Rebecca Willett

Ruoxi Jiang, Peter Y. Lu, Elena Orlova et al. · mit

Chaotic systems make long-horizon forecasts difficult because small perturbations in initial conditions cause trajectories to diverge at an exponential rate. In this setting, neural operators trained to minimize squared error losses, while capable of accurate short-term forecasts, often fail to reproduce statistical or structural properties of the dynamics over longer time horizons and can yield degenerate results. In this paper, we propose an alternative framework designed to preserve invariant measures of chaotic attractors that characterize the time-invariant statistical properties of the dynamics. Specifically, in the multi-environment setting (where each sample trajectory is governed by slightly different dynamics), we consider two novel approaches to training with noisy data. First, we propose a loss based on the optimal transport distance between the observed dynamics and the neural operator outputs. This approach requires expert knowledge of the underlying physics to determine what statistical features should be included in the optimal transport loss. Second, we show that a contrastive learning framework, which does not require any specialized prior knowledge, can preserve statistical properties of the dynamics nearly as well as the optimal transport approach. On a variety of chaotic systems, our method is shown empirically to preserve invariant measures of chaotic attractors.

Yi Ding, Avinash Rao, Hyebin Song et al. · mit

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.

Ryien Hosseini, Filippo Simini, Venkatram Vishwanath et al.

Graph Neural Networks learn on graph-structured data by iteratively aggregating local neighborhood information. While this local message passing paradigm imparts a powerful inductive bias and exploits graph sparsity, it also yields three key challenges: (i) oversquashing of long-range information, (ii) oversmoothing of node representations, and (iii) limited expressive power. In this work we inject randomized global embeddings of node features, which we term \textit{Sketched Random Features}, into standard GNNs, enabling them to efficiently capture long-range dependencies. The embeddings are unique, distance-sensitive, and topology-agnostic -- properties which we analytically and empirically show alleviate the aforementioned limitations when injected into GNNs. Experimental results on real-world graph learning tasks confirm that this strategy consistently improves performance over baseline GNNs, offering both a standalone solution and a complementary enhancement to existing techniques such as graph positional encodings. Our source code is available at \href{https://github.com/ryienh/sketched-random-features}{https://github.com/ryienh/sketched-random-features}.

Jake A. Soloff, Rina Foygel Barber, Rebecca Willett

Bagging is an important technique for stabilizing machine learning models. In this paper, we derive a finite-sample guarantee on the stability of bagging for any model. Our result places no assumptions on the distribution of the data, on the properties of the base algorithm, or on the dimensionality of the covariates. Our guarantee applies to many variants of bagging and is optimal up to a constant. Empirical results validate our findings, showing that bagging successfully stabilizes even highly unstable base algorithms.

Yuming Chen, Daniel Sanz-Alonso, Rebecca Willett

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.

Takuya Kurihana, Ian Foster, Rebecca Willett et al.

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.

Ruoxi Jiang, Rebecca Willett

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.

Owen Melia, Eric Jonas, Rebecca Willett

Rotational invariance is a popular inductive bias used by many fields in machine learning, such as computer vision and machine learning for quantum chemistry. Rotation-invariant machine learning methods set the state of the art for many tasks, including molecular property prediction and 3D shape classification. These methods generally either rely on task-specific rotation-invariant features, or they use general-purpose deep neural networks which are complicated to design and train. However, it is unclear whether the success of these methods is primarily due to the rotation invariance or the deep neural networks. To address this question, we suggest a simple and general-purpose method for learning rotation-invariant functions of three-dimensional point cloud data using a random features approach. Specifically, we extend the random features method of Rahimi & Recht 2007 by deriving a version that is invariant to three-dimensional rotations and showing that it is fast to evaluate on point cloud data. We show through experiments that our method matches or outperforms the performance of general-purpose rotation-invariant neural networks on standard molecular property prediction benchmark datasets QM7 and QM9. We also show that our method is general-purpose and provides a rotation-invariant baseline on the ModelNet40 shape classification task. Finally, we show that our method has an order of magnitude smaller prediction latency than competing kernel methods.

Elena Orlova, Haokun Liu, Raphael Rossellini et al.

