arXiv subjects

Balaji Lakshminarayanan

Publications and source records attributed to Balaji Lakshminarayanan.

61 records · Page 4Linked to original sources

arXiv2016

Mondrian Forests for Large-Scale Regression when Uncertainty Matters

Many real-world regression problems demand a measure of the uncertainty associated with each prediction. Standard decision forests deliver efficient state-of-the-art predictive performance, but high-quality uncertainty estimates are lacking. Gaussian processes (GPs) deliver uncertainty estimates, but scaling GPs to large-scale data sets comes at the cost of approximating the uncertainty estimates. We extend Mondrian forests, first proposed by Lakshminarayanan et al. (2014) for classification problems, to the large-scale non-parametric regression setting. Using a novel hierarchical Gaussian prior that dovetails with the Mondrian forest framework, we obtain principled uncertainty estimates, while still retaining the computational advantages of decision forests. Through a combination of illustrative examples, real-world large-scale datasets, and Bayesian optimization benchmarks, we demonstrate that Mondrian forests outperform approximate GPs on large-scale regression tasks and deliver better-calibrated uncertainty assessments than decision-forest-based methods.

stat.ML↗

arXiv2015

Approximate Inference with the Variational Holder Bound

We introduce the Variational Holder (VH) bound as an alternative to Variational Bayes (VB) for approximate Bayesian inference. Unlike VB which typically involves maximization of a non-convex lower bound with respect to the variational parameters, the VH bound involves minimization of a convex upper bound to the intractable integral with respect to the variational parameters. Minimization of the VH bound is a convex optimization problem; hence the VH method can be applied using off-the-shelf convex optimization algorithms and the approximation error of the VH bound can also be analyzed using tools from convex optimization literature. We present experiments on the task of integrating a truncated multivariate Gaussian distribution and compare our method to VB, EP and a state-of-the-art numerical integration method for this problem.

stat.ML↗

arXiv2015

Kernel-Based Just-In-Time Learning for Passing Expectation Propagation Messages

We propose an efficient nonparametric strategy for learning a message operator in expectation propagation (EP), which takes as input the set of incoming messages to a factor node, and produces an outgoing message as output. This learned operator replaces the multivariate integral required in classical EP, which may not have an analytic expression. We use kernel-based regression, which is trained on a set of probability distributions representing the incoming messages, and the associated outgoing messages. The kernel approach has two main advantages: first, it is fast, as it is implemented using a novel two-layer random feature representation of the input message distributions; second, it has principled uncertainty estimates, and can be cheaply updated online, meaning it can request and incorporate new training data when it encounters inputs on which it is uncertain. In experiments, our approach is able to solve learning problems where a single message operator is required for multiple, substantially different data sets (logistic regression for a variety of classification problems), where it is essential to accurately assess uncertainty and to efficiently and robustly update the message operator.

stat.ML↗

arXiv2015

Mondrian Forests: Efficient Online Random Forests

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.

stat.ML↗

arXiv2015

Particle Gibbs for Bayesian Additive Regression Trees

Additive regression trees are flexible non-parametric models and popular off-the-shelf tools for real-world non-linear regression. In application domains, such as bioinformatics, where there is also demand for probabilistic predictions with measures of uncertainty, the Bayesian additive regression trees (BART) model, introduced by Chipman et al. (2010), is increasingly popular. As data sets have grown in size, however, the standard Metropolis-Hastings algorithms used to perform inference in BART are proving inadequate. In particular, these Markov chains make local changes to the trees and suffer from slow mixing when the data are high-dimensional or the best fitting trees are more than a few layers deep. We present a novel sampler for BART based on the Particle Gibbs (PG) algorithm (Andrieu et al., 2010) and a top-down particle filtering algorithm for Bayesian decision trees (Lakshminarayanan et al., 2013). Rather than making local changes to individual trees, the PG sampler proposes a complete tree to fit the residual. Experiments show that the PG sampler outperforms existing samplers in many settings.

stat.ML↗

arXiv2013

Top-down particle filtering for Bayesian decision trees

Decision tree learning is a popular approach for classification and regression in machine learning and statistics, and Bayesian formulations---which introduce a prior distribution over decision trees, and formulate learning as posterior inference given data---have been shown to produce competitive performance. Unlike classic decision tree learning algorithms like ID3, C4.5 and CART, which work in a top-down manner, existing Bayesian algorithms produce an approximation to the posterior distribution by evolving a complete tree (or collection thereof) iteratively via local Monte Carlo modifications to the structure of the tree, e.g., using Markov chain Monte Carlo (MCMC). We present a sequential Monte Carlo (SMC) algorithm that instead works in a top-down manner, mimicking the behavior and speed of classic algorithms. We demonstrate empirically that our approach delivers accuracy comparable to the most popular MCMC method, but operates more than an order of magnitude faster, and thus represents a better computation-accuracy tradeoff.

stat.ML↗

arXiv2013

Inferring ground truth from multi-annotator ordinal data: a probabilistic approach

A popular approach for large scale data annotation tasks is crowdsourcing, wherein each data point is labeled by multiple noisy annotators. We consider the problem of inferring ground truth from noisy ordinal labels obtained from multiple annotators of varying and unknown expertise levels. Annotation models for ordinal data have been proposed mostly as extensions of their binary/categorical counterparts and have received little attention in the crowdsourcing literature. We propose a new model for crowdsourced ordinal data that accounts for instance difficulty as well as annotator expertise, and derive a variational Bayesian inference algorithm for parameter estimation. We analyze the ordinal extensions of several state-of-the-art annotator models for binary/categorical labels and evaluate the performance of all the models on two real world datasets containing ordinal query-URL relevance scores, collected through Amazon's Mechanical Turk. Our results indicate that the proposed model performs better or as well as existing state-of-the-art methods and is more resistant to `spammy' annotators (i.e., annotators who assign labels randomly without actually looking at the instance) than popular baselines such as mean, median, and majority vote which do not account for annotator expertise.

stat.ML↗