arXiv subjects

Rebecca Willett

Publications and source records attributed to Rebecca Willett.

At least 55 records · Page 3Linked to original sources

arXiv2023

Reduced-Order Autodifferentiable Ensemble Kalman Filters

This paper introduces a computational framework to reconstruct and forecast a partially observed state that evolves according to an unknown or expensive-to-simulate dynamical system. Our reduced-order autodifferentiable ensemble Kalman filters (ROAD-EnKFs) learn a latent low-dimensional surrogate model for the dynamics and a decoder that maps from the latent space to the state space. The learned dynamics and decoder are then used within an ensemble Kalman filter to reconstruct and forecast the state. Numerical experiments show that if the state dynamics exhibit a hidden low-dimensional structure, ROAD-EnKFs achieve higher accuracy at lower computational cost compared to existing methods. If such structure is not expressed in the latent state dynamics, ROAD-EnKFs achieve similar accuracy at lower cost, making them a promising approach for surrogate state reconstruction and forecasting.

stat.ML↗

arXiv2022

Embed and Emulate: Learning to estimate parameters of dynamical systems with uncertainty quantification

This paper explores learning emulators for parameter estimation with uncertainty estimation of high-dimensional dynamical systems. We assume access to a computationally complex simulator that inputs a candidate parameter and outputs a corresponding multichannel time series. Our task is to accurately estimate a range of likely values of the underlying parameters. Standard iterative approaches necessitate running the simulator many times, which is computationally prohibitive. This paper describes a novel framework for learning feature embeddings of observed dynamics jointly with an emulator that can replace high-cost simulators for parameter estimation. Leveraging a contrastive learning approach, our method exploits intrinsic data properties within and across parameter and trajectory domains. On a coupled 396-dimensional multiscale Lorenz 96 system, our method significantly outperforms a typical parameter estimation method based on predefined metrics and a classical numerical simulator, and with only 1.19% of the baseline's computation time. Ablation studies highlight the potential of explicitly designing learned emulators for parameter estimation by leveraging contrastive learning.

cs.LG↗

arXiv2022

Cloud Classification with Unsupervised Deep Learning

We present a framework for cloud characterization that leverages modern unsupervised deep learning technologies. While previous neural network-based cloud classification models have used supervised learning methods, unsupervised learning allows us to avoid restricting the model to artificial categories based on historical cloud classification schemes and enables the discovery of novel, more detailed classifications. Our framework learns cloud features directly from radiance data produced by NASA's Moderate Resolution Imaging Spectroradiometer (MODIS) satellite instrument, deriving cloud characteristics from millions of images without relying on pre-defined cloud types during the training process. We present preliminary results showing that our method extracts physically relevant information from radiance data and produces meaningful cloud classes.

physics.ao-ph↗

arXiv2022

Functional Linear Regression with Mixed Predictors

We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional vector predictors. We derive finite sample theoretical guarantees and show that the excess prediction risk of our estimator is minimax optimal. Furthermore, our analysis reveals an interesting phase transition phenomenon that the optimal excess risk is determined jointly by the smoothness and the sparsity of the functional regression coefficients. A novel efficient optimization algorithm based on iterative coordinate descent is devised to handle the smoothness and group penalties simultaneously. Simulation studies and real data applications illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.

math.ST↗

arXiv2022

NURD: Negative-Unlabeled Learning for Online Datacenter Straggler Prediction

Datacenters execute large computational jobs, which are composed of smaller tasks. A job completes when all its tasks finish, so stragglers -- rare, yet extremely slow tasks -- are a major impediment to datacenter performance. Accurately predicting stragglers would enable proactive intervention, allowing datacenter operators to mitigate stragglers before they delay a job. While much prior work applies machine learning to predict computer system performance, these approaches rely on complete labels -- i.e., sufficient examples of all possible behaviors, including straggling and non-straggling -- or strong assumptions about the underlying latency distributions -- e.g., whether Gaussian or not. Within a running job, however, none of this information is available until stragglers have revealed themselves when they have already delayed the job. To predict stragglers accurately and early without labeled positive examples or assumptions on latency distributions, this paper presents NURD, a novel Negative-Unlabeled learning approach with Reweighting and Distribution-compensation that only trains on negative and unlabeled streaming data. The key idea is to train a predictor using finished tasks of non-stragglers to predict latency for unlabeled running tasks, and then reweight each unlabeled task's prediction based on a weighting function of its feature space. We evaluate NURD on two production traces from Google and Alibaba, and find that compared to the best baseline approach, NURD produces 2--11 percentage point increases in the F1 score in terms of prediction accuracy, and 2.0--8.8 percentage point improvements in job completion time.

