arXiv subjects

Clayton Scott

Publications and source records attributed to Clayton Scott.

At least 37 records · Page 2Linked to original sources

arXiv2022

Estimating Phase from Observed Trajectories Using the Temporal 1-Form

Oscillators are ubiquitous in nature, and usually associated with the existence of an asymptotic phase that governs the long-term dynamics of the oscillator. % We show that asymptotic phase can be estimated using a carefully chosen series expansion which directly computes the phase response curve and provide an algorithm for estimating the co-efficients of this series. Unlike all previously available data driven phase estimation methods, our algorithm can: (i) use observations that are much shorter than a cycle; (ii) recover phase within any forward invariant region for which sufficient data are available; (iii) recover the phase response curves (PRC-s) that govern weak oscillator coupling; (iv) show isochron curvature, and recover nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series.

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arXiv2021

Calibrated Surrogate Losses for Adversarially Robust Classification

Adversarially robust classification seeks a classifier that is insensitive to adversarial perturbations of test patterns. This problem is often formulated via a minimax objective, where the target loss is the worst-case value of the 0-1 loss subject to a bound on the size of perturbation. Recent work has proposed convex surrogates for the adversarial 0-1 loss, in an effort to make optimization more tractable. A primary question is that of consistency, that is, whether minimization of the surrogate risk implies minimization of the adversarial 0-1 risk. In this work, we analyze this question through the lens of calibration, which is a pointwise notion of consistency. We show that no convex surrogate loss is calibrated with respect to the adversarial 0-1 loss when restricted to the class of linear models. We further introduce a class of nonconvex losses and offer necessary and sufficient conditions for losses in this class to be calibrated. We also show that if the underlying distribution satisfies Massart's noise condition, convex losses can also be calibrated in the adversarial setting.

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arXiv2021

Domain Generalization by Marginal Transfer Learning

In the problem of domain generalization (DG), there are labeled training data sets from several related prediction problems, and the goal is to make accurate predictions on future unlabeled data sets that are not known to the learner. This problem arises in several applications where data distributions fluctuate because of environmental, technical, or other sources of variation. We introduce a formal framework for DG, and argue that it can be viewed as a kind of supervised learning problem by augmenting the original feature space with the marginal distribution of feature vectors. While our framework has several connections to conventional analysis of supervised learning algorithms, several unique aspects of DG require new methods of analysis. This work lays the learning theoretic foundations of domain generalization, building on our earlier conference paper where the problem of DG was introduced (Blanchard et al., 2011). We present two formal models of data generation, corresponding notions of risk, and distribution-free generalization error analysis. By focusing our attention on kernel methods, we also provide more quantitative results and a universally consistent algorithm. An efficient implementation is provided for this algorithm, which is experimentally compared to a pooling strategy on one synthetic and three real-world data sets.

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arXiv2020

Learning from Label Proportions: A Mutual Contamination Framework

Learning from label proportions (LLP) is a weakly supervised setting for classification in which unlabeled training instances are grouped into bags, and each bag is annotated with the proportion of each class occurring in that bag. Prior work on LLP has yet to establish a consistent learning procedure, nor does there exist a theoretically justified, general purpose training criterion. In this work we address these two issues by posing LLP in terms of mutual contamination models (MCMs), which have recently been applied successfully to study various other weak supervision settings. In the process, we establish several novel technical results for MCMs, including unbiased losses and generalization error bounds under non-iid sampling plans. We also point out the limitations of a common experimental setting for LLP, and propose a new one based on our MCM framework.

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arXiv2020

Consistent Estimation of Identifiable Nonparametric Mixture Models from Grouped Observations

Recent research has established sufficient conditions for finite mixture models to be identifiable from grouped observations. These conditions allow the mixture components to be nonparametric and have substantial (or even total) overlap. This work proposes an algorithm that consistently estimates any identifiable mixture model from grouped observations. Our analysis leverages an oracle inequality for weighted kernel density estimators of the distribution on groups, together with a general result showing that consistent estimation of the distribution on groups implies consistent estimation of mixture components. A practical implementation is provided for paired observations, and the approach is shown to outperform existing methods, especially when mixture components overlap significantly.