Producing high-quality forecasts of key climate variables, such as temperature and precipitation, on subseasonal time scales has long been a gap in operational forecasting. This study explores an application of machine learning (ML) models as post-processing tools for subseasonal forecasting. Lagged numerical ensemble forecasts (i.e., an ensemble where the members have different initialization dates) and observational data, including relative humidity, pressure at sea level, and geopotential height, are incorporated into various ML methods to predict monthly average precipitation and two-meter temperature two weeks in advance for the continental United States. For regression, quantile regression, and tercile classification tasks, we consider using linear models, random forests, convolutional neural networks, and stacked models (a multi-model approach based on the prediction of the individual ML models). Unlike previous ML approaches that often use ensemble mean alone, we leverage information embedded in the ensemble forecasts to enhance prediction accuracy. Additionally, we investigate extreme event predictions that are crucial for planning and mitigation efforts. Considering ensemble members as a collection of spatial forecasts, we explore different approaches to using spatial information. Trade-offs between different approaches may be mitigated with model stacking. Our proposed models outperform standard baselines such as climatological forecasts and ensemble means. In addition, we investigate feature importance, trade-offs between using the full ensemble or only the ensemble mean, and different modes of accounting for spatial variability.

Olivia Tsang, Owen Melia, Vasileios Charisopoulos et al.

Inverse medium scattering is an ill-posed, nonlinear wave-based imaging problem arising in medical imaging, remote sensing, and non-destructive testing. Machine learning (ML) methods offer increased inference speed and flexibility in capturing prior knowledge of imaging targets relative to classical optimization-based approaches; however, they perform poorly in regimes where the scattering behavior is highly nonlinear. A key limitation is that ML methods struggle to incorporate the physics governing the scattering process, which are typically inferred implicitly from the training data or loosely enforced via architectural design. In this paper, we present a method that endows a machine learning framework with explicit knowledge of problem physics, in the form of a differentiable solver representing the forward model. The proposed method progressively refines reconstructions of the scattering potential using measurements at increasing wave frequencies, following a classical strategy to stabilize recovery. Empirically, we find that our method provides high-quality reconstructions at a fraction of the computational or sampling costs of competing approaches.

Ruoxi Jiang, Peter Y. Lu, Rebecca Willett

Scientific modeling and engineering applications rely heavily on parameter estimation methods to fit physical models and calibrate numerical simulations using real-world measurements. In the absence of analytic statistical models with tractable likelihoods, modern simulation-based inference (SBI) methods first use a numerical simulator to generate a dataset of parameters and simulated outputs. This dataset is then used to approximate the likelihood and estimate the system parameters given observation data. Several SBI methods employ machine learning emulators to accelerate data generation and parameter estimation. However, applying these approaches to high-dimensional physical systems remains challenging due to the cost and complexity of training high-dimensional emulators. This paper introduces Embed and Emulate (E&E): a new SBI method based on contrastive learning that efficiently handles high-dimensional data and complex, multimodal parameter posteriors. E&E learns a low-dimensional latent embedding of the data (i.e., a summary statistic) and a corresponding fast emulator in the latent space, eliminating the need to run expensive simulations or a high dimensional emulator during inference. We illustrate the theoretical properties of the learned latent space through a synthetic experiment and demonstrate superior performance over existing methods in a realistic, non-identifiable parameter estimation task using the high-dimensional, chaotic Lorenz 96 system.

Ryien Hosseini, Filippo Simini, Venkatram Vishwanath et al.

Deep generative models have recently achieved significant success in modeling graph data, including dynamic graphs, where topology and features evolve over time. However, unlike in vision and natural language domains, evaluating generative models for dynamic graphs is challenging due to the difficulty of visualizing their output, making quantitative metrics essential. In this work, we develop a new quality metric for evaluating generative models of dynamic graphs. Current metrics for dynamic graphs typically involve discretizing the continuous-evolution of graphs into static snapshots and then applying conventional graph similarity measures. This approach has several limitations: (a) it models temporally related events as i.i.d. samples, failing to capture the non-uniform evolution of dynamic graphs; (b) it lacks a unified measure that is sensitive to both features and topology; (c) it fails to provide a scalar metric, requiring multiple metrics without clear superiority; and (d) it requires explicitly instantiating each static snapshot, leading to impractical runtime demands that hinder evaluation at scale. We propose a novel metric based on the \textit{Johnson-Lindenstrauss} lemma, applying random projections directly to dynamic graph data. This results in an expressive, scalar, and application-agnostic measure of dynamic graph similarity that overcomes the limitations of traditional methods. We also provide a comprehensive empirical evaluation of metrics for continuous-time dynamic graphs, demonstrating the effectiveness of our approach compared to existing methods. Our implementation is available at https://github.com/ryienh/jl-metric.

Yang Meng, Ruoxi Jiang, Zhuokai Zhao et al.

