arXiv subjects

Mahdi Soltanolkotabi

Publications and source records attributed to Mahdi Soltanolkotabi.

At least 55 records · Page 3Linked to original sources

arXiv2022

Outlier-Robust Sparse Estimation via Non-Convex Optimization

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.

cs.LG↗

arXiv2022

Toward a Geometrical Understanding of Self-supervised Contrastive Learning

Self-supervised learning (SSL) is currently one of the premier techniques to create data representations that are actionable for transfer learning in the absence of human annotations. Despite their success, the underlying geometry of these representations remains elusive, which obfuscates the quest for more robust, trustworthy, and interpretable models. In particular, mainstream SSL techniques rely on a specific deep neural network architecture with two cascaded neural networks: the encoder and the projector. When used for transfer learning, the projector is discarded since empirical results show that its representation generalizes more poorly than the encoder's. In this paper, we investigate this curious phenomenon and analyze how the strength of the data augmentation policies affects the data embedding. We discover a non-trivial relation between the encoder, the projector, and the data augmentation strength: with increasingly larger augmentation policies, the projector, rather than the encoder, is more strongly driven to become invariant to the augmentations. It does so by eliminating crucial information about the data by learning to project it into a low-dimensional space, a noisy estimate of the data manifold tangent plane in the encoder representation. This analysis is substantiated through a geometrical perspective with theoretical and empirical results.

cs.LG↗

arXiv2022

Small random initialization is akin to spectral learning: Optimization and generalization guarantees for overparameterized low-rank matrix reconstruction

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.

cs.LG↗

arXiv2022

The Geometry of Self-supervised Learning Models and its Impact on Transfer Learning

Self-supervised learning (SSL) has emerged as a desirable paradigm in computer vision due to the inability of supervised models to learn representations that can generalize in domains with limited labels. The recent popularity of SSL has led to the development of several models that make use of diverse training strategies, architectures, and data augmentation policies with no existing unified framework to study or assess their effectiveness in transfer learning. We propose a data-driven geometric strategy to analyze different SSL models using local neighborhoods in the feature space induced by each. Unlike existing approaches that consider mathematical approximations of the parameters, individual components, or optimization landscape, our work aims to explore the geometric properties of the representation manifolds learned by SSL models. Our proposed manifold graph metrics (MGMs) provide insights into the geometric similarities and differences between available SSL models, their invariances with respect to specific augmentations, and their performances on transfer learning tasks. Our key findings are two fold: (i) contrary to popular belief, the geometry of SSL models is not tied to its training paradigm (contrastive, non-contrastive, and cluster-based); (ii) we can predict the transfer learning capability for a specific model based on the geometric properties of its semantic and augmentation manifolds.

cs.LG↗

arXiv2022

Theoretical insights into the optimization landscape of over-parameterized shallow neural networks

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 number of observations are fewer than the number of parameters in the model. We show that with 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.

cs.LG↗

arXiv2022

Neural Networks can Learn Representations with Gradient Descent

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 $n\asymp d^2 r + dr^p$. Furthermore, in a transfer learning setup where the data distributions in the source and target domain share the same representation $U$ but have different polynomial heads we show that a popular heuristic for transfer learning has a target sample complexity independent of $d$.

cs.LG↗

arXiv2022

FedNLP: Benchmarking Federated Learning Methods for Natural Language Processing Tasks

Increasing concerns and regulations about data privacy and sparsity necessitate the study of privacy-preserving, decentralized learning methods for natural language processing (NLP) tasks. Federated learning (FL) provides promising approaches for a large number of clients (e.g., personal devices or organizations) to collaboratively learn a shared global model to benefit all clients while allowing users to keep their data locally. Despite interest in studying FL methods for NLP tasks, a systematic comparison and analysis is lacking in the literature. Herein, we present the FedNLP, a benchmarking framework for evaluating federated learning methods on four different task formulations: text classification, sequence tagging, question answering, and seq2seq. We propose a universal interface between Transformer-based language models (e.g., BERT, BART) and FL methods (e.g., FedAvg, FedOPT, etc.) under various non-IID partitioning strategies. Our extensive experiments with FedNLP provide empirical comparisons between FL methods and helps us better understand the inherent challenges of this direction. The comprehensive analysis points to intriguing and exciting future research aimed at developing FL methods for NLP tasks.

cs.CL↗

arXiv2022

Precise Statistical Analysis of Classification Accuracies for Adversarial Training

Despite the wide empirical success of modern machine learning algorithms and models in a multitude of applications, they are known to be highly susceptible to seemingly small indiscernible perturbations to the input data known as \emph{adversarial attacks}. A variety of recent adversarial training procedures have been proposed to remedy this issue. Despite the success of such procedures at increasing accuracy on adversarially perturbed inputs or \emph{robust accuracy}, these techniques often reduce accuracy on natural unperturbed inputs or \emph{standard accuracy}. Complicating matters further, the effect and trend of adversarial training procedures on standard and robust accuracy is rather counter intuitive and radically dependent on a variety of factors including the perceived form of the perturbation during training, size/quality of data, model overparameterization, etc. In this paper we focus on binary classification problems where the data is generated according to the mixture of two Gaussians with general anisotropic covariance matrices and derive a precise characterization of the standard and robust accuracy for a class of minimax adversarially trained models. We consider a general norm-based adversarial model, where the adversary can add perturbations of bounded $\ell_p$ norm to each input data, for an arbitrary $p\ge 1$. Our comprehensive analysis allows us to theoretically explain several intriguing empirical phenomena and provide a precise understanding of the role of different problem parameters on standard and robust accuracies.