cs.LG↗

arXiv2022

Pure Exploration in Kernel and Neural Bandits

We study pure exploration in bandits, where the dimension of the feature representation can be much larger than the number of arms. To overcome the curse of dimensionality, we propose to adaptively embed the feature representation of each arm into a lower-dimensional space and carefully deal with the induced model misspecification. Our approach is conceptually very different from existing works that can either only handle low-dimensional linear bandits or passively deal with model misspecification. We showcase the application of our approach to two pure exploration settings that were previously under-studied: (1) the reward function belongs to a possibly infinite-dimensional Reproducing Kernel Hilbert Space, and (2) the reward function is nonlinear and can be approximated by neural networks. Our main results provide sample complexity guarantees that only depend on the effective dimension of the feature spaces in the kernel or neural representations. Extensive experiments conducted on both synthetic and real-world datasets demonstrate the efficacy of our methods.

stat.ML↗

arXiv2022

The Role of Linear Layers in Nonlinear Interpolating Networks

This paper explores the implicit bias of overparameterized neural networks of depth greater than two layers. Our framework considers a family of networks of varying depth that all have the same capacity but different implicitly defined representation costs. The representation cost of a function induced by a neural network architecture is the minimum sum of squared weights needed for the network to represent the function; it reflects the function space bias associated with the architecture. Our results show that adding linear layers to a ReLU network yields a representation cost that reflects a complex interplay between the alignment and sparsity of ReLU units. Specifically, using a neural network to fit training data with minimum representation cost yields an interpolating function that is constant in directions perpendicular to a low-dimensional subspace on which a parsimonious interpolant exists.

cs.LG↗

arXiv2021

Data-driven Cloud Clustering via a Rotationally Invariant Autoencoder

Advanced satellite-born remote sensing instruments produce high-resolution multi-spectral data for much of the globe at a daily cadence. These datasets open up the possibility of improved understanding of cloud dynamics and feedback, which remain the biggest source of uncertainty in global climate model projections. As a step towards answering these questions, we describe an automated rotation-invariant cloud clustering (RICC) method that leverages deep learning autoencoder technology to organize cloud imagery within large datasets in an unsupervised fashion, free from assumptions about predefined classes. We describe both the design and implementation of this method and its evaluation, which uses a sequence of testing protocols to determine whether the resulting clusters: (1) are physically reasonable, (i.e., embody scientifically relevant distinctions); (2) capture information on spatial distributions, such as textures; (3) are cohesive and separable in latent space; and (4) are rotationally invariant, (i.e., insensitive to the orientation of an image). Results obtained when these evaluation protocols are applied to RICC outputs suggest that the resultant novel cloud clusters capture meaningful aspects of cloud physics, are appropriately spatially coherent, and are invariant to orientations of input images. Our results support the possibility of using an unsupervised data-driven approach for automated clustering and pattern discovery in cloud imagery.

cs.CV↗

arXiv2021

Adaptive Differentially Private Empirical Risk Minimization

We propose an adaptive (stochastic) gradient perturbation method for differentially private empirical risk minimization. At each iteration, the random noise added to the gradient is optimally adapted to the stepsize; we name this process adaptive differentially private (ADP) learning. Given the same privacy budget, we prove that the ADP method considerably improves the utility guarantee compared to the standard differentially private method in which vanilla random noise is added. Our method is particularly useful for gradient-based algorithms with time-varying learning rates, including variants of AdaGrad (Duchi et al., 2011). We provide extensive numerical experiments to demonstrate the effectiveness of the proposed adaptive differentially private algorithm.

cs.LG↗

arXiv2021

Statistically and Computationally Efficient Change Point Localization in Regression Settings

