arXiv subjects

Zhihui Zhu

Publications and source records attributed to Zhihui Zhu.

At least 73 records · Page 4Linked to original sources

arXiv2023

OTOv3: Automatic Architecture-Agnostic Neural Network Training and Compression from Structured Pruning to Erasing Operators

Compressing a predefined deep neural network (DNN) into a compact sub-network with competitive performance is crucial in the efficient machine learning realm. This topic spans various techniques, from structured pruning to neural architecture search, encompassing both pruning and erasing operators perspectives. Despite advancements, existing methods suffers from complex, multi-stage processes that demand substantial engineering and domain knowledge, limiting their broader applications. We introduce the third-generation Only-Train-Once (OTOv3), which first automatically trains and compresses a general DNN through pruning and erasing operations, creating a compact and competitive sub-network without the need of fine-tuning. OTOv3 simplifies and automates the training and compression process, minimizes the engineering efforts required from users. It offers key technological advancements: (i) automatic search space construction for general DNNs based on dependency graph analysis; (ii) Dual Half-Space Projected Gradient (DHSPG) and its enhanced version with hierarchical search (H2SPG) to reliably solve (hierarchical) structured sparsity problems and ensure sub-network validity; and (iii) automated sub-network construction using solutions from DHSPG/H2SPG and dependency graphs. Our empirical results demonstrate the efficacy of OTOv3 across various benchmarks in structured pruning and neural architecture search. OTOv3 produces sub-networks that match or exceed the state-of-the-arts. The source code will be available at https://github.com/tianyic/only_train_once.

cs.LG↗

arXiv2023

Generalized Neural Collapse for a Large Number of Classes

Neural collapse provides an elegant mathematical characterization of learned last layer representations (a.k.a. features) and classifier weights in deep classification models. Such results not only provide insights but also motivate new techniques for improving practical deep models. However, most of the existing empirical and theoretical studies in neural collapse focus on the case that the number of classes is small relative to the dimension of the feature space. This paper extends neural collapse to cases where the number of classes are much larger than the dimension of feature space, which broadly occur for language models, retrieval systems, and face recognition applications. We show that the features and classifier exhibit a generalized neural collapse phenomenon, where the minimum one-vs-rest margins is maximized.We provide empirical study to verify the occurrence of generalized neural collapse in practical deep neural networks. Moreover, we provide theoretical study to show that the generalized neural collapse provably occurs under unconstrained feature model with spherical constraint, under certain technical conditions on feature dimension and number of classes.

cs.LG↗

arXiv2023

Error Analysis of Tensor-Train Cross Approximation

Tensor train decomposition is widely used in machine learning and quantum physics due to its concise representation of high-dimensional tensors, overcoming the curse of dimensionality. Cross approximation-originally developed for representing a matrix from a set of selected rows and columns-is an efficient method for constructing a tensor train decomposition of a tensor from few of its entries. While tensor train cross approximation has achieved remarkable performance in practical applications, its theoretical analysis, in particular regarding the error of the approximation, is so far lacking. To our knowledge, existing results only provide element-wise approximation accuracy guarantees, which lead to a very loose bound when extended to the entire tensor. In this paper, we bridge this gap by providing accuracy guarantees in terms of the entire tensor for both exact and noisy measurements. Our results illustrate how the choice of selected subtensors affects the quality of the cross approximation and that the approximation error caused by model error and/or measurement error may not grow exponentially with the order of the tensor. These results are verified by numerical experiments, and may have important implications for the usefulness of cross approximations for high-order tensors, such as those encountered in the description of quantum many-body states.

