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Robert D. Nowak

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23 papers

2 author rows

ICML Conference 2025 Conference Paper

Improved Algorithm for Deep Active Learning under Imbalance via Optimal Separation

Shyam Nuggehalli
Jifan Zhang
Lalit K. Jain
Robert D. Nowak

Class imbalance severely impacts machine learning performance on minority classes in real-world applications. While various solutions exist, active learning offers a fundamental fix by strategically collecting balanced, informative labeled examples from abundant unlabeled data. We introduce DIRECT, an algorithm that identifies class separation boundaries and selects the most uncertain nearby examples for annotation. By reducing the problem to one-dimensional active learning, DIRECT leverages established theory to handle batch labeling and label noise – another common challenge in data annotation that particularly affects active learning methods. Our work presents the first comprehensive study of active learning under both class imbalance and label noise. Extensive experiments on imbalanced datasets show DIRECT reduces annotation costs by over 60% compared to state-of-the-art active learning methods and over 80% versus random sampling, while maintaining robustness to label noise.

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UAI Conference 2025 Conference Paper

Metric Learning in an RKHS

Gokcan Tatli
Yi Chen
Blake Mason
Robert D. Nowak
Ramya Korlakai Vinayak

This paper investigates metric learning in a Reproducing Kernel Hilbert Space (RKHS) based on a set of random triplet comparisons in the form of *"Do you think item h is more similar to item i or item j? "* indicating similarity and differences between various items. The goal is to learn a metric in the RKHS that reflects the comparisons. Nonlinear metric learning using kernel methods and neural networks has shown great empirical promise. While previous works have addressed certain aspects of this problem, there is little or no theoretical understanding of such methods. The exception is the special (linear) case in which the RKHS is the standard $d$-dimensional Euclidean space; there is a comprehensive theory for metric learning in the $d$-dimensional Euclidean space. This paper develops a general RKHS framework for metric learning and provides novel generalization guarantees and sample complexity bounds. We validate our findings through a set of simulations and experiments on real datasets. Our code is publicly available at https: //github. com/RamyaLab/metric-learning-RKHS.

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NeurIPS Conference 2024 Conference Paper

A New Neural Kernel Regime: The Inductive Bias of Multi-Task Learning

Julia Nakhleh
Joseph Shenouda
Robert D. Nowak

This paper studies the properties of solutions to multi-task shallow ReLU neural network learning problems, wherein the network is trained to fit a dataset with minimal sum of squared weights. Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods. It is known that single-task neural network learning problems are equivalent to a minimum norm interpolation problem in a non-Hilbertian Banach space, and that the solutions of such problems are generally non-unique. In contrast, we prove that the solutions to univariate-input, multi-task neural network interpolation problems are almost always unique, and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space. We also demonstrate a similar phenomenon in the multivariate-input case; specifically, we show that neural network learning problems with large numbers of tasks are approximately equivalent to an $\ell^2$ (Hilbert space) minimization problem over a fixed kernel determined by the optimal neurons.

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ICLR Conference 2024 Conference Paper

Looped Transformers are Better at Learning Learning Algorithms

Liu Yang 0001
Kangwook Lee 0001
Robert D. Nowak
Dimitris S. Papailiopoulos

Transformers have demonstrated effectiveness in in-context solving data-fitting problems from various (latent) models, as reported by Garg et al. (2022). However, the absence of an inherent iterative structure in the transformer architecture presents a challenge in emulating the iterative algorithms, which are commonly employed in traditional machine learning methods. To address this, we propose the utilization of looped transformer architecture and its associated training methodology, with the aim of incorporating iterative characteristics into the transformer architectures. Experimental results suggest that the looped transformer achieves performance comparable to the standard transformer in solving various data-fitting problems, while utilizing less than 10% of the parameter count.