Detecting when the underlying distribution changes for the observed time series is a fundamental problem arising in a broad spectrum of applications. In this paper, we study multiple change-point localization in the high-dimensional regression setting, which is particularly challenging as no direct observations of the parameter of interest is available. Specifically, we assume we observe $\{ x_t, y_t\}_{t=1}^n$ where $ \{ x_t\}_{t=1}^n $ are $p$-dimensional covariates, $\{y_t\}_{t=1}^n$ are the univariate responses satisfying $\mathbb{E}(y_t) = x_t^\top β_t^* \text{ for } 1\le t \le n $ and $\{β_t^*\}_{t=1}^n $ are the unobserved regression coefficients that change over time in a piecewise constant manner. We propose a novel projection-based algorithm, Variance Projected Wild Binary Segmentation~(VPWBS), which transforms the original (difficult) problem of change-point detection in $p$-dimensional regression to a simpler problem of change-point detection in mean of a one-dimensional time series. VPWBS is shown to achieve sharp localization rate $O_p(1/n)$ up to a log factor, a significant improvement from the best rate $O_p(1/\sqrt{n})$ known in the existing literature for multiple change-point localization in high-dimensional regression. Extensive numerical experiments are conducted to demonstrate the robust and favorable performance of VPWBS over two state-of-the-art algorithms, especially when the size of change in the regression coefficients $\{β_t^*\}_{t=1}^n $ is small.

math.ST↗

arXiv2021

Auto-differentiable Ensemble Kalman Filters

Data assimilation is concerned with sequentially estimating a temporally-evolving state. This task, which arises in a wide range of scientific and engineering applications, is particularly challenging when the state is high-dimensional and the state-space dynamics are unknown. This paper introduces a machine learning framework for learning dynamical systems in data assimilation. Our auto-differentiable ensemble Kalman filters (AD-EnKFs) blend ensemble Kalman filters for state recovery with machine learning tools for learning the dynamics. In doing so, AD-EnKFs leverage the ability of ensemble Kalman filters to scale to high-dimensional states and the power of automatic differentiation to train high-dimensional surrogate models for the dynamics. Numerical results using the Lorenz-96 model show that AD-EnKFs outperform existing methods that use expectation-maximization or particle filters to merge data assimilation and machine learning. In addition, AD-EnKFs are easy to implement and require minimal tuning.

stat.ML↗

arXiv2021

Deep Equilibrium Architectures for Inverse Problems in Imaging

Recent efforts on solving inverse problems in imaging via deep neural networks use architectures inspired by a fixed number of iterations of an optimization method. The number of iterations is typically quite small due to difficulties in training networks corresponding to more iterations; the resulting solvers cannot be run for more iterations at test time without incurring significant errors. This paper describes an alternative approach corresponding to an infinite number of iterations, yielding a consistent improvement in reconstruction accuracy above state-of-the-art alternatives and where the computational budget can be selected at test time to optimize context-dependent trade-offs between accuracy and computation. The proposed approach leverages ideas from Deep Equilibrium Models, where the fixed-point iteration is constructed to incorporate a known forward model and insights from classical optimization-based reconstruction methods.

eess.IV↗

arXiv2021

Model Adaptation for Inverse Problems in Imaging

Deep neural networks have been applied successfully to a wide variety of inverse problems arising in computational imaging. These networks are typically trained using a forward model that describes the measurement process to be inverted, which is often incorporated directly into the network itself. However, these approaches are sensitive to changes in the forward model: if at test time the forward model varies (even slightly) from the one the network was trained for, the reconstruction performance can degrade substantially. Given a network trained to solve an initial inverse problem with a known forward model, we propose two novel procedures that adapt the network to a change in the forward model, even without full knowledge of the change. Our approaches do not require access to more labeled data (i.e., ground truth images). We show these simple model adaptation approaches achieve empirical success in a variety of inverse problems, including deblurring, super-resolution, and undersampled image reconstruction in magnetic resonance imaging.