Deep surrogate models for parametric partial differential equations (PDEs) can deliver high-fidelity approximations but remain prohibitively data-hungry: training often requires thousands of fine-grid simulations, each incurring substantial computational cost. To address this challenge, we introduce RLMesh, an end-to-end framework for efficient surrogate training under limited simulation budget. The key idea is to use reinforcement learning (RL) to adaptively allocate mesh grid points non-uniformly within each simulation domain, focusing numerical resolution in regions most critical for accurate PDE solutions. A lightweight proxy model further accelerates RL training by providing efficient reward estimates without full surrogate retraining. Experiments on PDE benchmarks demonstrate that RLMesh achieves competitive accuracy to baselines but with substantially fewer simulation queries. These results show that solver-level spatial adaptivity can dramatically improve the efficiency of surrogate training pipelines, enabling practical deployment of learning-based PDE surrogates across a wide range of problems.

Melissa Adrian, Daniel Sanz-Alonso, Rebecca Willett

Data assimilation algorithms estimate the state of a dynamical system from partial observations, where the successful performance of these algorithms hinges on costly parameter tuning and on employing an accurate model for the dynamics. This paper introduces a framework for jointly learning the state, dynamics, and parameters of filtering algorithms in data assimilation through a process we refer to as auto-differentiable filtering. The framework leverages a theoretically motivated loss function that enables learning from partial, noisy observations via gradient-based optimization using auto-differentiation. We further demonstrate how several well-known data assimilation methods can be learned or tuned within this framework. To underscore the versatility of auto-differentiable filtering, we perform experiments on dynamical systems spanning multiple scientific domains, such as the Clohessy-Wiltshire equations from aerospace engineering, the Lorenz-96 system from atmospheric science, and the generalized Lotka-Volterra equations from systems biology. Finally, we provide guidelines for practitioners to customize our framework according to their observation model, accuracy requirements, and computational budget.

Melissa Adrian, Daniel Sanz-Alonso, Rebecca Willett

Modern data-driven surrogate models for weather forecasting provide accurate short-term predictions but inaccurate and nonphysical long-term forecasts. This paper investigates online weather prediction using machine learning surrogates supplemented with partial and noisy observations. We empirically demonstrate and theoretically justify that, despite the long-time instability of the surrogates and the sparsity of the observations, filtering estimates can remain accurate in the long-time horizon. As a case study, we integrate FourCastNet, a weather surrogate model, within a variational data assimilation framework using partial, noisy ERA5 data. Our results show that filtering estimates remain accurate over a year-long assimilation window and provide effective initial conditions for forecasting tasks, including extreme event prediction.

Xiao Zhang, Ruoxi Jiang, William Gao et al.

We show that introducing a weighting factor to reduce the influence of identity shortcuts in residual networks significantly enhances semantic feature learning in generative representation learning frameworks, such as masked autoencoders (MAEs) and diffusion models. Our modification notably improves feature quality, raising ImageNet-1K K-Nearest Neighbor accuracy from 27.4% to 63.9% and linear probing accuracy from 67.8% to 72.7% for MAEs with a ViT-B/16 backbone, while also enhancing generation quality in diffusion models. This significant gap suggests that, while residual connection structure serves an essential role in facilitating gradient propagation, it may have a harmful side effect of reducing capacity for abstract learning by virtue of injecting an echo of shallower representations into deeper layers. We ameliorate this downside via a fixed formula for monotonically decreasing the contribution of identity connections as layer depth increases. Our design promotes the gradual development of feature abstractions, without impacting network trainability. Analyzing the representations learned by our modified residual networks, we find correlation between low effective feature rank and downstream task performance.

Jake A. Soloff, Rina Foygel Barber, Rebecca Willett

We propose a new framework for algorithmic stability in the context of multiclass classification. In practice, classification algorithms often operate by first assigning a continuous score (for instance, an estimated probability) to each possible label, then taking the maximizer -- i.e., selecting the class that has the highest score. A drawback of this type of approach is that it is inherently unstable, meaning that it is very sensitive to slight perturbations of the training data, since taking the maximizer is discontinuous. Motivated by this challenge, we propose a pipeline for constructing stable classifiers from data, using bagging (i.e., resampling and averaging) to produce stable continuous scores, and then using a stable relaxation of argmax, which we call the "inflated argmax," to convert these scores to a set of candidate labels. The resulting stability guarantee places no distributional assumptions on the data, does not depend on the number of classes or dimensionality of the covariates, and holds for any base classifier. Using a common benchmark data set, we demonstrate that the inflated argmax provides necessary protection against unstable classifiers, without loss of accuracy.

Ruoxi Jiang, Xiao Zhang, Karan Jakhar et al.