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ICLR Conference 2024 Conference Paper

On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation

Jeongyeol Kwon
Dohyun Kwon 0002
Stephen Wright
Robert D. Nowak

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter $\sigma > 0$. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be $O(\sigma)$-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as $O(\sigma)$-approximation of the original BO, we propose first-order algorithms that find an $\epsilon$-stationary solution by optimizing the penalty formulation with $\sigma = O(\epsilon)$. When the perturbed lower-level problem uniformly satisfies the {\it small-error} proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an $\epsilon$-stationary point of the penalty function using in total $O(\epsilon^{-7})$ accesses to first-order stochastic gradient oracles. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, {\it i.e.,} with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(\epsilon^{-5})$.

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ICML Conference 2024 Conference Paper

ReLUs Are Sufficient for Learning Implicit Neural Representations

Joseph Shenouda
Yamin Zhou
Robert D. Nowak

Motivated by the growing theoretical understanding of neural networks that employ the Rectified Linear Unit (ReLU) as their activation function, we revisit the use of ReLU activation functions for learning implicit neural representations (INRs). Inspired by second order B-spline wavelets, we incorporate a set of simple constraints to the ReLU neurons in each layer of a deep neural network (DNN) to remedy the spectral bias. This in turn enables its use for various INR tasks. Empirically, we demonstrate that, contrary to popular belief, one can learn state-of-the-art INRs based on a DNN composed of only ReLU neurons. Next, by leveraging recent theoretical works which characterize the kinds of functions ReLU neural networks learn, we provide a way to quantify the regularity of the learned function. This offers a principled approach to selecting the hyperparameters in INR architectures. We substantiate our claims through experiments in signal representation, super resolution, and computed tomography, demonstrating the versatility and effectiveness of our method. The code for all experiments can be found at https: //github. com/joeshenouda/relu-inrs.

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ICML Conference 2024 Conference Paper

SaVeR: Optimal Data Collection Strategy for Safe Policy Evaluation in Tabular MDP

Subhojyoti Mukherjee
Josiah P. Hanna
Robert D. Nowak

In this paper, we study safe data collection for the purpose of policy evaluation in tabular Markov decision processes (MDPs). In policy evaluation, we are given a target policy and asked to estimate the expected cumulative reward it will obtain. Policy evaluation requires data and we are interested in the question of what behavior policy should collect the data for the most accurate evaluation of the target policy. While prior work has considered behavior policy selection, in this paper, we additionally consider a safety constraint on the behavior policy. Namely, we assume there exists a known default policy that incurs a particular expected cost when run and we enforce that the cumulative cost of all behavior policies ran is better than a constant factor of the cost that would be incurred had we always run the default policy. We first show that there exists a class of intractable MDPs where no safe oracle algorithm with knowledge about problem parameters can efficiently collect data and satisfy the safety constraints. We then define the tractability condition for an MDP such that a safe oracle algorithm can efficiently collect data and using that we prove the first lower bound for this setting. We then introduce an algorithm SaVeR for this problem that approximates the safe oracle algorithm and bound the finite-sample mean squared error of the algorithm while ensuring it satisfies the safety constraint. Finally, we show in simulations that SaVeR produces low MSE policy evaluation while satisfying the safety constraint.

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JMLR Journal 2024 Journal Article

Variation Spaces for Multi-Output Neural Networks: Insights on Multi-Task Learning and Network Compression

Joseph Shenouda
Rahul Parhi
Kangwook Lee
Robert D. Nowak

This paper introduces a novel theoretical framework for the analysis of vector-valued neural networks through the development of vector-valued variation spaces, a new class of reproducing kernel Banach spaces. These spaces emerge from studying the regularization effect of weight decay in training networks with activation functions like the rectified linear unit (ReLU). This framework offers a deeper understanding of multi-output networks and their function-space characteristics. A key contribution of this work is the development of a representer theorem for the vector-valued variation spaces. This representer theorem establishes that shallow vector-valued neural networks are the solutions to data-fitting problems over these infinite-dimensional spaces, where the network widths are bounded by the square of the number of training data. This observation reveals that the norm associated with these vector-valued variation spaces encourages the learning of features that are useful for multiple tasks, shedding new light on multi-task learning with neural networks. Finally, this paper develops a connection between weight-decay regularization and the multi-task lasso problem. This connection leads to novel bounds for layer widths in deep networks that depend on the intrinsic dimensions of the training data representations. This insight not only deepens the understanding of the deep network architectural requirements, but also yields a simple convex optimization method for deep neural network compression. The performance of this compression procedure is evaluated on various architectures. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

