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Daniel Roy

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9 papers

1 author row

NeurIPS Conference 2019 Conference Paper

Fast-rate PAC-Bayes Generalization Bounds via Shifted Rademacher Processes

Jun Yang
Shengyang Sun
Daniel Roy

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecué and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates. Our analysis establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields new insights on PAC-Bayesian theory.

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NeurIPS Conference 2019 Conference Paper

Information-Theoretic Generalization Bounds for SGLD via Data-Dependent Estimates

Jeffrey Negrea
Mahdi Haghifam
Gintare Karolina Dziugaite
Ashish Khisti
Daniel Roy

In this work, we improve upon the stepwise analysis of noisy iterative learning algorithms initiated by Pensia, Jog, and Loh (2018) and recently extended by Bu, Zou, and Veeravalli (2019). Our main contributions are significantly improved mutual information bounds for Stochastic Gradient Langevin Dynamics via data-dependent estimates. Our approach is based on the variational characterization of mutual information and the use of data-dependent priors that forecast the mini-batch gradient based on a subset of the training samples. Our approach is broadly applicable within the information-theoretic framework of Russo and Zou (2015) and Xu and Raginsky (2017). Our bound can be tied to a measure of flatness of the empirical risk surface. As compared with other bounds that depend on the squared norms of gradients, empirical investigations show that the terms in our bounds are orders of magnitude smaller.

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NeurIPS Conference 2018 Conference Paper

Data-dependent PAC-Bayes priors via differential privacy

Gintare Karolina Dziugaite
Daniel Roy

The Probably Approximately Correct (PAC) Bayes framework (McAllester, 1999) can incorporate knowledge about the learning algorithm and (data) distribution through the use of distribution-dependent priors, yielding tighter generalization bounds on data-dependent posteriors. Using this flexibility, however, is difficult, especially when the data distribution is presumed to be unknown. We show how a differentially private data-dependent prior yields a valid PAC-Bayes bound, and then show how non-private mechanisms for choosing priors can also yield generalization bounds. As an application of this result, we show that a Gaussian prior mean chosen via stochastic gradient Langevin dynamics (SGLD; Welling and Teh, 2011) leads to a valid PAC-Bayes bound due to control of the 2-Wasserstein distance to a differentially private stationary distribution. We study our data-dependent bounds empirically, and show that they can be nonvacuous even when other distribution-dependent bounds are vacuous.

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NeurIPS Conference 2016 Conference Paper

Measuring the reliability of MCMC inference with bidirectional Monte Carlo

Roger Grosse
Siddharth Ancha
Daniel Roy

Markov chain Monte Carlo (MCMC) is one of the main workhorses of probabilistic inference, but it is notoriously hard to measure the quality of approximate posterior samples. This challenge is particularly salient in black box inference methods, which can hide details and obscure inference failures. In this work, we extend the recently introduced bidirectional Monte Carlo technique to evaluate MCMC-based posterior inference algorithms. By running annealed importance sampling (AIS) chains both from prior to posterior and vice versa on simulated data, we upper bound in expectation the symmetrized KL divergence between the true posterior distribution and the distribution of approximate samples. We integrate our method into two probabilistic programming languages, WebPPL and Stan, and validate it on several models and datasets. As an example of how our method be used to guide the design of inference algorithms, we apply it to study the effectiveness of different model representations in WebPPL and Stan.

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NeurIPS Conference 2014 Conference Paper

Mondrian Forests: Efficient Online Random Forests

Balaji Lakshminarayanan
Daniel Roy
Yee Whye Teh

Ensembles of randomized decision trees, usually referred to as random forests, are widely used for classification and regression tasks in machine learning and statistics. Random forests achieve competitive predictive performance and are computationally efficient to train and test, making them excellent candidates for real-world prediction tasks. The most popular random forest variants (such as Breiman's random forest and extremely randomized trees) operate on batches of training data. Online methods are now in greater demand. Existing online random forests, however, require more training data than their batch counterpart to achieve comparable predictive performance. In this work, we use Mondrian processes (Roy and Teh, 2009) to construct ensembles of random decision trees we call Mondrian forests. Mondrian forests can be grown in an incremental/online fashion and remarkably, the distribution of online Mondrian forests is the same as that of batch Mondrian forests. Mondrian forests achieve competitive predictive performance comparable with existing online random forests and periodically re-trained batch random forests, while being more than an order of magnitude faster, thus representing a better computation vs accuracy tradeoff.

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NeurIPS Conference 2012 Conference Paper

Random function priors for exchangeable arrays with applications to graphs and relational data

James Lloyd
Peter Orbanz
Zoubin Ghahramani
Daniel Roy

A fundamental problem in the analysis of structured relational data like graphs, networks, databases, and matrices is to extract a summary of the common struc- ture underlying relations between individual entities. Relational data are typically encoded in the form of arrays; invariance to the ordering of rows and columns corresponds to exchangeable arrays. Results in probability theory due to Aldous, Hoover and Kallenberg show that exchangeable arrays can be represented in terms of a random measurable function which constitutes the natural model parameter in a Bayesian model. We obtain a ﬂexible yet simple Bayesian nonparametric model by placing a Gaussian process prior on the parameter function. Efﬁcient inference utilises elliptical slice sampling combined with a random sparse approximation to the Gaussian process. We demonstrate applications of the model to network data and clarify its relation to models in the literature, several of which emerge as special cases.

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NeurIPS Conference 2008 Conference Paper

The Mondrian Process

Daniel Roy
Yee Teh

We describe a novel stochastic process that can be used to construct a multidimensional generalization of the stick-breaking process and which is related to the classic stick breaking process described by Sethuraman1994 in one dimension. We describe how the process can be applied to relational data modeling using the de Finetti representation for infinitely and partially exchangeable arrays.

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NeurIPS Conference 2007 Conference Paper

Bayesian Agglomerative Clustering with Coalescents

Yee Teh
Hal Daume III
Daniel Roy

We introduce a new Bayesian model for hierarchical clustering based on a prior over trees called Kingman’s coalescent. We develop novel greedy and sequential Monte Carlo inferences which operate in a bottom-up agglomerative fashion. We show experimentally the superiority of our algorithms over the state-of-the-art, and demonstrate our approach in document clustering and phylolinguistics.

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NeurIPS Conference 2006 Conference Paper

Learning annotated hierarchies from relational data

Daniel Roy
Charles Kemp
Vikash Mansinghka
Joshua Tenenbaum

The objects in many real-world domains can be organized into hierarchies, where each internal node picks out a category of objects. Given a collection of fea- tures and relations deﬁned over a set of objects, an annotated hierarchy includes a speciﬁcation of the categories that are most useful for describing each individual feature and relation. We deﬁne a generative model for annotated hierarchies and the features and relations that they describe, and develop a Markov chain Monte Carlo scheme for learning annotated hierarchies. We show that our model discov- ers interpretable structure in several real-world data sets.

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