In this paper, we study the gradient descent dynamics for jointly training both layers of a one-hidden-layer ReLU network to fit a linear target function. Concretely, we consider a realizable setting where inputs are drawn i.i.d. from a Gaussian distribution and labels follow a planted linear model. This stylized framework captures salient features of end-to-end training in inverse problems and certain auto-encoder models. Despite its apparent simplicity, the dynamics remain poorly understood, in part because the loss landscape contains multiple non-strict saddle points, making it unclear why gradient descent from random initialization reliably escapes bad stationary regions. We provide a detailed characterization of the optimization landscape and prove that gradient descent from a moderately small random initialization-simultaneously training both layers-converges to a global minimizer at a linear rate with order-wise optimal sample complexity. Our analysis tracks the trajectory through three phases: an alignment phase in which hidden weights progressively align with the planted direction while the output weights maintain the correct sign pattern; a growth phase in which the norms of both layers increase while preserving alignment; and a local refinement phase in which the aligned neurons rapidly converge to the planted direction, yielding fast local convergence. To rigorously show that GD avoids non-strict saddles, we develop trajectory-level control arguments for the end-to-end dynamics. In addition, we establish novel uniform concentration results that hold along the entire trajectory, and are essential for obtaining order-wise optimal sample complexity. We corroborate our theory with extensive experiments across a range of configurations.

In computed tomography (CT), the forward model consists of a linear Radon transform followed by an exponential nonlinearity based on the attenuation of light according to the Beer–Lambert Law. Conventional reconstruction often involves inverting this nonlinearity and then solving a linear inverse problem. However, this nonlinear measurement preprocessing is poorly conditioned in the vicinity of high-density materials, such as metal. This preprocessing makes CT reconstruction methods numerically sensitive and susceptible to artifacts near high-density regions. In this paper, we study a technique where the signal is directly reconstructed from raw measurements through the nonlinear forward model. Though this optimization is nonconvex, we show that gradient descent provably converges to the global optimum at a geometric rate, perfectly reconstructing the underlying signal with a near minimal number of random measurements. We also prove similar results in the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal. This is achieved by enforcing prior structural information about the signal through constraints on the optimization variables. We illustrate the benefits of direct nonlinear CT reconstruction with cone-beam CT experiments on synthetic and real 3D volumes, in which metal artifacts are reduced compared to standard linear reconstruction methods. Our experiments also demonstrate that logarithmic preprocessing alone is sufficient to produce metal artifacts, even in the absence of other causes such as beam hardening.

We address the problem of learning a binary classifier given partially labeled data where all labeled samples come from only one of the classes, commonly known as Positive Unlabeled (PU) learning. Classical methods such as clustering, out-of-distribution detection, and positive density estimation, while effective in low-dimensional scenarios, lose their efficacy as the dimensionality of data increases, because of the increasing complexity. This has led to the development of methods that address the problem in high-dimensional spaces; however, many of these methods are also impacted by the increased complexity inherent in high-dimensional data. The contribution of this paper is the learning of a neural network-based data representation by employing a loss function that enables the projection of unlabeled data into two distinct clusters - positive and negative - facilitating their identification through basic clustering techniques and mirroring the simplicity of the problem seen in low-dimensional settings. We further enhance this separation of unlabeled data clusters by implementing a vector quantization strategy. Our experimental results on benchmarking PU datasets validate the superiority of our method over existing state-of-the-art techniques. Additionally, we provide theoretical justification to support our cluster-based approach and algorithmic choices.

Multimodal Large Language Models (MLLMs) have tremendous potential to improve the accuracy, availability, and cost-effectiveness of healthcare by providing automated solutions or serving as aids to medical professionals. Despite promising first steps in developing medical MLLMs in the past few years, their capabilities and limitations are not well understood. Recently, many benchmark datasets have been proposed that test the general medical knowledge of such models across a variety of medical areas. However, the systematic failure modes and vulnerabilities of such models are severely underexplored with most medical benchmarks failing to expose the shortcomings of existing models in this safety-critical domain. In this paper, we introduce MediConfusion, a challenging medical Visual Question Answering (VQA) benchmark dataset, that probes the failure modes of medical MLLMs from a vision perspective. We reveal that state-of-the-art models are easily confused by image pairs that are otherwise visually dissimilar and clearly distinct for medical experts. Strikingly, all available models (open-source or proprietary) achieve performance below random guessing on MediConfusion, raising serious concerns about the reliability of existing medical MLLMs for healthcare deployment. We also extract common patterns of model failure that may help the design of a new generation of more trustworthy and reliable MLLMs in healthcare.

Test-time training (TTT) methods explicitly update the weights of a model to adapt to the specific test instance, and they have found success in a variety of settings, including most recently language modeling and reasoning. To demystify this success, we investigate a gradient-based TTT algorithm for in-context learning, where we train a transformer model on the in-context demonstrations provided in the test prompt. Specifically, we provide a comprehensive theoretical characterization of linear transformers when the update rule is a single gradient step. Our theory (i) delineates the role of alignment between pretraining distribution and target task, (ii) demystifies how TTT can alleviate distribution shift, and (iii) quantifies the sample complexity of TTT including how it can significantly reduce the eventual sample size required for in-context learning. As our empirical contribution, we study the benefits of TTT for TabPFN, a tabular foundation model. In line with our theory, we demonstrate that TTT significantly reduces the required sample size for tabular classification (3 to 5 times fewer) unlocking substantial inference efficiency with a negligible training cost.