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ICML Conference 2023 Conference Paper

A Fully First-Order Method for Stochastic Bilevel Optimization

Jeongyeol Kwon
Dohyun Kwon 0002
Stephen Wright
Robert D. Nowak

We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an $\epsilon$-stationary solution of the bilevel problem after $\epsilon^{-7/2}, \epsilon^{-5/2}$, and $\epsilon^{-3/2}$ iterations (each iteration using $O(1)$ samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to $\epsilon^{-5/2}, \epsilon^{-4/2}$, and $\epsilon^{-3/2}$, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

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ICML Conference 2023 Conference Paper

Feed Two Birds with One Scone: Exploiting Wild Data for Both Out-of-Distribution Generalization and Detection

Haoyue Bai 0001
Gregory Canal
Xuefeng Du
Jeongyeol Kwon
Robert D. Nowak
Yixuan Li 0001

Modern machine learning models deployed in the wild can encounter both covariate and semantic shifts, giving rise to the problems of out-of-distribution (OOD) generalization and OOD detection respectively. While both problems have received significant research attention lately, they have been pursued independently. This may not be surprising, since the two tasks have seemingly conflicting goals. This paper provides a new unified approach that is capable of simultaneously generalizing to covariate shifts while robustly detecting semantic shifts. We propose a margin-based learning framework that exploits freely available unlabeled data in the wild that captures the environmental test-time OOD distributions under both covariate and semantic shifts. We show both empirically and theoretically that the proposed margin constraint is the key to achieving both OOD generalization and detection. Extensive experiments show the superiority of our framework, outperforming competitive baselines that specialize in either OOD generalization or OOD detection. Code is publicly available at https: //github. com/deeplearning-wisc/scone.

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ICML Conference 2022 Conference Paper

GALAXY: Graph-based Active Learning at the Extreme

Jifan Zhang
Julian Katz-Samuels
Robert D. Nowak

Active learning is a label-efficient approach to train highly effective models while interactively selecting only small subsets of unlabelled data for labelling and training. In “open world" settings, the classes of interest can make up a small fraction of the overall dataset – most of the data may be viewed as an out-of-distribution or irrelevant class. This leads to extreme class-imbalance, and our theory and methods focus on this core issue. We propose a new strategy for active learning called GALAXY (Graph-based Active Learning At the eXtrEme), which blends ideas from graph-based active learning and deep learning. GALAXY automatically and adaptively selects more class-balanced examples for labeling than most other methods for active learning. Our theory shows that GALAXY performs a refined form of uncertainty sampling that gathers a much more class-balanced dataset than vanilla uncertainty sampling. Experimentally, we demonstrate GALAXY’s superiority over existing state-of-art deep active learning algorithms in unbalanced vision classification settings generated from popular datasets.

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UAI Conference 2022 Conference Paper

ReVar: Strengthening policy evaluation via reduced variance sampling

Subhojyoti Mukherjee
Josiah P. Hanna
Robert D. Nowak

This paper studies the problem of data collection for policy evaluation in Markov decision processes (MDPs). In policy evaluation, we are given a \textit{target} policy and asked to estimate the expected cumulative reward it will obtain in an environment formalized as an MDP. We develop theory for optimal data collection within the class of tree-structured MDPs by first deriving an oracle exploration strategy that uses knowledge of the variance of the reward distributions. We then introduce the \textbf{Re}duced \textbf{Var}iance Sampling (\rev\!) algorithm that approximates the oracle strategy when the reward variances are unknown a priori and bound its sub-optimality compared to the oracle strategy. Finally, we empirically validate that \rev leads to policy evaluation with mean squared error comparable to the oracle strategy and significantly lower than simply running the target policy.