cs.LG↗

arXiv2023

OTOV2: Automatic, Generic, User-Friendly

The existing model compression methods via structured pruning typically require complicated multi-stage procedures. Each individual stage necessitates numerous engineering efforts and domain-knowledge from the end-users which prevent their wider applications onto broader scenarios. We propose the second generation of Only-Train-Once (OTOv2), which first automatically trains and compresses a general DNN only once from scratch to produce a more compact model with competitive performance without fine-tuning. OTOv2 is automatic and pluggable into various deep learning applications, and requires almost minimal engineering efforts from the users. Methodologically, OTOv2 proposes two major improvements: (i) Autonomy: automatically exploits the dependency of general DNNs, partitions the trainable variables into Zero-Invariant Groups (ZIGs), and constructs the compressed model; and (ii) Dual Half-Space Projected Gradient (DHSPG): a novel optimizer to more reliably solve structured-sparsity problems. Numerically, we demonstrate the generality and autonomy of OTOv2 on a variety of model architectures such as VGG, ResNet, CARN, ConvNeXt, DenseNet and StackedUnets, the majority of which cannot be handled by other methods without extensive handcrafting efforts. Together with benchmark datasets including CIFAR10/100, DIV2K, Fashion-MNIST, SVNH and ImageNet, its effectiveness is validated by performing competitively or even better than the state-of-the-arts. The source code is available at https://github.com/tianyic/only_train_once.

cs.CV↗

arXiv2023

The Law of Parsimony in Gradient Descent for Learning Deep Linear Networks

Over the past few years, an extensively studied phenomenon in training deep networks is the implicit bias of gradient descent towards parsimonious solutions. In this work, we investigate this phenomenon by narrowing our focus to deep linear networks. Through our analysis, we reveal a surprising "law of parsimony" in the learning dynamics when the data possesses low-dimensional structures. Specifically, we show that the evolution of gradient descent starting from orthogonal initialization only affects a minimal portion of singular vector spaces across all weight matrices. In other words, the learning process happens only within a small invariant subspace of each weight matrix, despite the fact that all weight parameters are updated throughout training. This simplicity in learning dynamics could have significant implications for both efficient training and a better understanding of deep networks. First, the analysis enables us to considerably improve training efficiency by taking advantage of the low-dimensional structure in learning dynamics. We can construct smaller, equivalent deep linear networks without sacrificing the benefits associated with the wider counterparts. Second, it allows us to better understand deep representation learning by elucidating the linear progressive separation and concentration of representations from shallow to deep layers. We also conduct numerical experiments to support our theoretical results. The code for our experiments can be found at https://github.com/cjyaras/lawofparsimony.

cs.LG↗

arXiv2023

Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold

When training overparameterized deep networks for classification tasks, it has been widely observed that the learned features exhibit a so-called "neural collapse" phenomenon. More specifically, for the output features of the penultimate layer, for each class the within-class features converge to their means, and the means of different classes exhibit a certain tight frame structure, which is also aligned with the last layer's classifier. As feature normalization in the last layer becomes a common practice in modern representation learning, in this work we theoretically justify the neural collapse phenomenon for normalized features. Based on an unconstrained feature model, we simplify the empirical loss function in a multi-class classification task into a nonconvex optimization problem over the Riemannian manifold by constraining all features and classifiers over the sphere. In this context, we analyze the nonconvex landscape of the Riemannian optimization problem over the product of spheres, showing a benign global landscape in the sense that the only global minimizers are the neural collapse solutions while all other critical points are strict saddles with negative curvature. Experimental results on practical deep networks corroborate our theory and demonstrate that better representations can be learned faster via feature normalization.

cs.LG↗

arXiv2023

A Provable Splitting Approach for Symmetric Nonnegative Matrix Factorization

The symmetric Nonnegative Matrix Factorization (NMF), a special but important class of the general NMF, has found numerous applications in data analysis such as various clustering tasks. Unfortunately, designing fast algorithms for the symmetric NMF is not as easy as for its nonsymmetric counterpart, since the latter admits the splitting property that allows state-of-the-art alternating-type algorithms. To overcome this issue, we first split the decision variable and transform the symmetric NMF to a penalized nonsymmetric one, paving the way for designing efficient alternating-type algorithms. We then show that solving the penalized nonsymmetric reformulation returns a solution to the original symmetric NMF. Moreover, we design a family of alternating-type algorithms and show that they all admit strong convergence guarantee: the generated sequence of iterates is convergent and converges at least sublinearly to a critical point of the original symmetric NMF. Finally, we conduct experiments on both synthetic data and real image clustering to support our theoretical results and demonstrate the performance of the alternating-type algorithms.