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arXiv2020

Simple Regret Minimization for Contextual Bandits

There are two variants of the classical multi-armed bandit (MAB) problem that have received considerable attention from machine learning researchers in recent years: contextual bandits and simple regret minimization. Contextual bandits are a sub-class of MABs where, at every time step, the learner has access to side information that is predictive of the best arm. Simple regret minimization assumes that the learner only incurs regret after a pure exploration phase. In this work, we study simple regret minimization for contextual bandits. Motivated by applications where the learner has separate training and autonomous modes, we assume that the learner experiences a pure exploration phase, where feedback is received after every action but no regret is incurred, followed by a pure exploitation phase in which regret is incurred but there is no feedback. We present the Contextual-Gap algorithm and establish performance guarantees on the simple regret, i.e., the regret during the pure exploitation phase. Our experiments examine a novel application to adaptive sensor selection for magnetic field estimation in interplanetary spacecraft, and demonstrate considerable improvement over algorithms designed to minimize the cumulative regret.

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arXiv2019

PAC Reinforcement Learning without Real-World Feedback

This work studies reinforcement learning in the Sim-to-Real setting, in which an agent is first trained on a number of simulators before being deployed in the real world, with the aim of decreasing the real-world sample complexity requirement. Using a dynamic model known as a rich observation Markov decision process (ROMDP), we formulate a theoretical framework for Sim-to-Real in the situation where feedback in the real world is not available. We establish real-world sample complexity guarantees that are smaller than what is currently known for directly (i.e., without access to simulators) learning a ROMDP with feedback.

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arXiv2019

Learning from Multiple Corrupted Sources, with Application to Learning from Label Proportions

We study binary classification in the setting where the learner is presented with multiple corrupted training samples, with possibly different sample sizes and degrees of corruption, and introduce an approach based on minimizing a weighted combination of corruption-corrected empirical risks. We establish a generalization error bound, and further show that the bound is optimized when the weights are certain interpretable and intuitive functions of the sample sizes and degrees of corruptions. We then apply this setting to the problem of learning with label proportions (LLP), and propose an algorithm that enjoys the most general statistical performance guarantees known for LLP. Experiments demonstrate the utility of our theory.

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arXiv2019

Local inversion-free estimation of spatial Gaussian processes

Maximizing the likelihood has been widely used for estimating the unknown covariance parameters of spatial Gaussian processes. However, evaluating and optimizing the likelihood function can be computationally intractable, particularly for large number of (possibly) irregularly spaced observations, due to the need to handle the inverse of ill-conditioned and large covariance matrices. Extending the "inversion-free" method of Anitescu, Chen and Stein \cite{anitescu2017inversion}, we investigate a broad class of covariance parameter estimators based on inversion-free surrogate losses and block diagonal approximation schemes of the covariance structure. This class of estimators yields a spectrum for negotiating the trade-off between statistical accuracy and computational cost. We present fixed-domain asymptotic properties of our proposed method, establishing $\sqrt{n}$-consistency and asymptotic normality results for isotropic Matern Gaussian processes observed on a multi-dimensional and irregular lattice. Simulation studies are also presented for assessing the scalability and statistical efficiency of the proposed algorithm for large data sets.

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arXiv2019

A Generalization Error Bound for Multi-class Domain Generalization

Domain generalization is the problem of assigning labels to an unlabeled data set, given several similar data sets for which labels have been provided. Despite considerable interest in this problem over the last decade, there has been no theoretical analysis in the setting of multi-class classification. In this work, we study a kernel-based learning algorithm and establish a generalization error bound that scales logarithmically in the number of classes, matching state-of-the-art bounds for multi-class classification in the conventional learning setting. We also demonstrate empirically that the proposed algorithm achieves significant performance gains compared to a pooling strategy.

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arXiv2019

Decontamination of Mutual Contamination Models

Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees.

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arXiv2019

A Generalized Neyman-Pearson Criterion for Optimal Domain Adaptation

In the problem of domain adaptation for binary classification, the learner is presented with labeled examples from a source domain, and must correctly classify unlabeled examples from a target domain, which may differ from the source. Previous work on this problem has assumed that the performance measure of interest is the expected value of some loss function. We introduce a new Neyman-Pearson-like criterion and argue that, for this optimality criterion, stronger domain adaptation results are possible than what has previously been established. In particular, we study a class of domain adaptation problems that generalizes both the covariate shift assumption and a model for feature-dependent label noise, and establish optimal classification on the target domain despite not having access to labelled data from this domain.