Neural PDE solvers offer a powerful tool for modeling complex dynamical systems, but often struggle with error accumulation over long time horizons and maintaining stability and physical consistency. We introduce a multiscale implicit neural emulator that enhances long-term prediction accuracy by conditioning on a hierarchy of lower-dimensional future state representations. Drawing inspiration from the stability properties of numerical implicit time-stepping methods, our approach leverages predictions several steps ahead in time at increasing compression rates for next-timestep refinements. By actively adjusting the temporal downsampling ratios, our design enables the model to capture dynamics across multiple granularities and enforce long-range temporal coherence. Experiments on turbulent fluid dynamics show that our method achieves high short-term accuracy and produces long-term stable forecasts, significantly outperforming autoregressive baselines while adding minimal computational overhead.

Suzanna Parkinson, Greg Ongie, Rebecca Willett et al.

We study depth separation in infinite-width neural networks, where complexity is controlled by the overall squared $\ell_2$-norm of the weights (sum of squares of all weights in the network). Whereas previous depth separation results focused on separation in terms of width, such results do not give insight into whether depth determines if it is possible to learn a network that generalizes well even when the network width is unbounded. Here, we study separation in terms of the sample complexity required for learnability. Specifically, we show that there are functions that are learnable with sample complexity polynomial in the input dimension by norm-controlled depth-3 ReLU networks, yet are not learnable with sub-exponential sample complexity by norm-controlled depth-2 ReLU networks (with any value for the norm). We also show that a similar statement in the reverse direction is not possible: any function learnable with polynomial sample complexity by a norm-controlled depth-2 ReLU network with infinite width is also learnable with polynomial sample complexity by a norm-controlled depth-3 ReLU network.

Ruiting Liang, Jake A. Soloff, Rina Foygel Barber et al.

In this work, we consider ranking problems among a finite set of candidates: for instance, selecting the top-$k$ items among a larger list of candidates or obtaining the full ranking of all items in the set. These problems are often unstable, in the sense that estimating a ranking from noisy data can exhibit high sensitivity to small perturbations. Concretely, if we use data to provide a score for each item (say, by aggregating preference data over a sample of users), then for two items with similar scores, small fluctuations in the data can alter the relative ranking of those items. Many existing theoretical results for ranking problems assume a separation condition to avoid this challenge, but real-world data often contains items whose scores are approximately tied, limiting the applicability of existing theory. To address this gap, we develop a new algorithmic stability framework for ranking problems, and propose two novel ranking operators for achieving stable ranking: the \emph{inflated top-$k$} for the top-$k$ selection problem and the \emph{inflated full ranking} for ranking the full list. To enable stability, each method allows for expressing some uncertainty in the output. For both of these two problems, our proposed methods provide guaranteed stability, with no assumptions on data distributions and no dependence on the total number of candidates to be ranked. Experiments on real-world data confirm that the proposed methods offer stability without compromising the informativeness of the output.

Hannah Laus, Suzanna Parkinson, Vasileios Charisopoulos et al.

Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few measurements generated via a known acquisition procedure. In particular, neural networks perform well empirically but have limited theoretical guarantees. In this work, we study an underdetermined linear inverse problem that admits several possible solution mappings. A standard remedy (e.g., in compressed sensing) establishing uniqueness of the solution mapping is to assume knowledge of latent low-dimensional structure in the source signal. We ask the following question: do deep neural networks adapt to this low-dimensional structure when trained by gradient descent with weight decay regularization? We prove that mildly overparameterized deep linear networks trained in this manner converge to an approximate solution that accurately solves the inverse problem while implicitly encoding latent subspace structure. To our knowledge, this is the first result to rigorously show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data under practical stepsize and weight initialization schemes. Our work highlights that regularization and overparameterization improve generalization, while overparameterization also accelerates convergence during training.

Adela DePavia, Vasileios Charisopoulos, Rebecca Willett

Adaptive optimization algorithms -- such as Adagrad, Adam, and their variants -- have found widespread use in machine learning, signal processing and many other settings. Several methods in this family are not rotationally equivariant, meaning that simple reparameterizations (i.e. change of basis) can drastically affect their convergence. However, their sensitivity to the choice of parameterization has not been systematically studied; it is not clear how to identify a "favorable" change of basis in which these methods perform best. In this paper we propose a reparameterization method and demonstrate both theoretically and empirically its potential to improve their convergence behavior. Our method is an orthonormal transformation based on the expected gradient outer product (EGOP) matrix, which can be approximated using either full-batch or stochastic gradient oracles. We show that for a broad class of functions, the sensitivity of adaptive algorithms to choice-of-basis is influenced by the decay of the EGOP matrix spectrum. We illustrate the potential impact of EGOP reparameterization by presenting empirical evidence and theoretical arguments that common machine learning tasks with "natural" data exhibit EGOP spectral decay.