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ICML Conference 2022 Conference Paper

Training OOD Detectors in their Natural Habitats

Julian Katz-Samuels
Julia B. Nakhleh
Robert D. Nowak
Yixuan Li 0001

Out-of-distribution (OOD) detection is important for machine learning models deployed in the wild. Recent methods use auxiliary outlier data to regularize the model for improved OOD detection. However, these approaches make a strong distributional assumption that the auxiliary outlier data is completely separable from the in-distribution (ID) data. In this paper, we propose a novel framework that leverages wild mixture data—that naturally consists of both ID and OOD samples. Such wild data is abundant and arises freely upon deploying a machine learning classifier in their natural habitats. Our key idea is to formulate a constrained optimization problem and to show how to tractably solve it. Our learning objective maximizes the OOD detection rate, subject to constraints on the classification error of ID data and on the OOD error rate of ID examples. We extensively evaluate our approach on common OOD detection tasks and demonstrate superior performance. Code is available at https: //github. com/jkatzsam/woods_ood.

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JMLR Journal 2021 Journal Article

Banach Space Representer Theorems for Neural Networks and Ridge Splines

Rahul Parhi
Robert D. Nowak

We develop a variational framework to understand the properties of the functions learned by neural networks fit to data. We propose and study a family of continuous-domain linear inverse problems with total variation-like regularization in the Radon domain subject to data fitting constraints. We derive a representer theorem showing that finite-width, single-hidden layer neural networks are solutions to these inverse problems. We draw on many techniques from variational spline theory and so we propose the notion of polynomial ridge splines, which correspond to single-hidden layer neural networks with truncated power functions as the activation function. The representer theorem is reminiscent of the classical reproducing kernel Hilbert space representer theorem, but we show that the neural network problem is posed over a non-Hilbertian Banach space. While the learning problems are posed in the continuous-domain, similar to kernel methods, the problems can be recast as finite-dimensional neural network training problems. These neural network training problems have regularizers which are related to the well-known weight decay and path-norm regularizers. Thus, our result gives insight into functional characteristics of trained neural networks and also into the design neural network regularizers. We also show that these regularizers promote neural network solutions with desirable generalization properties. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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UAI Conference 2021 Conference Paper

Nearest neighbor search under uncertainty

Blake Mason
Ardhendu Tripathy
Robert D. Nowak

Nearest Neighbor Search (NNS) is a central task in knowledge representation, learning, and reasoning. There is vast literature on efficient algorithms for constructing data structures and performing exact and approximate NNS. This paper studies NNS under Uncertainty (NNSU). Specifically, consider the setting in which an NNS algorithm has access only to a stochastic distance oracle that provides a noisy, unbiased estimate of the distance between any pair of points, rather than the exact distance. This models many situations of practical importance, including NNS based on human similarity judgements, physical measurements, or fast, randomized approximations to exact distances. A naive approach to NNSU could employ any standard NNS algorithm and repeatedly query and average results from the stochastic oracle (to reduce noise) whenever it needs a pairwise distance. The problem is that a sufficient number of repeated queries is unknown in advance; e. g. , a point may be distant from all but one other point (crude distance estimates suffice) or it may be close to a large number of other points (accurate estimates are necessary). This paper shows how ideas from cover trees and multi-armed bandits can be leveraged to develop an NNSU algorithm that has optimal dependence on the dataset size and the (unknown) geometry of the dataset.

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ICML Conference 2020 Conference Paper

Robust Outlier Arm Identification

Yinglun Zhu
Sumeet Katariya
Robert D. Nowak

We study the problem of Robust Outlier Arm Identification (ROAI), where the goal is to identify arms whose expected rewards deviate substantially from the majority, by adaptively sampling from their reward distributions. We compute the outlier threshold using the median and median absolute deviation of the expected rewards. This is a robust choice for the threshold compared to using the mean and standard deviation, since it can identify outlier arms even in the presence of extreme outlier values. Our setting is different from existing pure exploration problems where the threshold is pre-specified as a given value or rank. This is useful in applications where the goal is to identify the set of promising items but the cardinality of this set is unknown, such as finding promising drugs for a new disease or identifying items favored by a population. We propose two $\delta$-PAC algorithms for ROAI, which includes the first UCB-style algorithm for outlier detection, and derive upper bounds on their sample complexity. We also prove a matching, up to logarithmic factors, worst case lower bound for the problem, indicating that our upper bounds are generally unimprovable. Experimental results show that our algorithms are both robust and about $5$x sample efficient compared to state-of-the-art.