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arXiv2018

Nonparametric Preference Completion

We consider the task of collaborative preference completion: given a pool of items, a pool of users and a partially observed item-user rating matrix, the goal is to recover the \emph{personalized ranking} of each user over all of the items. Our approach is nonparametric: we assume that each item $i$ and each user $u$ have unobserved features $x_i$ and $y_u$, and that the associated rating is given by $g_u(f(x_i,y_u))$ where $f$ is Lipschitz and $g_u$ is a monotonic transformation that depends on the user. We propose a $k$-nearest neighbors-like algorithm and prove that it is consistent. To the best of our knowledge, this is the first consistency result for the collaborative preference completion problem in a nonparametric setting. Finally, we demonstrate the performance of our algorithm with experiments on the Netflix and Movielens datasets.

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arXiv2017

Dictionary-Free MRI PERK: Parameter Estimation via Regression with Kernels

This paper introduces a fast, general method for dictionary-free parameter estimation in quantitative magnetic resonance imaging (QMRI) via regression with kernels (PERK). PERK first uses prior distributions and the nonlinear MR signal model to simulate many parameter-measurement pairs. Inspired by machine learning, PERK then takes these parameter-measurement pairs as labeled training points and learns from them a nonlinear regression function using kernel functions and convex optimization. PERK admits a simple implementation as per-voxel nonlinear lifting of MRI measurements followed by linear minimum mean-squared error regression. We demonstrate PERK for $T_1,T_2$ estimation, a well-studied application where it is simple to compare PERK estimates against dictionary-based grid search estimates. Numerical simulations as well as single-slice phantom and in vivo experiments demonstrate that PERK and grid search produce comparable $T_1,T_2$ estimates in white and gray matter, but PERK is consistently at least $23\times$ faster. This acceleration factor will increase by several orders of magnitude for full-volume QMRI estimation problems involving more latent parameters per voxel.

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arXiv2017

Consistent Kernel Density Estimation with Non-Vanishing Bandwidth

Consistency of the kernel density estimator requires that the kernel bandwidth tends to zero as the sample size grows. In this paper we investigate the question of whether consistency is possible when the bandwidth is fixed, if we consider a more general class of weighted KDEs. To answer this question in the affirmative, we introduce the fixed-bandwidth KDE (fbKDE), obtained by solving a quadratic program, and prove that it consistently estimates any continuous square-integrable density. We also establish rates of convergence for the fbKDE with radial kernels and the box kernel under appropriate smoothness assumptions. Furthermore, in an experimental study we demonstrate that the fbKDE compares favorably to the standard KDE and the previously proposed variable bandwidth KDE.

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arXiv2017

Multi-Task Learning for Contextual Bandits

Contextual bandits are a form of multi-armed bandit in which the agent has access to predictive side information (known as the context) for each arm at each time step, and have been used to model personalized news recommendation, ad placement, and other applications. In this work, we propose a multi-task learning framework for contextual bandit problems. Like multi-task learning in the batch setting, the goal is to leverage similarities in contexts for different arms so as to improve the agent's ability to predict rewards from contexts. We propose an upper confidence bound-based multi-task learning algorithm for contextual bandits, establish a corresponding regret bound, and interpret this bound to quantify the advantages of learning in the presence of high task (arm) similarity. We also describe an effective scheme for estimating task similarity from data, and demonstrate our algorithm's performance on several data sets.

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arXiv2017

Optimal change point detection in Gaussian processes

We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.

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arXiv2017

Adaptive Questionnaires for Direct Identification of Optimal Product Design

We consider the problem of identifying the most profitable product design from a finite set of candidates under unknown consumer preference. A standard approach to this problem follows a two-step strategy: First, estimate the preference of the consumer population, represented as a point in part-worth space, using an adaptive discrete-choice questionnaire. Second, integrate the estimated part-worth vector with engineering feasibility and cost models to determine the optimal design. In this work, we (1) demonstrate that accurate preference estimation is neither necessary nor sufficient for identifying the optimal design, (2) introduce a novel adaptive questionnaire that leverages knowledge about engineering feasibility and manufacturing costs to directly determine the optimal design, and (3) interpret product design in terms of a nonlinear segmentation of part-worth space, and use this interpretation to illuminate the intrinsic difficulty of optimal design in the presence of noisy questionnaire responses. We establish the superiority of the proposed approach using a well-documented optimal product design task. This study demonstrates how the identification of optimal product design can be accelerated by integrating marketing and manufacturing knowledge into the adaptive questionnaire.

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