Mental visualization, the ability to construct and manipulate visual representations internally, is a core component of human cognition and plays a vital role in tasks involving reasoning, prediction, and abstraction. Despite the rapid progress of Multimodal Large Language Models (MLLMs), current benchmarks primarily assess passive visual perception, offering limited insight into the more active capability of internally constructing visual patterns to support problem solving. Yet mental visualization is a critical cognitive skill in humans, supporting abilities such as spatial navigation, predicting physical trajectories, and solving complex visual problems through imaginative simulation. To bridge this gap, we introduce Hyperphantasia, a synthetic benchmark designed to evaluate the mental visualization abilities of MLLMs through four carefully constructed puzzles. Each task is procedurally generated and presented at three difficulty levels, enabling controlled analysis of model performance across increasing complexity. Our comprehensive evaluation of state-of-the-art models reveals a substantial gap between the performance of humans and MLLMs. Additionally, we explore the potential of reinforcement learning to improve visual simulation capabilities. Our findings suggest that while some models exhibit partial competence in recognizing visual patterns, robust mental visualization remains an open challenge for current MLLMs.

Diffusion models have become the go-to method for text-to-image generation, producing high-quality images from pure noise. However, the inner workings of diffusion models is still largely a mystery due to their black-box nature and complex, multi-step generation process. Mechanistic interpretability techniques, such as Sparse Autoencoders (SAEs), have been successful in understanding and steering the behavior of large language models at scale. However, the great potential of SAEs has not yet been applied toward gaining insight into the intricate generative process of diffusion models. In this work, we leverage the SAE framework to probe the inner workings of a popular text-to-image diffusion model, and uncover a variety of human-interpretable concepts in its activations. Interestingly, we find that *even before the first reverse diffusion step* is completed, the final composition of the scene can be predicted surprisingly well by looking at the spatial distribution of activated concepts. Moreover, going beyond correlational analysis, we design intervention techniques aimed at manipulating image composition and style, and demonstrate that (1) in early stages of diffusion image composition can be effectively controlled, (2) in the middle stages image composition is finalized, however stylistic interventions are effective, and (3) in the final stages only minor textural details are subject to change.

Adam is the de facto optimization algorithm for several deep learning applications, but an understanding of its implicit bias and how it differs from other algorithms, particularly standard first-order methods such as (stochastic) gradient descent (GD), remains limited. In practice, neural networks (NNs) trained with SGD are known to exhibit simplicity bias --- a tendency to find simple solutions. In contrast, we show that Adam is more resistant to such simplicity bias. First, we investigate the differences in the implicit biases of Adam and GD when training two-layer ReLU NNs on a binary classification task with Gaussian data. We find that GD exhibits a simplicity bias, resulting in a linear decision boundary with a suboptimal margin, whereas Adam leads to much richer and more diverse features, producing a nonlinear boundary that is closer to the Bayes' optimal predictor. This richer decision boundary also allows Adam to achieve higher test accuracy both in-distribution and under certain distribution shifts. We theoretically prove these results by analyzing the population gradients. Next, to corroborate our theoretical findings, we present extensive empirical results showing that this property of Adam leads to superior generalization across various datasets with spurious correlations where NNs trained with SGD are known to show simplicity bias and do not generalize well under certain distributional shifts.

Recently, there has been significant progress in understanding the convergence and generalization properties of gradient-based methods for training overparameterized learning models. However, many aspects including the role of small random initialization and how the various parameters of the model are coupled during gradient-based updates to facilitate good generalization remain largely mysterious. A series of recent papers have begun to study this role for non-convex formulations of symmetric Positive Semi-Definite (PSD) matrix sensing problems which involve reconstructing a low-rank PSD matrix from a few linear measurements. The underlying symmetry/PSDness is crucial to existing convergence and generalization guarantees for this problem. In this paper, we study a general overparameterized low-rank matrix sensing problem where one wishes to reconstruct an asymmetric rectangular low-rank matrix from a few linear measurements. We prove that an overparameterized model trained via factorized gradient descent converges to the low-rank matrix generating the measurements. We show that in this setting, factorized gradient descent enjoys two implicit properties: (1) coupling of the trajectory of gradient descent where the factors are coupled in various ways throughout the gradient update trajectory and (2) an algorithmic regularization property where the iterates show a propensity towards low-rank models despite the overparameterized nature of the factorized model. These two implicit properties in turn allow us to show that the gradient descent trajectory from small random initialization moves towards solutions that are both globally optimal and generalize well.

An increasingly popular machine learning paradigm is to pretrain a neural network (NN) on many tasks offline, then adapt it to downstream tasks, often by re-training only the last linear layer of the network. This approach yields strong downstream performance in a variety of contexts, demonstrating that multitask pretraining leads to effective feature learning. Although several recent theoretical studies have shown that shallow NNs learn meaningful features when either (i) they are trained on a *single* task or (ii) they are *linear*, very little is known about the closer-to-practice case of *nonlinear* NNs trained on *multiple* tasks. In this work, we present the first results proving that feature learning occurs during training with a nonlinear model on multiple tasks. Our key insight is that multi-task pretraining induces a pseudo-contrastive loss that favors representations that align points that typically have the same label across tasks. Using this observation, we show that when the tasks are binary classification tasks with labels depending on the projection of the data onto an $r$-dimensional subspace within the $d\gg r$-dimensional input space, a simple gradient-based multitask learning algorithm on a two-layer ReLU NN recovers this projection, allowing for generalization to downstream tasks with sample and neuron complexity independent of $d$. In contrast, we show that with high probability over the draw of a single task, training on this single task cannot guarantee to learn all $r$ ground-truth features.