Xiao Zhang, Ruoxi Jiang, Rebecca Willett et al.

We introduce nested diffusion models, an efficient and powerful hierarchical generative framework that substantially enhances the generation quality of diffusion models, particularly for images of complex scenes. Our approach employs a series of diffusion models to progressively generate latent variables at different semantic levels. Each model in this series is conditioned on the output of the preceding higher-level models, culminating in image generation. Hierarchical latent variables guide the generation process along predefined semantic pathways, allowing our approach to capture intricate structural details while significantly improving image quality. To construct these latent variables, we leverage a pre-trained visual encoder, which learns strong semantic visual representations, and modulate its capacity via dimensionality reduction and noise injection. Across multiple datasets, our system demonstrates significant enhancements in image quality for both unconditional and class/text conditional generation. Moreover, our unconditional generation system substantially outperforms the baseline conditional system. These advancements incur minimal computational overhead as the more abstract levels of our hierarchy work with lower-dimensional representations.

Adela DePavia, Vasileios Charisopoulos, Rebecca Willett

Adaptive gradient methods such as Adam and Adagrad are widely used in machine learning, yet their effect on the generalization of learned models -- relative to methods like gradient descent -- remains poorly understood. Prior work on binary classification suggests that Adam exhibits a ``richness bias,'' which can help it learn nonlinear decision boundaries closer to the Bayes-optimal decision boundary relative to gradient descent. However, the coordinate-wise preconditioning scheme employed by Adam renders the overall method sensitive to orthogonal transformations of feature space. We show that this sensitivity can manifest as a reversal of Adam's competitive advantage: even small rotations of the underlying data distribution can make Adam forfeit its richness bias and converge to a linear decision boundary that is farther from the Bayes-optimal decision boundary than the one learned by gradient descent. To alleviate this issue, we show that a recently proposed reparameterization method -- which applies an orthogonal transformation to the optimization objective -- endows any first-order method with equivariance to data rotations, and we empirically demonstrate its ability to restore Adam's bias towards rich decision boundaries.

Melissa Adrian, Jake A. Soloff, Rebecca Willett

Model selection is the process of choosing from a class of candidate models given data. For instance, methods such as the LASSO and sparse identification of nonlinear dynamics (SINDy) formulate model selection as finding a sparse solution to a linear system of equations determined by training data. However, absent strong assumptions, such methods are highly unstable: if a single data point is removed from the training set, a different model may be selected. In this paper, we present a new approach to stabilizing model selection with theoretical stability guarantees that leverages a combination of bagging and an ''inflated'' argmax operation. Our method selects a small collection of models that all fit the data, and it is stable in that, with high probability, the removal of any training point will result in a collection of selected models that overlaps with the original collection. We illustrate this method in (a) a simulation in which strongly correlated covariates make standard LASSO model selection highly unstable, (b) a Lotka-Volterra model selection problem focused on identifying how competition in an ecosystem influences species' abundances, and (c) a graph subset selection problem using cell-signaling data from proteomics. In these settings, the proposed method yields stable, compact, and accurate collections of selected models, outperforming a variety of benchmarks.

Elena Orlova, Aleksei Ustimenko, Ruoxi Jiang et al.

This paper introduces a novel deep-learning-based approach for numerical simulation of a time-evolving Schrödinger equation inspired by stochastic mechanics and generative diffusion models. Unlike existing approaches, which exhibit computational complexity that scales exponentially in the problem dimension, our method allows us to adapt to the latent low-dimensional structure of the wave function by sampling from the Markovian diffusion. Depending on the latent dimension, our method may have far lower computational complexity in higher dimensions. Moreover, we propose novel equations for stochastic quantum mechanics, resulting in quadratic computational complexity with respect to the number of dimensions. Numerical simulations verify our theoretical findings and show a significant advantage of our method compared to other deep-learning-based approaches used for quantum mechanics.