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ICML Conference 2019 Conference Paper

Bilinear Bandits with Low-rank Structure

Kwang-Sung Jun
Rebecca Willett
Stephen J. Wright 0001
Robert D. Nowak

We introduce the bilinear bandit problem with low-rank structure in which an action takes the form of a pair of arms from two different entity types, and the reward is a bilinear function of the known feature vectors of the arms. The unknown in the problem is a $d_1$ by $d_2$ matrix $\mathbf{\Theta}^*$ that defines the reward, and has low rank $r \ll \min\{d_1, d_2\}$. Determination of $\mathbf{\Theta}^*$ with this low-rank structure poses a significant challenge in finding the right exploration-exploitation tradeoff. In this work, we propose a new two-stage algorithm called “Explore-Subspace-Then-Refine” (ESTR). The first stage is an explicit subspace exploration, while the second stage is a linear bandit algorithm called “almost-low-dimensional OFUL” (LowOFUL) that exploits and further refines the estimated subspace via a regularization technique. We show that the regret of ESTR is $\widetilde{\mathcal{O}}((d_1+d_2)^{3/2} \sqrt{r T})$ where $\widetilde{\mathcal{O}}$ hides logarithmic factors and $T$ is the time horizon, which improves upon the regret of $\widetilde{\mathcal{O}}(d_1d_2\sqrt{T})$ attained for a naïve linear bandit reduction. We conjecture that the regret bound of ESTR is unimprovable up to polylogarithmic factors, and our preliminary experiment shows that ESTR outperforms a naïve linear bandit reduction.

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ICML Conference 2017 Conference Paper

Algebraic Variety Models for High-Rank Matrix Completion

Greg Ongie
Rebecca Willett
Robert D. Nowak
Laura Balzano

We consider a non-linear generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i. e. , each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly high-rank, but it becomes low-rank after mapping each column to a higher dimensional space of monomial features. Algebraic varieties capture a range of well-studied linear models, including affine subspaces and their union, but also quadratic and higher degree curves and surfaces. We study the sampling requirements for a general variety model with a focus on the union of affine subspaces. We propose an efficient matrix completion algorithm that minimizes a convex or non-convex surrogate of the rank of the lifted matrix. Our algorithm uses the well-known “kernel trick” to avoid working directly with the high-dimensional lifted data matrix and scales efficiently with data size. We show the proposed algorithm is able to recover synthetically generated data up to the predicted sampling complexity bounds. The algorithm also outperforms standard techniques in experiments with real data.

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ICML Conference 2016 Conference Paper

Anytime Exploration for Multi-armed Bandits using Confidence Information

Kwang-Sung Jun
Robert D. Nowak

We introduce anytime Explore-m, a pure exploration problem for multi-armed bandits (MAB) that requires making a prediction of the top-m arms at every time step. Anytime Explore-m is more practical than fixed budget or fixed confidence formulations of the top-m problem, since many applications involve a finite, but unpredictable, budget. However, the development and analysis of anytime algorithms present many challenges. We propose AT-LUCB (AnyTime Lower and Upper Confidence Bound), the first nontrivial algorithm that provably solves anytime Explore-m. Our analysis shows that the sample complexity of AT-LUCB is competitive to anytime variants of existing algorithms. Moreover, our empirical evaluation on AT-LUCB shows that AT-LUCB performs as well as or better than state-of-the-art baseline methods for anytime Explore-m.