Inverse problems arise in a multitude of applications, where the goal is to recover a clean signal from noisy and possibly (non)linear observations. The difficulty of a reconstruction problem depends on multiple factors, such as the ground truth signal structure, the severity of the degradation and the complex interactions between the above. This results in natural sample-by-sample variation in the difficulty of a reconstruction problem. Our key observation is that most existing inverse problem solvers lack the ability to adapt their compute power to the difficulty of the reconstruction task, resulting in subpar performance and wasteful resource allocation. We propose a novel method, severity encoding, to estimate the degradation severity of corrupted signals in the latent space of an autoencoder. We show that the estimated severity has strong correlation with the true corruption level and can provide useful hints on the difficulty of reconstruction problems on a sample-by-sample basis. Furthermore, we propose a reconstruction method based on latent diffusion models that leverages the predicted degradation severities to fine-tune the reverse diffusion sampling trajectory and thus achieve sample-adaptive inference times. Our framework, Flash-Diffusion, acts as a wrapper that can be combined with any latent diffusion-based baseline solver, imbuing it with sample-adaptivity and acceleration. We perform experiments on both linear and nonlinear inverse problems and demonstrate that our technique greatly improves the performance of the baseline solver and achieves up to $10\times$ acceleration in mean sampling speed.

Diffusion models have established new state of the art in a multitude of computer vision tasks, including image restoration. Diffusion-based inverse problem solvers generate reconstructions of exceptional visual quality from heavily corrupted measurements. However, in what is widely known as the perception-distortion trade-off, the price of perceptually appealing reconstructions is often paid in declined distortion metrics, such as PSNR. Distortion metrics measure faithfulness to the observation, a crucial requirement in inverse problems. In this work, we propose a novel framework for inverse problem solving, namely we assume that the observation comes from a stochastic degradation process that gradually degrades and noises the original clean image. We learn to reverse the degradation process in order to recover the clean image. Our technique maintains consistency with the original measurement throughout the reverse process, and allows for great flexibility in trading off perceptual quality for improved distortion metrics and sampling speedup via early-stopping. We demonstrate the efficiency of our method on different high-resolution datasets and inverse problems, achieving great improvements over other state-of-the-art diffusion-based methods with respect to both perceptual and distortion metrics.

We initiate a study of supervised learning from many independent sequences ("trajectories") of non-independent covariates, reflecting tasks in sequence modeling, control, and reinforcement learning. Conceptually, our multi-trajectory setup sits between two traditional settings in statistical learning theory: learning from independent examples and learning from a single auto-correlated sequence. Our conditions for efficient learning generalize the former setting---trajectories must be non-degenerate in ways that extend standard requirements for independent examples. Notably, we do not require that trajectories be ergodic, long, nor strictly stable. For linear least-squares regression, given $n$-dimensional examples produced by $m$ trajectories, each of length $T$, we observe a notable change in statistical efficiency as the number of trajectories increases from a few (namely $m \lesssim n$) to many (namely $m \gtrsim n$). Specifically, we establish that the worst-case error rate of this problem is $\Theta(n / m T)$ whenever $m \gtrsim n$. Meanwhile, when $m \lesssim n$, we establish a (sharp) lower bound of $\Omega(n^2 / m^2 T)$ on the worst-case error rate, realized by a simple, marginally unstable linear dynamical system. A key upshot is that, in domains where trajectories regularly reset, the error rate eventually behaves as if all of the examples were independent, drawn from their marginals. As a corollary of our analysis, we also improve guarantees for the linear system identification problem.

Recently, there has been significant progress in understanding the convergence and generalization properties of gradient-based methods for training overparameterized learning models. However, many aspects including the role of small random initialization and how the various parameters of the model are coupled during gradient-based updates to facilitate good generalization, remain largely mysterious. A series of recent papers have begun to study this role for non-convex formulations of symmetric Positive Semi-Definite (PSD) matrix sensing problems which involve reconstructing a low-rank PSD matrix from a few linear measurements. The underlying symmetry/PSDness is crucial to existing convergence and generalization guarantees for this problem. In this paper, we study a general overparameterized low-rank matrix sensing problem where one wishes to reconstruct an asymmetric rectangular low-rank matrix from a few linear measurements. We prove that an overparameterized model trained via factorized gradient descent converges to the low-rank matrix generating the measurements. We show that in this setting, factorized gradient descent enjoys two implicit properties: (1) coupling of the trajectory of gradient descent where the factors are coupled in various ways throughout the gradient update trajectory and (2) an algorithmic regularization property where the iterates show a propensity towards low-rank models despite the overparameterized nature of the factorized model. These two implicit properties in turn allow us to show that the gradient descent trajectory from small random initialization moves towards solutions that are both globally optimal and generalize well.

Large-scale training of modern deep learning models heavily relies on publicly available data on the web. This potentially unauthorized usage of online data leads to concerns regarding data privacy. Recent works aim to make unlearnable data for deep learning models by adding small, specially designed noises to tackle this issue. However, these methods are vulnerable to adversarial training (AT) and/or are computationally heavy. In this work, we propose a novel, model-free, Convolution-based Unlearnable DAtaset (CUDA) generation technique. CUDA is generated using controlled class-wise convolutions with filters that are randomly generated via a private key. CUDA encourages the network to learn the relation between filters and labels rather than informative features for classifying the clean data. We develop some theoretical analysis demonstrating that CUDA can successfully poison Gaussian mixture data by reducing the clean data performance of the optimal Bayes classifier. We also empirically demonstrate the effectiveness of CUDA with various datasets (CIFAR-10, CIFAR-100, ImageNet-100, and Tiny-ImageNet), and architectures (ResNet-18, VGG-16, Wide ResNet-34-10, DenseNet-121, DeIT, EfficientNetV2-S, and MobileNetV2). Our experiments show that CUDA is robust to various data augmentations and training approaches such as smoothing, AT with different budgets, transfer learning, and fine-tuning. For instance, training a ResNet-18 on ImageNet-100 CUDA achieves only 8.96%, 40.08%, and 20.58% clean test accuracies with empirical risk minimization (ERM), L∞AT, and L2 AT, respectively. Here, ERM on the clean training data achieves a clean test accuracy of 80.66%. CUDA exhibits unlearnability effect with ERM even when only a fraction of the training dataset is perturbed. Furthermore, we also show that CUDA is robust to adaptive defenses designed specifically to break it.