Suzanna Parkinson, Greg Ongie, Rebecca Willett

Neural networks often operate in the overparameterized regime, in which there are far more parameters than training samples, allowing the training data to be fit perfectly. That is, training the network effectively learns an interpolating function, and properties of the interpolant affect predictions the network will make on new samples. This manuscript explores how properties of such functions learned by neural networks of depth greater than two layers. Our framework considers a family of networks of varying depths that all have the same capacity but different 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 additional linear layers to the input side of a shallow ReLU network yields a representation cost favoring functions with low mixed variation -- that is, it has limited variation in directions orthogonal to a low-dimensional subspace and can be well approximated by a single- or multi-index model. This bias occurs because minimizing the sum of squared weights of the linear layers is equivalent to minimizing a low-rank promoting Schatten quasi-norm of a single "virtual" weight matrix. Our experiments confirm this behavior in standard network training regimes. They additionally show that linear layers can improve generalization and the learned network is well-aligned with the true latent low-dimensional linear subspace when data is generated using a multi-index model.

Greg Ongie, Rebecca Willett

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.

Xiaoxia Wu, Lingxiao Wang, Irina Cristali et al.

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.

Yuming Chen, Daniel Sanz-Alonso, Rebecca Willett

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.

Yinglun Zhu, Dongruo Zhou, Ruoxi Jiang et al.

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.

Hyebin Song, Garvesh Raskutti, Rebecca Willett

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.

Takuya Kurihana, Elisabeth Moyer, Rebecca Willett et al.

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.

Davis Gilton, Gregory Ongie, Rebecca Willett

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.

Davis Gilton, Gregory Ongie, Rebecca Willett

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.

Gregory Ongie, Ajil Jalal, Christopher A. Metzler et al.

Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging. We explore the central prevailing themes of this emerging area and present a taxonomy that can be used to categorize different problems and reconstruction methods. Our taxonomy is organized along two central axes: (1) whether or not a forward model is known and to what extent it is used in training and testing, and (2) whether or not the learning is supervised or unsupervised, i.e., whether or not the training relies on access to matched ground truth image and measurement pairs. We also discuss the trade-offs associated with these different reconstruction approaches, caveats and common failure modes, plus open problems and avenues for future work.

Davis Gilton, Ruotian Luo, Rebecca Willett et al.

This paper presents a framework for the analysis of changes in visual streams: ordered sequences of images, possibly separated by significant time gaps. We propose a new approach to incorporating unlabeled data into training to generate natural language descriptions of change. We also develop a framework for estimating the time of change in visual stream. We use learned representations for change evidence and consistency of perceived change, and combine these in a regularized graph cut based change detector. Experimental evaluation on visual stream datasets, which we release as part of our contribution, shows that representation learning driven by natural language descriptions significantly improves change detection accuracy, compared to methods that do not rely on language.

Lili Zheng, Garvesh Raskutti, Rebecca Willett et al.

High-dimensional autoregressive point processes model how current events trigger or inhibit future events, such as activity by one member of a social network can affect the future activity of his or her neighbors. While past work has focused on estimating the underlying network structure based solely on the times at which events occur on each node of the network, this paper examines the more nuanced problem of estimating context-dependent networks that reflect how features associated with an event (such as the content of a social media post) modulate the strength of influences among nodes. Specifically, we leverage ideas from compositional time series and regularization methods in machine learning to conduct network estimation for high-dimensional marked point processes. Two models and corresponding estimators are considered in detail: an autoregressive multinomial model suited to categorical marks and a logistic-normal model suited to marks with mixed membership in different categories. Importantly, the logistic-normal model leads to a convex negative log-likelihood objective and captures dependence across categories. We provide theoretical guarantees for both estimators, which we validate by simulations and a synthetic data-generating model. We further validate our methods through two real data examples and demonstrate the advantages and disadvantages of both approaches.

Rungang Han, Rebecca Willett, Anru R. Zhang

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.

Greg Ongie, Rebecca Willett, Daniel Soudry et al.

A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ as a single hidden-layer ReLU network with an unbounded number of units (infinite width), but where the Euclidean norm of the weights is bounded, including precisely characterizing which functions can be realized with finite norm. This was settled for univariate univariate functions in Savarese et al. (2019), where it was shown that the required norm is determined by the L1-norm of the second derivative of the function. We extend the characterization to multivariate functions (i.e., networks with d input units), relating the required norm to the L1-norm of the Radon transform of a (d+1)/2-power Laplacian of the function. This characterization allows us to show that all functions in Sobolev spaces $W^{s,1}(\mathbb{R})$, $s\geq d+1$, can be represented with bounded norm, to calculate the required norm for several specific functions, and to obtain a depth separation result. These results have important implications for understanding generalization performance and the distinction between neural networks and more traditional kernel learning.