stat.ML↗

arXiv2022

Generalization Guarantees for Neural Architecture Search with Train-Validation Split

Neural Architecture Search (NAS) is a popular method for automatically designing optimized architectures for high-performance deep learning. In this approach, it is common to use bilevel optimization where one optimizes the model weights over the training data (inner problem) and various hyperparameters such as the configuration of the architecture over the validation data (outer problem). This paper explores the statistical aspects of such problems with train-validation splits. In practice, the inner 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 the role of train-validation split. To this aim this work establishes the following results. (1) We show that refined properties of the validation loss such as risk and hyper-gradients are indicative of those of the true test loss. This reveals that the outer problem helps select the most generalizable model and prevent overfitting with a near-minimal validation sample size. This is established for continuous search spaces which are relevant for differentiable schemes. Extensions to transfer learning are developed in terms of the mismatch between training & validation distributions. (2) We establish generalization bounds for NAS problems with an emphasis on an activation search problem. When optimized with gradient-descent, we show that the train-validation procedure returns the best (model, architecture) pair even if all architectures can perfectly fit the training data to achieve zero error. (3) Finally, we highlight connections between NAS, multiple kernel learning, and low-rank matrix learning. The latter leads to novel insights where the solution of the outer problem can be accurately learned via efficient spectral methods to achieve near-minimal risk.

stat.ML↗

arXiv2021

FedCV: A Federated Learning Framework for Diverse Computer Vision Tasks

Federated Learning (FL) is a distributed learning paradigm that can learn a global or personalized model from decentralized datasets on edge devices. However, in the computer vision domain, model performance in FL is far behind centralized training due to the lack of exploration in diverse tasks with a unified FL framework. FL has rarely been demonstrated effectively in advanced computer vision tasks such as object detection and image segmentation. To bridge the gap and facilitate the development of FL for computer vision tasks, in this work, we propose a federated learning library and benchmarking framework, named FedCV, to evaluate FL on the three most representative computer vision tasks: image classification, image segmentation, and object detection. We provide non-I.I.D. benchmarking datasets, models, and various reference FL algorithms. Our benchmark study suggests that there are multiple challenges that deserve future exploration: centralized training tricks may not be directly applied to FL; the non-I.I.D. dataset actually downgrades the model accuracy to some degree in different tasks; improving the system efficiency of federated training is challenging given the huge number of parameters and the per-client memory cost. We believe that such a library and benchmark, along with comparable evaluation settings, is necessary to make meaningful progress in FL on computer vision tasks. FedCV is publicly available: https://github.com/FedML-AI/FedCV.

cs.CV↗

arXiv2021

SSFL: Tackling Label Deficiency in Federated Learning via Personalized Self-Supervision

Federated Learning (FL) is transforming the ML training ecosystem from a centralized over-the-cloud setting to distributed training over edge devices in order to strengthen data privacy. An essential but rarely studied challenge in FL is label deficiency at the edge. This problem is even more pronounced in FL compared to centralized training due to the fact that FL users are often reluctant to label their private data. Furthermore, due to the heterogeneous nature of the data at edge devices, it is crucial to develop personalized models. In this paper we propose self-supervised federated learning (SSFL), a unified self-supervised and personalized federated learning framework, and a series of algorithms under this framework which work towards addressing these challenges. First, under the SSFL framework, we demonstrate that the standard FedAvg algorithm is compatible with recent breakthroughs in centralized self-supervised learning such as SimSiam networks. Moreover, to deal with data heterogeneity at the edge devices in this framework, we have innovated a series of algorithms that broaden existing supervised personalization algorithms into the setting of self-supervised learning. We further propose a novel personalized federated self-supervised learning algorithm, Per-SSFL, which balances personalization and consensus by carefully regulating the distance between the local and global representations of data. To provide a comprehensive comparative analysis of all proposed algorithms, we also develop a distributed training system and related evaluation protocol for SSFL. Our findings show that the gap of evaluation accuracy between supervised learning and unsupervised learning in FL is both small and reasonable. The performance comparison indicates the representation regularization-based personalization method is able to outperform other variants.

cs.LG↗

arXiv2021

A Field Guide to Federated Optimization

Federated learning and analytics are a distributed approach for collaboratively learning models (or statistics) from decentralized data, motivated by and designed for privacy protection. The distributed learning process can be formulated as solving federated optimization problems, which emphasize communication efficiency, data heterogeneity, compatibility with privacy and system requirements, and other constraints that are not primary considerations in other problem settings. This paper provides recommendations and guidelines on formulating, designing, evaluating and analyzing federated optimization algorithms through concrete examples and practical implementation, with a focus on conducting effective simulations to infer real-world performance. The goal of this work is not to survey the current literature, but to inspire researchers and practitioners to design federated learning algorithms that can be used in various practical applications.