cs.LG↗

arXiv2022

Incremental Methods for Weakly Convex Optimization

Incremental methods are widely utilized for solving finite-sum optimization problems in machine learning and signal processing. In this paper, we study a family of incremental methods -- including incremental subgradient, incremental proximal point, and incremental prox-linear methods -- for solving weakly convex optimization problems. Such a problem class covers many nonsmooth nonconvex instances that arise in engineering fields. We show that the three said incremental methods have an iteration complexity of $O(\varepsilon^{-4})$ for driving a natural stationarity measure to below $\varepsilon$. Moreover, we show that if the weakly convex function satisfies a sharpness condition, then all three incremental methods, when properly initialized and equipped with geometrically diminishing stepsizes, can achieve a local linear rate of convergence. Our work is the first to extend the convergence rate analysis of incremental methods from the nonsmooth convex regime to the weakly convex regime. Lastly, we conduct numerical experiments on the robust matrix sensing problem to illustrate the convergence performance of the three incremental methods.

math.OC↗

arXiv2022

Revisiting Sparse Convolutional Model for Visual Recognition

Despite strong empirical performance for image classification, deep neural networks are often regarded as ``black boxes'' and they are difficult to interpret. On the other hand, sparse convolutional models, which assume that a signal can be expressed by a linear combination of a few elements from a convolutional dictionary, are powerful tools for analyzing natural images with good theoretical interpretability and biological plausibility. However, such principled models have not demonstrated competitive performance when compared with empirically designed deep networks. This paper revisits the sparse convolutional modeling for image classification and bridges the gap between good empirical performance (of deep learning) and good interpretability (of sparse convolutional models). Our method uses differentiable optimization layers that are defined from convolutional sparse coding as drop-in replacements of standard convolutional layers in conventional deep neural networks. We show that such models have equally strong empirical performance on CIFAR-10, CIFAR-100, and ImageNet datasets when compared to conventional neural networks. By leveraging stable recovery property of sparse modeling, we further show that such models can be much more robust to input corruptions as well as adversarial perturbations in testing through a simple proper trade-off between sparse regularization and data reconstruction terms. Source code can be found at https://github.com/Delay-Xili/SDNet.

cs.CV↗

arXiv2022

Are All Losses Created Equal: A Neural Collapse Perspective

While cross entropy (CE) is the most commonly used loss to train deep neural networks for classification tasks, many alternative losses have been developed to obtain better empirical performance. Among them, which one is the best to use is still a mystery, because there seem to be multiple factors affecting the answer, such as properties of the dataset, the choice of network architecture, and so on. This paper studies the choice of loss function by examining the last-layer features of deep networks, drawing inspiration from a recent line work showing that the global optimal solution of CE and mean-square-error (MSE) losses exhibits a Neural Collapse phenomenon. That is, for sufficiently large networks trained until convergence, (i) all features of the same class collapse to the corresponding class mean and (ii) the means associated with different classes are in a configuration where their pairwise distances are all equal and maximized. We extend such results and show through global solution and landscape analyses that a broad family of loss functions including commonly used label smoothing (LS) and focal loss (FL) exhibits Neural Collapse. Hence, all relevant losses(i.e., CE, LS, FL, MSE) produce equivalent features on training data. Based on the unconstrained feature model assumption, we provide either the global landscape analysis for LS loss or the local landscape analysis for FL loss and show that the (only!) global minimizers are neural collapse solutions, while all other critical points are strict saddles whose Hessian exhibit negative curvature directions either in the global scope for LS loss or in the local scope for FL loss near the optimal solution. The experiments further show that Neural Collapse features obtained from all relevant losses lead to largely identical performance on test data as well, provided that the network is sufficiently large and trained until convergence.