Davis Gilton, Greg Ongie, Rebecca Willett

Many challenging image processing tasks can be described by an ill-posed linear inverse problem: deblurring, deconvolution, inpainting, compressed sensing, and superresolution all lie in this framework. Traditional inverse problem solvers minimize a cost function consisting of a data-fit term, which measures how well an image matches the observations, and a regularizer, which reflects prior knowledge and promotes images with desirable properties like smoothness. Recent advances in machine learning and image processing have illustrated that it is often possible to learn a regularizer from training data that can outperform more traditional regularizers. We present an end-to-end, data-driven method of solving inverse problems inspired by the Neumann series, which we call a Neumann network. Rather than unroll an iterative optimization algorithm, we truncate a Neumann series which directly solves the linear inverse problem with a data-driven nonlinear regularizer. The Neumann network architecture outperforms traditional inverse problem solution methods, model-free deep learning approaches, and state-of-the-art unrolled iterative methods on standard datasets. Finally, when the images belong to a union of subspaces and under appropriate assumptions on the forward model, we prove there exists a Neumann network configuration that well-approximates the optimal oracle estimator for the inverse problem and demonstrate empirically that the trained Neumann network has the form predicted by theory.

Kwang-Sung Jun, Rebecca Willett, Stephen Wright et al.

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 $\mathbfΘ^*$ that defines the reward, and has low rank $r \ll \min\{d_1,d_2\}$. Determination of $\mathbfΘ^*$ 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.

Benjamin Mark, Garvesh Raskutti, Rebecca Willett

Multivariate Bernoulli autoregressive (BAR) processes model time series of events in which the likelihood of current events is determined by the times and locations of past events. These processes can be used to model nonlinear dynamical systems corresponding to criminal activity, responses of patients to different medical treatment plans, opinion dynamics across social networks, epidemic spread, and more. Past work examines this problem under the assumption that the event data is complete, but in many cases only a fraction of events are observed. Incomplete observations pose a significant challenge in this setting because the unobserved events still govern the underlying dynamical system. In this work, we develop a novel approach to estimating the parameters of a BAR process in the presence of unobserved events via an unbiased estimator of the complete data log-likelihood function. We propose a computationally efficient estimation algorithm which approximates this estimator via Taylor series truncation and establish theoretical results for both the statistical error and optimization error of our algorithm. We further justify our approach by testing our method on both simulated data and a real data set consisting of crimes recorded by the city of Chicago.

Greg Ongie, Daniel Pimentel-Alarcón, Laura Balzano et al.

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.

Yuan Li, Benjamin Mark, Garvesh Raskutti et al.

Sparse models for high-dimensional linear regression and machine learning have received substantial attention over the past two decades. Model selection, or determining which features or covariates are the best explanatory variables, is critical to the interpretability of a learned model. Much of the current literature assumes that covariates are only mildly correlated. However, in many modern applications covariates are highly correlated and do not exhibit key properties (such as the restricted eigenvalue condition, restricted isometry property, or other related assumptions). This work considers a high-dimensional regression setting in which a graph governs both correlations among the covariates and the similarity among regression coefficients -- meaning there is \emph{alignment} between the covariates and regression coefficients. Using side information about the strength of correlations among features, we form a graph with edge weights corresponding to pairwise covariances. This graph is used to define a graph total variation regularizer that promotes similar weights for correlated features. This work shows how the proposed graph-based regularization yields mean-squared error guarantees for a broad range of covariance graph structures. These guarantees are optimal for many specific covariance graphs, including block and lattice graphs. Our proposed approach outperforms other methods for highly-correlated design in a variety of experiments on synthetic data and real biochemistry data.

Amin Jalali, Rebecca Willett

High-dimensional time series data exist in numerous areas such as finance, genomics, healthcare, and neuroscience. An unavoidable aspect of all such datasets is missing data, and dealing with this issue has been an important focus in statistics, control, and machine learning. In this work, we consider a high-dimensional estimation problem where a dynamical system, governed by a stable vector autoregressive model, is randomly and only partially observed at each time point. Our task amounts to estimating the transition matrix, which is assumed to be sparse. In such a scenario, where covariates are highly interdependent and partially missing, new theoretical challenges arise. While transition matrix estimation in vector autoregressive models has been studied previously, the missing data scenario requires separate efforts. Moreover, while transition matrix estimation can be studied from a high-dimensional sparse linear regression perspective, the covariates are highly dependent and existing results on regularized estimation with missing data from i.i.d.~covariates are not applicable. At the heart of our analysis lies 1) a novel concentration result when the innovation noise satisfies the convex concentration property, as well as 2) a new quantity for characterizing the interactions of the time-varying observation process with the underlying dynamical system.