Prompt-tuning is an emerging strategy to adapt large language models (LLM) to downstream tasks by learning a (soft-)prompt parameter from data. Despite its success in LLMs, there is limited theoretical understanding of the power of prompt-tuning and the role of the attention mechanism in prompting. In this work, we explore prompt-tuning for one-layer attention architectures and study contextual mixture-models where each input token belongs to a context-relevant or -irrelevant set. We isolate the role of prompt-tuning through a self-contained prompt-attention model. Our contributions are as follows: (1) We show that softmax-prompt-attention is provably more expressive than softmax-self-attention and linear-prompt-attention under our contextual data model. (2) We analyze the initial trajectory of gradient descent and show that it learns the prompt and prediction head with near-optimal sample complexity and demonstrate how the prompt can provably attend to sparse context-relevant tokens. (3) Assuming a known prompt but an unknown prediction head, we characterize the exact finite sample performance of prompt-attention which reveals the fundamental performance limits and the precise benefit of the context information. We also provide experiments that verify our theoretical insights on real datasets and demonstrate how prompt-tuning enables the model to attend to context-relevant information.

Deep learning models often suffer from forgetting previously learned information when trained on new data. This problem is exacerbated in federated learning (FL), where the data is distributed and can change independently for each user. Many solutions are proposed to resolve this catastrophic forgetting in a centralized setting. However, they do not apply directly to FL because of its unique complexities, such as privacy concerns and resource limitations. To overcome these challenges, this paper presents a framework for \textbf{federated class incremental learning} that utilizes a generative model to synthesize samples from past distributions. This data can be later exploited alongside the training data to mitigate catastrophic forgetting. To preserve privacy, the generative model is trained on the server using data-free methods at the end of each task without requesting data from clients. Moreover, our solution does not demand the users to store old data or models, which gives them the freedom to join/leave the training at any time. Additionally, we introduce SuperImageNet, a new regrouping of the ImageNet dataset specifically tailored for federated continual learning. We demonstrate significant improvements compared to existing baselines through extensive experiments on multiple datasets.

Deep neural networks provide excellent performance for inverse problems such as denoising. However, neural networks can be sensitive to adversarial or worst-case perturbations. This raises the question of whether such networks can be trained efficiently to be worst-case robust. In this paper, we investigate whether jittering, a simple regularization technique that adds isotropic Gaussian noise during training, is effective for learning worst-case robust estimators for inverse problems. While well studied for prediction in classification tasks, the effectiveness of jittering for inverse problems has not been systematically investigated. In this paper, we present a novel analytical characterization of the optimal $\ell_2$-worst-case robust estimator for linear denoising and show that jittering yields optimal robust denoisers. Furthermore, we examine jittering empirically via training deep neural networks (U-nets) for natural image denoising, deconvolution, and accelerated magnetic resonance imaging (MRI). The results show that jittering significantly enhances the worst-case robustness, but can be suboptimal for inverse problems beyond denoising. Moreover, our results imply that training on real data which often contains slight noise is somewhat robustness enhancing.

Significant theoretical work has established that in specific regimes, neural networks trained by gradient descent behave like kernel methods. However, in practice, it is known that neural networks strongly outperform their associated kernels. In this work, we explain this gap by demonstrating that there is a large class of functions which cannot be efficiently learned by kernel methods but can be easily learned with gradient descent on a two layer neural network outside the kernel regime by learning representations that are relevant to the target task. We also demonstrate that these representations allow for efficient transfer learning, which is impossible in the kernel regime. Specifically, we consider the problem of learning polynomials which depend on only a few relevant directions, i.e. of the form $f^\star(x) = g(Ux)$ where $U: \R^d \to \R^r$ with $d \gg r$. When the degree of $f^\star$ is $p$, it is known that $n \asymp d^p$ samples are necessary to learn $f^\star$ in the kernel regime. Our primary result is that gradient descent learns a representation of the data which depends only on the directions relevant to $f^\star$. This results in an improved sample complexity of $n\asymp d^2$ and enables transfer learning with sample complexity independent of $d$.

Active learning aims to reduce the cost of labeling through selective sampling. Despite reported empirical success over passive learning, many popular active learning heuristics such as uncertainty sampling still lack satisfying theoretical guarantees. Towards closing the gap between practical use and theoretical understanding in active learning, we propose to characterize the exact behavior of uncertainty sampling for high-dimensional Gaussian mixture data, in a modern regime of big data where the numbers of samples and features are commensurately large. Through a sharp characterization of the learning results, our analysis sheds light on the important question of when uncertainty sampling works better than passive learning. Our results show that the effectiveness of uncertainty sampling is not always ensured. In fact it depends crucially on the choice of i) an adequate initial classifier used to start the active sampling process and ii) a proper loss function that allows an adaptive treatment of samples queried at various steps.