cs.LG↗

arXiv2022

Sparsity-guided Network Design for Frame Interpolation

DNN-based frame interpolation, which generates intermediate frames from two consecutive frames, is often dependent on model architectures with a large number of features, preventing their deployment on systems with limited resources, such as mobile devices. We present a compression-driven network design for frame interpolation that leverages model pruning through sparsity-inducing optimization to greatly reduce the model size while attaining higher performance. Concretely, we begin by compressing the recently proposed AdaCoF model and demonstrating that a 10 times compressed AdaCoF performs similarly to its original counterpart, where different strategies for using layerwise sparsity information as a guide are comprehensively investigated under a variety of hyperparameter settings. We then enhance this compressed model by introducing a multi-resolution warping module, which improves visual consistency with multi-level details. As a result, we achieve a considerable performance gain with a quarter of the size of the original AdaCoF. In addition, our model performs favorably against other state-of-the-art approaches on a wide variety of datasets. We note that the suggested compression-driven framework is generic and can be easily transferred to other DNN-based frame interpolation algorithms. The source code is available at https://github.com/tding1/CDFI.

cs.CV↗

arXiv2022

Robust Training under Label Noise by Over-parameterization

Recently, over-parameterized deep networks, with increasingly more network parameters than training samples, have dominated the performances of modern machine learning. However, when the training data is corrupted, it has been well-known that over-parameterized networks tend to overfit and do not generalize. In this work, we propose a principled approach for robust training of over-parameterized deep networks in classification tasks where a proportion of training labels are corrupted. The main idea is yet very simple: label noise is sparse and incoherent with the network learned from clean data, so we model the noise and learn to separate it from the data. Specifically, we model the label noise via another sparse over-parameterization term, and exploit implicit algorithmic regularizations to recover and separate the underlying corruptions. Remarkably, when trained using such a simple method in practice, we demonstrate state-of-the-art test accuracy against label noise on a variety of real datasets. Furthermore, our experimental results are corroborated by theory on simplified linear models, showing that exact separation between sparse noise and low-rank data can be achieved under incoherent conditions. The work opens many interesting directions for improving over-parameterized models by using sparse over-parameterization and implicit regularization.

cs.LG↗

arXiv2022

Quantum state tomography with tensor train cross approximation

It has been recently shown that a state generated by a one-dimensional noisy quantum computer is well approximated by a matrix product operator with a finite bond dimension independent of the number of qubits. We show that full quantum state tomography can be performed for such a state with a minimal number of measurement settings using a method known as tensor train cross approximation. The method works for reconstructing full rank density matrices and only requires measuring local operators, which are routinely performed in state-of-art experimental quantum platforms. Our method requires exponentially fewer state copies than the best known tomography method for unstructured states and local measurements. The fidelity of our reconstructed state can be further improved via supervised machine learning, without demanding more experimental data. Scalable tomography is achieved if the full state can be reconstructed from local reductions.

quant-ph↗

arXiv2022

On the Optimization Landscape of Neural Collapse under MSE Loss: Global Optimality with Unconstrained Features

When training deep neural networks for classification tasks, an intriguing empirical phenomenon has been widely observed in the last-layer classifiers and features, where (i) the class means and the last-layer classifiers all collapse to the vertices of a Simplex Equiangular Tight Frame (ETF) up to scaling, and (ii) cross-example within-class variability of last-layer activations collapses to zero. This phenomenon is called Neural Collapse (NC), which seems to take place regardless of the choice of loss functions. In this work, we justify NC under the mean squared error (MSE) loss, where recent empirical evidence shows that it performs comparably or even better than the de-facto cross-entropy loss. Under a simplified unconstrained feature model, we provide the first global landscape analysis for vanilla nonconvex MSE loss and show that the (only!) global minimizers are neural collapse solutions, while all other critical points are strict saddles whose Hessian exhibit negative curvature directions. Furthermore, we justify the usage of rescaled MSE loss by probing the optimization landscape around the NC solutions, showing that the landscape can be improved by tuning the rescaling hyperparameters. Finally, our theoretical findings are experimentally verified on practical network architectures.

cs.LG↗

arXiv2022

Convolutional Normalization: Improving Deep Convolutional Network Robustness and Training