cs.LG↗

arXiv2021

Data augmentation for deep learning based accelerated MRI reconstruction with limited data

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 can improve the robustness of the model against various shifts in the test distribution.

eess.IV↗

arXiv2021

Understanding Overparameterization in Generative Adversarial Networks

A broad class of unsupervised deep learning methods such as Generative Adversarial Networks (GANs) involve training of overparameterized models where the number of parameters of the model exceeds a certain threshold. A large body of work in supervised learning have shown the importance of model overparameterization in the convergence of the gradient descent (GD) to globally optimal solutions. In contrast, the unsupervised setting and GANs in particular involve non-convex concave mini-max optimization problems that are often trained using Gradient Descent/Ascent (GDA). The role and benefits of model overparameterization in the convergence of GDA to a global saddle point in non-convex concave problems is far less understood. In this work, we present a comprehensive analysis of the importance of model overparameterization in GANs both theoretically and empirically. We theoretically show that in an overparameterized GAN model with a $1$-layer neural network generator and a linear discriminator, GDA converges to a global saddle point of the underlying non-convex concave min-max problem. To the best of our knowledge, this is the first result for global convergence of GDA in such settings. Our theory is based on a more general result that holds for a broader class of nonlinear generators and discriminators that obey certain assumptions (including deeper generators and random feature discriminators). We also empirically study the role of model overparameterization in GANs using several large-scale experiments on CIFAR-10 and Celeb-A datasets. Our experiments show that overparameterization improves the quality of generated samples across various model architectures and datasets. Remarkably, we observe that overparameterization leads to faster and more stable convergence behavior of GDA across the board.

cs.LG↗

arXiv2021

Convergence and sample complexity of gradient methods for the model-free linear quadratic regulator problem

Model-free reinforcement learning attempts to find an optimal control action for an unknown dynamical system by directly searching over the parameter space of controllers. The convergence behavior and statistical properties of these approaches are often poorly understood because of the nonconvex nature of the underlying optimization problems and the lack of exact gradient computation. In this paper, we take a step towards demystifying the performance and efficiency of such methods by focusing on the standard infinite-horizon linear quadratic regulator problem for continuous-time systems with unknown state-space parameters. We establish exponential stability for the ordinary differential equation (ODE) that governs the gradient-flow dynamics over the set of stabilizing feedback gains and show that a similar result holds for the gradient descent method that arises from the forward Euler discretization of the corresponding ODE. We also provide theoretical bounds on the convergence rate and sample complexity of the random search method with two-point gradient estimates. We prove that the required simulation time for achieving $ε$-accuracy in the model-free setup and the total number of function evaluations both scale as $\log \, (1/ε)$.

math.OC↗

arXiv2021

PipeTransformer: Automated Elastic Pipelining for Distributed Training of Transformers

The size of Transformer models is growing at an unprecedented pace. It has only taken less than one year to reach trillion-level parameters after 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 and elastic pipelining and data parallelism for efficient distributed training of Transformer models. PipeTransformer automatically adjusts the pipelining and data parallelism by identifying and freezing some layers during the training, and instead allocates resources for training of the remaining active layers. More specifically, PipeTransformer dynamically excludes converged 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 GLUE and SQuAD datasets. Our results show that PipeTransformer attains a 2.4 fold speedup compared to the state-of-the-art baseline. We also provide various performance analyses for a more comprehensive understanding of our algorithmic and system-wise design. We also develop open-sourced flexible APIs for PipeTransformer, which offer a clean separation among the freeze algorithm, model definitions, and training accelerations, hence allowing it to be applied to other algorithms that require similar freezing strategies.

cs.LG↗

arXiv2021

Super-Resolution Radar

In this paper we study the identification of a time-varying linear system from its response to a known input signal. More specifically, we consider systems whose response to the input signal is given by a weighted superposition of delayed and Doppler shifted versions of the input. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth B, and the received signal is observed over a finite time interval of length T only. This gives rise to a delay and Doppler resolution of 1/B and 1/T. We show that this resolution limit can be overcome, i.e., we can exactly recover the continuous delay-Doppler pairs and the corresponding attenuation factors, by solving a convex optimization problem. This result holds provided that the distance between the delay-Doppler pairs is at least 2.37/B in time or 2.37/T in frequency. Furthermore, this result allows the total number of delay-Doppler pairs to be linear up to a log-factor in BT, the dimensionality of the response of the system, and thereby the limit for identifiability. Stated differently, we show that we can estimate the time-frequency components of a signal that is S-sparse in the continuous dictionary of time-frequency shifts of a random window function, from a number of measurements, that is linear up to a log-factor in S.

cs.IT↗

arXiv2020

Theoretical Insights Into Multiclass Classification: A High-dimensional Asymptotic View

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.

cs.LG↗