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Matérn to the widely-used squared exponential Gaussian process. By allowing Riemannian Matérn Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.Gaussian Processes and Statistical Decision-making in Non-Euclidean spaces2022-02-21T00:00:00+00:002022-02-21T00:00:00+00:00https://avt.im/publications/2022/02/21/PhD-Thesis<p>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.</p>Alexander TereninBayesian 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.