Modern machine learning models require a large amount of labeled data for training to perform well. A recently emerging paradigm for reducing the reliance of large model training on massive labeled data is to take advantage of abundantly available labeled data from a related source task to boost the performance of the model in a desired target task where there may not be a lot of data available. This approach, which is called transfer learning, has been applied successfully in many application domains. However, despite the fact that many transfer learning algorithms have been developed, the fundamental understanding of "when" and "to what extent" transfer learning can reduce sample complexity is still limited. In this work, we take a step towards foundational understanding of transfer learning by focusing on binary classification with linear models and Gaussian features and develop statistical minimax lower bounds in terms of the number of source and target samples and an appropriate notion of similarity between source and target tasks. To derive this bound, we reduce the transfer learning problem to hypothesis testing via constructing a packing set of source and target parameters by exploiting Gilbert-Varshamov bound, which in turn leads to a lower bound on sample complexity. We also evaluate our theoretical results by experiments on real data sets.

We explore the connection between outlier-robust high-dimensional statistics and non-convex optimization in the presence of sparsity constraints, with a focus on the fundamental tasks of robust sparse mean estimation and robust sparse PCA. We develop novel and simple optimization formulations for these problems such that any approximate stationary point of the associated optimization problem yields a near-optimal solution for the underlying robust estimation task. As a corollary, we obtain that any first-order method that efficiently converges to stationarity yields an efficient algorithm for these tasks. The obtained algorithms are simple, practical, and succeed under broader distributional assumptions compared to prior work.

In accelerated MRI reconstruction, the anatomy of a patient is recovered from a set of undersampled and noisy measurements. Deep learning approaches have been proven to be successful in solving this ill-posed inverse problem and are capable of producing very high quality reconstructions. However, current architectures heavily rely on convolutions, that are content-independent and have difficulties modeling long-range dependencies in images. Recently, Transformers, the workhorse of contemporary natural language processing, have emerged as powerful building blocks for a multitude of vision tasks. These models split input images into non-overlapping patches, embed the patches into lower-dimensional tokens and utilize a self-attention mechanism that does not suffer from the aforementioned weaknesses of convolutional architectures. However, Transformers incur extremely high compute and memory cost when 1) the input image resolution is high and 2) when the image needs to be split into a large number of patches to preserve fine detail information, both of which are typical in low-level vision problems such as MRI reconstruction, having a compounding effect. To tackle these challenges, we propose HUMUS-Net, a hybrid architecture that combines the beneficial implicit bias and efficiency of convolutions with the power of Transformer blocks in an unrolled and multi-scale network. HUMUS-Net extracts high-resolution features via convolutional blocks and refines low-resolution features via a novel Transformer-based multi-scale feature extractor. Features from both levels are then synthesized into a high-resolution output reconstruction. Our network establishes new state of the art on the largest publicly available MRI dataset, the fastMRI dataset. We further demonstrate the performance of HUMUS-Net on two other popular MRI datasets and perform fine-grained ablation studies to validate our design.

The size of Transformer models is growing at an unprecedented rate. It has taken less than one year to reach trillion-level parameters since the release of GPT-3 (175B). Training such models requires both substantial engineering efforts and enormous computing resources, which are luxuries most research teams cannot afford. In this paper, we propose PipeTransformer, which leverages automated elastic pipelining for efficient distributed training of Transformer models. In PipeTransformer, we design an adaptive on the fly freeze algorithm that can identify and freeze some layers gradually during training, and an elastic pipelining system that can dynamically allocate resources to train the remaining active layers. More specifically, PipeTransformer automatically excludes frozen layers from the pipeline, packs active layers into fewer GPUs, and forks more replicas to increase data-parallel width. We evaluate PipeTransformer using Vision Transformer (ViT) on ImageNet and BERT on SQuAD and GLUE datasets. Our results show that compared to the state-of-the-art baseline, PipeTransformer attains up to 2.83-fold speedup without losing accuracy. We also provide various performance analyses for a more comprehensive understanding of our algorithmic and system-wise design. Finally, we have modularized our training system with flexible APIs and made the source code publicly available at https://DistML.ai.

Deep neural networks have emerged as very successful tools for image restoration and reconstruction tasks. These networks are often trained end-to-end to directly reconstruct an image from a noisy or corrupted measurement of that image. To achieve state-of-the-art performance, training on large and diverse sets of images is considered critical. However, it is often difficult and/or expensive to collect large amounts of training images. Inspired by the success of Data Augmentation (DA) for classification problems, in this paper, we propose a pipeline for data augmentation for accelerated MRI reconstruction and study its effectiveness at reducing the required training data in a variety of settings. Our DA pipeline, MRAugment, is specifically designed to utilize the invariances present in medical imaging measurements as naive DA strategies that neglect the physics of the problem fail. Through extensive studies on multiple datasets we demonstrate that in the low-data regime DA prevents overfitting and can match or even surpass the state of the art while using significantly fewer training data, whereas in the high-data regime it has diminishing returns. Furthermore, our findings show that DA improves the robustness of the model against various shifts in the test distribution.

Neural Architecture Search (NAS) is a popular method for automatically designing optimized deep-learning architectures. NAS methods commonly use bilevel optimization where one optimizes the weights over the training data (lower-level problem) and hyperparameters - such as the architecture - over the validation data (upper-level problem). This paper explores the statistical aspects of such problems with train-validation splits. In practice, the lower-level problem is often overparameterized and can easily achieve zero loss. Thus, a-priori, it seems impossible to distinguish the right hyperparameters based on training loss alone which motivates a better understanding of train-validation split. To this aim, we first show that refined properties of the validation loss such as risk and hyper-gradients are indicative of those of the true test loss and help prevent overfitting with a near-minimal validation sample size. Importantly, this is established for continuous search spaces which are relevant for differentiable search schemes. We then establish generalization bounds for NAS problems with an emphasis on an activation search problem and gradient-based methods. Finally, we show rigorous connections between NAS and low-rank matrix learning which leads to algorithmic insights where the solution of the upper problem can be accurately learned via spectral methods to achieve near-minimal risk.