eess.IV↗

arXiv2021

Prediction in the presence of response-dependent missing labels

In a variety of settings, limitations of sensing technologies or other sampling mechanisms result in missing labels, where the likelihood of a missing label in the training set is an unknown function of the data. For example, satellites used to detect forest fires cannot sense fires below a certain size threshold. In such cases, training datasets consist of positive and pseudo-negative observations where pseudo-negative observations can be either true negatives or undetected positives with small magnitudes. We develop a new methodology and non-convex algorithm P(ositive) U(nlabeled) - O(ccurrence) M(agnitude) M(ixture) which jointly estimates the occurrence and detection likelihood of positive samples, utilizing prior knowledge of the detection mechanism. Our approach uses ideas from positive-unlabeled (PU)-learning and zero-inflated models that jointly estimate the magnitude and occurrence of events. We provide conditions under which our model is identifiable and prove that even though our approach leads to a non-convex objective, any local minimizer has optimal statistical error (up to a log term) and projected gradient descent has geometric convergence rates. We demonstrate on both synthetic data and a California wildfire dataset that our method out-performs existing state-of-the-art approaches.

stat.ML↗

arXiv2021

An Optimal Statistical and Computational Framework for Generalized Tensor Estimation

This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator consists of finding a low-rank tensor fit to the data under generalized parametric models. To overcome the difficulty of non-convexity in these problems, we introduce a unified approach of projected gradient descent that adapts to the underlying low-rank structure. Under mild conditions on the loss function, we establish both an upper bound on statistical error and the linear rate of computational convergence through a general deterministic analysis. Then we further consider a suite of generalized tensor estimation problems, including sub-Gaussian tensor PCA, tensor regression, and Poisson and binomial tensor PCA. We prove that the proposed algorithm achieves the minimax optimal rate of convergence in estimation error. Finally, we demonstrate the superiority of the proposed framework via extensive experiments on both simulated and real data.

math.ST↗

arXiv2020

Functional Autoregressive Processes in Reproducing Kernel Hilbert Spaces

We study the estimation and prediction of functional autoregressive~(FAR) processes, a statistical tool for modeling functional time series data. Due to the infinite-dimensional nature of FAR processes, the existing literature addresses its inference via dimension reduction and theoretical results therein require the (unrealistic) assumption of fully observed functional time series. We propose an alternative inference framework based on Reproducing Kernel Hilbert Spaces~(RKHS). Specifically, a nuclear norm regularization method is proposed for estimating the transition operators of the FAR process directly from discrete samples of the functional time series. We derive a representer theorem for the FAR process, which enables infinite-dimensional inference without dimension reduction. Sharp theoretical guarantees are established under the (more realistic) assumption that we only have finite discrete samples of the FAR process. Extensive numerical experiments and a real data application of energy consumption prediction are further conducted to illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.

stat.ME↗

arXiv2020

Localizing Changes in High-Dimensional Regression Models

This paper addresses the problem of localizing change points in high-dimensional linear regression models with piecewise constant regression coefficients. We develop a dynamic programming approach to estimate the locations of the change points whose performance improves upon the current state-of-the-art, even as the dimensionality, the sparsity of the regression coefficients, the temporal spacing between two consecutive change points, and the magnitude of the difference of two consecutive regression coefficient vectors are allowed to vary with the sample size. Furthermore, we devise a computationally-efficient refinement procedure that provably reduces the localization error of preliminary estimates of the change points. We demonstrate minimax lower bounds on the localization error that nearly match the upper bound on the localization error of our methodology and show that the signal-to-noise condition we impose is essentially the weakest possible based on information-theoretic arguments. Extensive numerical results support our theoretical findings, and experiments on real air quality data reveal change points supported by historical information not used by the algorithm.

stat.ME↗

arXiv2020

Tensor Methods for Nonlinear Matrix Completion

In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases where the columns are points on a low-dimensional nonlinear algebraic variety, a problem we call Low Algebraic Dimension Matrix Completion (LADMC). Matrices whose columns belong to a union of subspaces are an important special case. We propose a LADMC algorithm that leverages existing LRMC methods on a tensorized representation of the data. For example, a second-order tensorized representation is formed by taking the Kronecker product of each column with itself, and we consider higher order tensorizations as well. This approach will succeed in many cases where traditional LRMC is guaranteed to fail because the data are low-rank in the tensorized representation but not in the original representation. We provide a formal mathematical justification for the success of our method. In particular, we give bounds of the rank of these data in the tensorized representation, and we prove sampling requirements to guarantee uniqueness of the solution. We also provide experimental results showing that the new approach outperforms existing state-of-the-art methods for matrix completion under a union of subspaces model.

stat.ML↗