Normalization techniques have become a basic component in modern convolutional neural networks (ConvNets). In particular, many recent works demonstrate that promoting the orthogonality of the weights helps train deep models and improve robustness. For ConvNets, most existing methods are based on penalizing or normalizing weight matrices derived from concatenating or flattening the convolutional kernels. These methods often destroy or ignore the benign convolutional structure of the kernels; therefore, they are often expensive or impractical for deep ConvNets. In contrast, we introduce a simple and efficient "Convolutional Normalization" (ConvNorm) method that can fully exploit the convolutional structure in the Fourier domain and serve as a simple plug-and-play module to be conveniently incorporated into any ConvNets. Our method is inspired by recent work on preconditioning methods for convolutional sparse coding and can effectively promote each layer's channel-wise isometry. Furthermore, we show that our ConvNorm can reduce the layerwise spectral norm of the weight matrices and hence improve the Lipschitzness of the network, leading to easier training and improved robustness for deep ConvNets. Applied to classification under noise corruptions and generative adversarial network (GAN), we show that the ConvNorm improves the robustness of common ConvNets such as ResNet and the performance of GAN. We verify our findings via numerical experiments on CIFAR and ImageNet.

cs.CV↗

arXiv2021

Only Train Once: A One-Shot Neural Network Training And Pruning Framework

Structured pruning is a commonly used technique in deploying deep neural networks (DNNs) onto resource-constrained devices. However, the existing pruning methods are usually heuristic, task-specified, and require an extra fine-tuning procedure. To overcome these limitations, we propose a framework that compresses DNNs into slimmer architectures with competitive performances and significant FLOPs reductions by Only-Train-Once (OTO). OTO contains two keys: (i) we partition the parameters of DNNs into zero-invariant groups, enabling us to prune zero groups without affecting the output; and (ii) to promote zero groups, we then formulate a structured-sparsity optimization problem and propose a novel optimization algorithm, Half-Space Stochastic Projected Gradient (HSPG), to solve it, which outperforms the standard proximal methods on group sparsity exploration and maintains comparable convergence. To demonstrate the effectiveness of OTO, we train and compress full models simultaneously from scratch without fine-tuning for inference speedup and parameter reduction, and achieve state-of-the-art results on VGG16 for CIFAR10, ResNet50 for CIFAR10 and Bert for SQuAD and competitive result on ResNet50 for ImageNet. The source code is available at https://github.com/tianyic/only_train_once.

cs.LG↗

arXiv2021

Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery

We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust $\ell_1$ loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank. On the other hand, we show that the RDPP condition holds under generic settings, such as Gaussian measurements under independent or adversarial sparse corruptions, where the result could be of independent interest. Both the exact recovery and the convergence rate of the proposed subgradient method are numerically verified in the overspecified regime. Moreover, our experiment further shows that our particular design of diminishing stepsize effectively prevents overfitting for robust recovery under overparameterized models, such as robust matrix sensing and learning robust deep image prior. This regularization effect is worth further investigation.

math.OC↗

arXiv2021

The Global Optimization Geometry of Low-Rank Matrix Optimization

This paper considers general rank-constrained optimization problems that minimize a general objective function $f(X)$ over the set of rectangular $n\times m$ matrices that have rank at most $r$. To tackle the rank constraint and also to reduce the computational burden, we factorize $X$ into $UV^T$ where $U$ and $V$ are $n\times r$ and $m\times r$ matrices, respectively, and then optimize over the small matrices $U$ and $V$. We characterize the global optimization geometry of the nonconvex factored problem and show that the corresponding objective function satisfies the robust strict saddle property as long as the original objective function $f$ satisfies restricted strong convexity and smoothness properties, ensuring global convergence of many local search algorithms (such as noisy gradient descent) in polynomial time for solving the factored problem. We also provide a comprehensive analysis for the optimization geometry of a matrix factorization problem where we aim to find $n\times r$ and $m\times r$ matrices $U$ and $V$ such that $UV^T$ approximates a given matrix $X^\star$. Aside from the robust strict saddle property, we show that the objective function of the matrix factorization problem has no spurious local minima and obeys the strict saddle property not only for the exact-parameterization case where $rank(X^\star) = r$, but also for the over-parameterization case where $rank(X^\star) < r$ and the under-parameterization case where $rank(X^\star) > r$. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) converge to a global solution with random initialization.

cs.IT↗