Recently there has been significant theoretical progress on understanding the convergence and generalization of gradient-based methods on nonconvex losses with overparameterized models. Nevertheless, many aspects of optimization and generalization and in particular the critical role of small random initialization are not fully understood. In this paper, we take a step towards demystifying this role by proving that small random initialization followed by a few iterations of gradient descent behaves akin to popular spectral methods. We also show that this implicit spectral bias from small random initialization, which is provably more prominent for overparameterized models, also puts the gradient descent iterations on a particular trajectory towards solutions that are not only globally optimal but also generalize well. Concretely, we focus on the problem of reconstructing a low-rank matrix from a few measurements via a natural nonconvex formulation. In this setting, we show that the trajectory of the gradient descent iterations from small random initialization can be approximately decomposed into three phases: (I) a spectral or alignment phase where we show that that the iterates have an implicit spectral bias akin to spectral initialization allowing us to show that at the end of this phase the column space of the iterates and the underlying low-rank matrix are sufficiently aligned, (II) a saddle avoidance/refinement phase where we show that the trajectory of the gradient iterates moves away from certain degenerate saddle points, and (III) a local refinement phase where we show that after avoiding the saddles the iterates converge quickly to the underlying low-rank matrix. Underlying our analysis are insights for the analysis of overparameterized nonconvex optimization schemes that may have implications for computational problems beyond low-rank reconstruction.

Modern neural networks are typically trained in an over-parameterized regime where the parameters of the model far exceed the size of the training data. Such neural networks in principle have the capacity to (over)fit any set of labels including significantly corrupted ones. Despite this (over)fitting capacity in this paper we demonstrate that such overparameterized networks have an intriguing robustness capability: they are surprisingly robust to label noise when first order methods with early stopping is used to train them. This paper also takes a step towards demystifying this phenomena. Under a rich dataset model, we show that gradient descent is provably robust to noise/corruption on a constant fraction of the labels. In particular, we prove that: (i) In the first few iterations where the updates are still in the vicinity of the initialization gradient descent only fits to the correct labels essentially ignoring the noisy labels. (ii) To start to overfit to the noisy labels network must stray rather far from the initialization which can only occur after many more iterations. Together, these results show that gradient descent with early stopping is provably robust to label noise and shed light on the empirical robustness of deep networks as well as commonly adopted heuristics to prevent overfitting.

We consider the fundamental problem of ReLU regression, where the goal is to output the best fitting ReLU with respect to square loss given access to draws from some unknown distribution. We give the first efficient, constant-factor approximation algorithm for this problem assuming the underlying distribution satisfies some weak concentration and anti-concentration conditions (and includes, for example, all log-concave distributions). This solves the main open problem of Goel et al., who proved hardness results for any exact algorithm for ReLU regression (up to an additive $\epsilon$). Using more sophisticated techniques, we can improve our results and obtain a polynomial-time approximation scheme for any subgaussian distribution. Given the aforementioned hardness results, these guarantees can not be substantially improved. Our main insight is a new characterization of {\em surrogate losses} for nonconvex activations. While prior work had established the existence of convex surrogates for monotone activations, we show that properties of the underlying distribution actually induce strong convexity for the loss, allowing us to relate the global minimum to the activation’s {\em Chow parameters}.

Despite breakthrough performance, modern learning models are known to be highly vulnerable to small adversarial perturbations in their inputs. While a wide variety of recent \emph{adversarial training} methods have been effective at improving robustness to perturbed inputs (robust accuracy), often this benefit is accompanied by a decrease in accuracy on benign inputs (standard accuracy), leading to a tradeoff between often competing objectives. Complicating matters further, recent empirical evidence suggest that a variety of other factors (size and quality of training data, model size, etc.) affect this tradeoff in somewhat surprising ways. In this paper we provide a precise and comprehensive understanding of the role of adversarial training in the context of linear regression with Gaussian features. In particular, we characterize the fundamental tradeoff between the accuracies achievable by any algorithm regardless of computational power or size of the training data. Furthermore, we precisely characterize the standard/robust accuracy and the corresponding tradeoff achieved by a contemporary mini-max adversarial training approach in a high-dimensional regime where the number of data points and the parameters of the model grow in proportion to each other. Our theory for adversarial training algorithms also facilitates the rigorous study of how a variety of factors (size and quality of training data, model overparametrization etc.) affect the tradeoff between these two competing accuracies.

We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomial-time algorithms for this problem with dimension-independent error guarantees for a range of natural distribution families. In this work, we show that a natural non-convex formulation of the problem can be solved directly by gradient descent. Our approach leverages a novel structural lemma, roughly showing that any approximate stationary point of our non-convex objective gives a near-optimal solution to the underlying robust estimation task. Our work establishes an intriguing connection between algorithmic high-dimensional robust statistics and non-convex optimization, which may have broader applications to other robust estimation tasks.

Un-trained convolutional neural networks have emerged as highly successful tools for image recovery and restoration. They are capable of solving standard inverse problems such as denoising and compressive sensing with excellent results by simply fitting a neural network model to measurements from a single image or signal without the need for any additional training data. For some applications, this critically requires additional regularization in the form of early stopping the optimization. For signal recovery from a few measurements, however, un-trained convolutional networks have an intriguing self-regularizing property: Even though the network can perfectly fit any image, the network recovers a natural image from few measurements when trained with gradient descent until convergence. In this paper, we provide numerical evidence for this property and study it theoretically. We show that—without any further regularization—an un-trained convolutional neural network can approximately reconstruct signals and images that are sufficiently structured, from a near minimal number of random measurements.