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ICML Conference 2016 Conference Paper

Representational Similarity Learning with Application to Brain Networks

Urvashi Oswal
Christopher R. Cox
Matthew A. Lambon Ralph
Timothy T. Rogers
Robert D. Nowak

Representational Similarity Learning (RSL) aims to discover features that are important in representing (human-judged) similarities among objects. RSL can be posed as a sparsity-regularized multi-task regression problem. Standard methods, like group lasso, may not select important features if they are strongly correlated with others. To address this shortcoming we present a new regularizer for multitask regression called Group Ordered Weighted \ell_1 (GrOWL). Another key contribution of our paper is a novel application to fMRI brain imaging. Representational Similarity Analysis (RSA) is a tool for testing whether localized brain regions encode perceptual similarities. Using GrOWL, we propose a new approach called Network RSA that can discover arbitrarily structured brain networks (possibly widely distributed and non-local) that encode similarity information. We show, in theory and fMRI experiments, how GrOWL deals with strongly correlated covariates.

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ICML Conference 2016 Conference Paper

The Information-Theoretic Requirements of Subspace Clustering with Missing Data

Daniel L. Pimentel-Alarcón
Robert D. Nowak

Subspace clustering with missing data (SCMD) is a useful tool for analyzing incomplete datasets. Let d be the ambient dimension, and r the dimension of the subspaces. Existing theory shows that Nk = O(r d) columns per subspace are necessary for SCMD, and Nk =O(min d^(log d), d^(r+1) ) are sufficient. We close this gap, showing that Nk =O(r d) is also sufficient. To do this we derive deterministic sampling conditions for SCMD, which give precise information theoretic requirements and determine sampling regimes. These results explain the performance of SCMD algorithms from the literature. Finally, we give a practical algorithm to certify the output of any SCMD method deterministically.

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JMLR Journal 2014 Journal Article

A Junction Tree Framework for Undirected Graphical Model Selection

Divyanshu Vats
Robert D. Nowak

An undirected graphical model is a joint probability distribution defined on an undirected graph $G^*$, where the vertices in the graph index a collection of random variables and the edges encode conditional independence relationships among random variables. The undirected graphical model selection (UGMS) problem is to estimate the graph $G^*$ given observations drawn from the undirected graphical model. This paper proposes a framework for decomposing the UGMS problem into multiple subproblems over clusters and subsets of the separators in a junction tree. The junction tree is constructed using a graph that contains a superset of the edges in $G^*$. We highlight three main properties of using junction trees for UGMS. First, different regularization parameters or different UGMS algorithms can be used to learn different parts of the graph. This is possible since the subproblems we identify can be solved independently of each other. Second, under certain conditions, a junction tree based UGMS algorithm can produce consistent results with fewer observations than the usual requirements of existing algorithms. Third, both our theoretical and experimental results show that the junction tree framework does a significantly better job at finding the weakest edges in a graph than existing methods. This property is a consequence of both the first and second properties. Finally, we note that our framework is independent of the choice of the UGMS algorithm and can be used as a wrapper around standard UGMS algorithms for more accurate graph estimation. [abs] [ pdf ][ bib ] &copy JMLR 2014. ( edit, beta )

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JMLR Journal 2006 Journal Article

Learning Minimum Volume Sets

Clayton D. Scott
Robert D. Nowak

Given a probability measure P and a reference measure μ, one is often interested in the minimum μ -measure set with P -measure at least α. Minimum volume sets of this type summarize the regions of greatest probability mass of P, and are useful for detecting anomalies and constructing confidence regions. This paper addresses the problem of estimating minimum volume sets based on independent samples distributed according to P. Other than these samples, no other information is available regarding P, but the reference measure μ is assumed to be known. We introduce rules for estimating minimum volume sets that parallel the empirical risk minimization and structural risk minimization principles in classification. As in classification, we show that the performances of our estimators are controlled by the rate of uniform convergence of empirical to true probabilities over the class from which the estimator is drawn. Thus we obtain finite sample size performance bounds in terms of VC dimension and related quantities. We also demonstrate strong universal consistency, an oracle inequality, and rates of convergence. The proposed estimators are illustrated with histogram and decision tree set estimation rules. [abs] [ pdf ][ bib ] &copy JMLR 2006. ( edit, beta )

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