Benjamin Mark, Garvesh Raskutti, Rebecca Willett

Consider observing a collection of discrete events within a network that reflect how network nodes influence one another. Such data are common in spike trains recorded from biological neural networks, interactions within a social network, and a variety of other settings. Data of this form may be modeled as self-exciting point processes, in which the likelihood of future events depends on the past events. This paper addresses the problem of estimating self-excitation parameters and inferring the underlying functional network structure from self-exciting point process data. Past work in this area was limited by strong assumptions which are addressed by the novel approach here. Specifically, in this paper we (1) incorporate saturation in a point process model which both ensures stability and models non-linear thresholding effects; (2) impose general low-dimensional structural assumptions that include sparsity, group sparsity and low-rankness that allows bounds to be developed in the high-dimensional setting; and (3) incorporate long-range memory effects through moving average and higher-order auto-regressive components. Using our general framework, we provide a number of novel theoretical guarantees for high-dimensional self-exciting point processes that reflect the role played by the underlying network structure and long-term memory. We also provide simulations and real data examples to support our methodology and main results.

Kwang-Sung Jun, Francesco Orabona, Stephen Wright et al.

A key challenge in online learning is that classical algorithms can be slow to adapt to changing environments. Recent studies have proposed "meta" algorithms that convert any online learning algorithm to one that is adaptive to changing environments, where the adaptivity is analyzed in a quantity called the strongly-adaptive regret. This paper describes a new meta algorithm that has a strongly-adaptive regret bound that is a factor of $\sqrt{\log(T)}$ better than other algorithms with the same time complexity, where $T$ is the time horizon. We also extend our algorithm to achieve a first-order (i.e., dependent on the observed losses) strongly-adaptive regret bound for the first time, to our knowledge. At its heart is a new parameter-free algorithm for the learning with expert advice (LEA) problem in which experts sometimes do not output advice for consecutive time steps (i.e., \emph{sleeping} experts). This algorithm is derived by a reduction from optimal algorithms for the so-called coin betting problem. Empirical results show that our algorithm outperforms state-of-the-art methods in both learning with expert advice and metric learning scenarios.

Zachary Charles, Amin Jalali, Rebecca Willett

Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process of clustering the points according to their subspace and identifying the subspaces. One popular approach, sparse subspace clustering (SSC), represents each sample as a weighted combination of the other samples, with weights of minimal $\ell_1$ norm, and then uses those learned weights to cluster the samples. SSC is stable in settings where each sample is contaminated by a relatively small amount of noise. However, when there is a significant amount of additive noise, or a considerable number of entries are missing, theoretical guarantees are scarce. In this paper, we study a robust variant of SSC and establish clustering guarantees in the presence of corrupted or missing data. We give explicit bounds on amount of noise and missing data that the algorithm can tolerate, both in deterministic settings and in a random generative model. Notably, our approach provides guarantees for higher tolerance to noise and missing data than existing analyses for this method. By design, the results hold even when we do not know the locations of the missing data; e.g., as in presence-only data.

Kwang-Sung Jun, Aniruddha Bhargava, Robert Nowak et al.

Generalized Linear Bandits (GLBs), a natural extension of the stochastic linear bandits, has been popular and successful in recent years. However, existing GLBs scale poorly with the number of rounds and the number of arms, limiting their utility in practice. This paper proposes new, scalable solutions to the GLB problem in two respects. First, unlike existing GLBs, whose per-time-step space and time complexity grow at least linearly with time $t$, we propose a new algorithm that performs online computations to enjoy a constant space and time complexity. At its heart is a novel Generalized Linear extension of the Online-to-confidence-set Conversion (GLOC method) that takes \emph{any} online learning algorithm and turns it into a GLB algorithm. As a special case, we apply GLOC to the online Newton step algorithm, which results in a low-regret GLB algorithm with much lower time and memory complexity than prior work. Second, for the case where the number $N$ of arms is very large, we propose new algorithms in which each next arm is selected via an inner product search. Such methods can be implemented via hashing algorithms (i.e., "hash-amenable") and result in a time complexity sublinear in $N$. While a Thompson sampling extension of GLOC is hash-amenable, its regret bound for $d$-dimensional arm sets scales with $d^{3/2}$, whereas GLOC's regret bound scales with $d$. Towards closing this gap, we propose a new hash-amenable algorithm whose regret bound scales with $d^{5/4}$. Finally, we propose a fast approximate hash-key computation (inner product) with a better accuracy than the state-of-the-art, which can be of independent interest. We conclude the paper with preliminary experimental results confirming the merits of our methods.