Transfer learning has emerged as a powerful technique for improving the performance of machine learning models on new domains where labeled training data may be scarce. In this approach a model trained for a source task, where plenty of labeled training data is available, is used as a starting point for training a model on a related target task with only few labeled training data. Despite recent empirical success of transfer learning approaches, the benefits and fundamental limits of transfer learning are poorly understood. In this paper we develop a statistical minimax framework to characterize the fundamental limits of transfer learning in the context of regression with linear and one-hidden layer neural network models. Specifically, we derive a lower-bound for the target generalization error achievable by any algorithm as a function of the number of labeled source and target data as well as appropriate notions of similarity between the source and target tasks. Our lower bound provides new insights into the benefits and limitations of transfer learning. We further corroborate our theoretical finding with various experiments.

Contemporary machine learning applications often involve classification tasks with many classes. Despite their extensive use, a precise understanding of the statistical properties and behavior of classification algorithms is still missing, especially in modern regimes where the number of classes is rather large. In this paper, we take a step in this direction by providing the first asymptotically precise analysis of linear multiclass classification. Our theoretical analysis allows us to precisely characterize how the test error varies over different training algorithms, data distributions, problem dimensions as well as number of classes, inter/intra class correlations and class priors. Specifically, our analysis reveals that the classification accuracy is highly distribution-dependent with different algorithms achieving optimal performance for different data distributions and/or training/features sizes. Unlike linear regression/binary classification, the test error in multiclass classification relies on intricate functions of the trained model (e.g., correlation between some of the trained weights) whose asymptotic behavior is difficult to characterize. This challenge is already present in simple classifiers, such as those minimizing a square loss. Our novel theoretical techniques allow us to overcome some of these challenges. The insights gained may pave the way for a precise understanding of other classification algorithms beyond those studied in this paper.

We consider a scenario involving computations over a massive dataset stored distributedly across multiple workers, which is at the core of distributed learning algorithms. We propose Lagrange Coded Computing (LCC), a new framework to simultaneously provide (1) resiliency against stragglers that may prolong computations; (2) security against Byzantine (or malicious) workers that deliberately modify the computation for their benefit; and (3) (information-theoretic) privacy of the dataset amidst possible collusion of workers. LCC, which leverages the well-known Lagrange polynomial to create computation redundancy in a novel coded form across workers, can be applied to any computation scenario in which the function of interest is an arbitrary multivariate polynomial of the input dataset, hence covering many computations of interest in machine learning. LCC significantly generalizes prior works to go beyond linear computations. It also enables secure and private computing in distributed settings, improving the computation and communication efficiency of the state-of-the-art. Furthermore, we prove the optimality of LCC by showing that it achieves the optimal tradeoff between resiliency, security, and privacy, i.e., in terms of tolerating the maximum number of stragglers and adversaries, and providing data privacy against the maximum number of colluding workers. Finally, we show via experiments on Amazon EC2 that LCC speeds up the conventional uncoded implementation of distributed least-squares linear regression by up to $13.43\times$, and also achieves a $2.36\times$-$12.65\times$ speedup over the state-of-the-art straggler mitigation strategies.

Many modern learning tasks involve fitting nonlinear models which are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Due to this overparameterization, the training loss may have infinitely many global minima and it is critical to understand the properties of the solutions found by first-order optimization schemes such as (stochastic) gradient descent starting from different initializations. In this paper we demonstrate that when the loss has certain properties over a minimally small neighborhood of the initial point, first order methods such as (stochastic) gradient descent have a few intriguing properties: (1) the iterates converge at a geometric rate to a global optima even when the loss is nonconvex, (2) among all global optima of the loss the iterates converge to one with a near minimal distance to the initial point, (3) the iterates take a near direct route from the initial point to this global optimum. As part of our proof technique, we introduce a new potential function which captures the tradeoff between the loss function and the distance to the initial point as the iterations progress. The utility of our general theory is demonstrated for a variety of problem domains spanning low-rank matrix recovery to shallow neural network training.

In this paper we focus on the problem of finding the optimal weights of the shallowest of neural networks consisting of a single Rectified Linear Unit (ReLU). These functions are of the form x → max(0, 〈w, x〉) with w ∈ ℝddenoting the weight vector. We focus on a planted i model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to a planted weight vector. We first show that mini-batch stochastic gradient descent when suitably initialized, converges at a geometric rate to the planted model with a number of samples that is optimal up to numerical constants. Next we focus on a parallel implementation where in each iteration the mini-batch gradient is calculated in a distributed manner across multiple processors and then broadcast to a master or all other processors. To reduce the communication cost in this setting we utilize a Quanitzed Stochastic Gradient Scheme (QSGD) where the partial gradients are quantized. Perhaps unexpectedly, we show that QSGD maintains the fast convergence of SGD to a globally optimal model while significantly reducing the communication cost. We further corroborate our numerical findings via various experiments including distributed implementations over Amazon EC2.

This paper concerns the problem of recovering an unknown but structured signal x ∈ ℝnfrom m-quadratic measurements of the form yr= |(ar, x)|2for r = 1, 2, . . . , m. We focus on the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal (m <;<; n). We formulate the recovery problem as a nonconvex optimization problem where prior structural information about the signal is enforced through constrains on the optimization variables. We prove the projected gradient descent, when initialized in a neighborhood of the desired signal, converges to the unknown signal at a linear rate. These results hold for any closed constraint set (convex or non-convex) providing convergence guarantees to the global optimum even when the objective function and constraint set are nonconvex. Furthermore, these results hold with a number of measurements that are only a constant factor away from the minimal number of measurements required to uniquely identify the unknown signal. Our results provide the first provably tractable algorithm for this data-poor regime, breaking local sample complexity barriers that have emerged in this paper. In this paper, we utilize and further develop powerful tools for uniform convergence of empirical processes that may have broader implications for rigorous understanding of constrained nonconvex optimization heuristics.

In this paper, we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the numbers of observations are fewer than the number of parameters in the model. We show that with the quadratic activations, the optimization landscape of training, such shallow neural networks, has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. This result holds for an arbitrary training data of input/output pairs. For differentiable activation functions, we also show that gradient descent, when suitably initialized, converges at a linear rate to a globally optimal model. This result focuses on a realizable model where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted weight coefficients.

Fourier ptychographic microscopy enables gigapixel-scale imaging, with both large field-of-view and high resolution. Using a set of low-resolution images that are recorded under varying illumination angles, the goal is to computationally reconstruct high-resolution phase and amplitude images. To increase temporal resolution, one may use multiplexed measurements where the sample is illuminated simultaneously from a subset of the angles. In this paper, we develop an algorithm for Fourier ptychographic microscopy with such multiplexed illumination. Specifically, we consider gradient descent type updates and propose an analytical step size that ensures the convergence of the iterates to a stationary point. Furthermore, we propose an accelerated version of our algorithm (with the same step size) which significantly improves the convergence speed. We demonstrate that the practical performance of our algorithm is identical to the case where the step size is manually tuned. Finally, we apply our parameter-free approach to real data and validate its applicability.

Line spectral estimation is the problem of recovering the frequencies and amplitudes of a mixture of a few sinusoids from equispaced samples. However, in a variety of signal processing problems arising in imaging, radar, and localization, we do not have access directly to such equispaced samples. Rather, we only observe a severely undersampled version of these observations through linear measurements. This paper is about such generalized line spectral estimation problems. We reformulate these problems as sparse signal recovery problems over a continuously indexed dictionary, which can be solved via a convex program. We prove that the frequencies and amplitudes of the components of the mixture can be recovered perfectly from a near-minimal number of observations via this convex program. This result holds provided the frequencies are sufficiently separated, and the linear measurements obey natural conditions that are satisfied in a variety of applications.

In this paper, we characterize sharp time-data tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a least-squares objective subject to a constraint defined as the sub-level set of a penalty function. We present a unified convergence analysis of the gradient projection algorithm applied to such problems. We sharply characterize the convergence rate associated with a wide variety of random measurement ensembles in terms of the number of measurements and structural complexity of the signal with respect to the chosen penalty function. The results apply to both convex and nonconvex constraints, demonstrating that a linear convergence rate is attainable even though the least squares objective is not strongly convex in these settings. When specialized to Gaussian measurements our results show that such linear convergence occurs when the number of measurements is merely four times the minimal number required to recover the desired signal at all (also known as the phase transition). We also achieve a slower but geometric rate of convergence precisely above the phase transition point. Extensive numerical results suggest that the derived rates exactly match the empirical performance.

In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor $1/2$ approximation to the global maxima. We also study stochastic gradient methods and show that after $\mathcal{O}(1/\epsilon^2)$ iterations these methods reach solutions which achieve in expectation objective values exceeding $(\frac{\text{OPT}}{2}-\epsilon)$. An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor $1/2$ approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort.

In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form $\vct{x}\mapsto \max(0,\langle \vct{w},\vct{x}\rangle)$ with $\vct{w}\in\R^d$ denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d.~from a Gaussian distribution and the labels are generated according to a planted weight vector. We show that projected gradient descent, when initialized at $\vct{0}$, converges at a linear rate to the planted model with a number of samples that is optimal up to numerical constants. Our results on the dynamics of convergence of these very shallow neural nets may provide some insights towards understanding the dynamics of deeper architectures.

In this paper we study the problem of recovering a low-rank matrix from linear measurements. Our algorithm, which we call Procrustes Flow, starts from an initial estimate obtained by a thresholding scheme followed by gradient descent on a non-convex objective. We show that as long as the measurements obey a standard restricted isometry property, our algorithm converges to the unknown matrix at a geometric rate. In the case of Gaussian measurements, such convergence occurs for a n1 \times n2 matrix of rank r when the number of measurements exceeds a constant times (n1 + n2)r.

We study the problem of recovering the phase from magnitude measurements; specifically, we wish to reconstruct a complex-valued signal$ \boldsymbol {x}\in \mathbb {C}^{n}$about which we have phaseless samples of the form$y_{r} = \left |{\langle \boldsymbol {a}_{r}, \boldsymbol {x} \rangle }\right |^{2}$,$r = 1,\ldots , m$(knowledge of the phase of these samples would yield a linear system). This paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm. In a nutshell, this algorithm starts with a careful initialization obtained by means of a spectral method, and then refines this initial estimate by iteratively applying novel update rules, which have low computational complexity, much like in a gradient descent scheme. The main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements. Indeed, the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources. In theory, a variation on this scheme leads to a near-linear time algorithm for a physically realizable model based on coded diffraction patterns. We illustrate the effectiveness of our methods with various experiments on image data. Underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval.

In this paper we introduce a new class of codes for over-loaded synchronous wireless CDMA systems which increases the number of users for a fixed number of chips without introducing any errors. In addition these codes support active user detection. We derive an upper bound on the number of users with a fixed spreading factor. Also we propose an ML decoder for a subclass of these codes that is computationally implementable. Although for our simulations we consider a scenario that is worse than what occurs in practice, simulation results indicate that this coding/decoding scheme is robust against additive noise. As an example, for 64 chips and 88 users we propose a coding/decoding scheme that can obtain an arbitrary small probability of error which is computationally feasible and can detect active users. Furthermore, we prove that for this to be possible the number of users cannot be beyond 230.