Deep networks are typically trained with many more parameters than the size of the training dataset. Recent empirical evidence indicates that the practice of overparameterization not only benefits training large models, but also assists - perhaps counterintuitively - building lightweight models. Specifically, it suggests that overparameterization benefits model pruning / sparsification. This paper sheds light on these empirical findings by theoretically characterizing the high-dimensional asymptotics of model pruning in the overparameterized regime. The theory presented addresses the following core question: "should one train a small model from the beginning, or first train a large model and then prune?". We analytically identify regimes in which, even if the location of the most informative features is known, we are better off fitting a large model and then pruning rather than simply training with the known informative features. This leads to a new double descent in the training of sparse models: growing the original model, while preserving the target sparsity, improves the test accuracy as one moves beyond the overparameterization threshold. Our analysis further reveals the benefit of retraining by relating it to feature correlations. We find that the above phenomena are already present in linear and random-features models. Our technical approach advances the toolset of high-dimensional analysis and precisely characterizes the asymptotic distribution of over-parameterized least-squares. The intuition gained by analytically studying simpler models is numerically verified on neural networks.
A surprising phenomenon in modern machine learning is the ability of a highly overparameterized model to generalize well (small error on the test data) even when it is trained to memorize the training data (zero error on the training data). This has led to an arms race towards increasingly overparameterized models (c.f., deep learning). In this paper, we study an underexplored hidden cost of overparameterization: the fact that overparameterized models may be more vulnerable to privacy attacks, in particular the membership inference attack that predicts the (potentially sensitive) examples used…
Network pruning in Convolutional Neural Networks (CNNs) has been extensively investigated in recent years. To determine the impact of pruning a group of filters on a network's accuracy, state-of-the-art pruning methods consistently assume filters of a CNN are independent. This allows the importance of a group of filters to be estimated as the sum of importances of individual filters. However, overparameterization in modern networks results in highly correlated filters that invalidate this assumption, thereby resulting in incorrect importance estimates. To address this issue, we propose OrthoRe…
Beihang University（北京航空航天大学）；The University of Tokyo（东京大学）；RIKEN Center for AIP（理化学研究所人工智能研究中心）
People usually believe that network pruning not only reduces the computational cost of deep networks, but also prevents overfitting by decreasing model capacity. However, our work surprisingly discovers that network pruning sometimes even aggravates overfitting. We report an unexpected sparse double descent phenomenon that, as we increase model sparsity via network pruning, test performance first gets worse (due to overfitting), then gets better (due to relieved overfitting), and gets worse at last (due to forgetting useful information). While recent studies focused on the deep double descent …
Modern deep neural networks rely on overparameterization to achieve state-of-the-art generalization. But overparameterized models are computationally expensive. Network pruning is often employed to obtain less demanding models for deployment. Fine-grained pruning removes individual weights in parameter tensors and can achieve a high model compression ratio with little accuracy degradation. However, it introduces irregularity into the computing dataflow and often does not yield improved model inference efficiency in practice. Coarse-grained model pruning, while realizing satisfactory inference …
Provable Separations between Memorization and Generalization in Diffusion Models
Department of Industrial Engineering and Management Sciences, Northwestern University（工业工程与管理科学系，西北大学）；Department of Statistics and Data Science, Northwestern University（统计与数据科学系，西北大学）；School of Mathematics, Georgia Institute of Technology（数学系，佐治亚理工学院）
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
Overparameterization has been shown to benefit both the optimization and generalization of neural networks, but large networks are resource hungry at both training and test time. Network pruning can reduce test-time resource requirements, but is typically applied to trained networks and therefore cannot avoid the expensive training process. We aim to prune networks at initialization, thereby saving resources at training time as well. Specifically, we argue that efficient training requires preserving the gradient flow through the network. This leads to a simple but effective pruning criterion w…
CSAIL, MIT（麻省理工学院计算机科学与人工智能实验室）；University of Haifa（海法大学）
We present a provable, sampling-based approach for generating compact Convolutional Neural Networks (CNNs) by identifying and removing redundant filters from an over-parameterized network. Our algorithm uses a small batch of input data points to assign a saliency score to each filter and constructs an importance sampling distribution where filters that highly affect the output are sampled with correspondingly high probability. In contrast to existing filter pruning approaches, our method is simultaneously data-informed, exhibits provable guarantees on the size and performance of the pruned net…
Training deep neural networks with stochastic gradient descent (SGD) can often achieve zero training loss on real-world tasks although the optimization landscape is known to be highly non-convex. To understand the success of SGD for training deep neural networks, this work presents a mean-field analysis of deep residual networks, based on a line of works that interpret the continuum limit of the deep residual network as an ordinary differential equation when the network capacity tends to infinity. Specifically, we propose a new continuum limit of deep residual networks, which enjoys a good lan…
Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology（麻省理工学院计算机科学与人工智能实验室）；Robotics and Big Data Laboratory, University of Haifa（海法大学机器人与大数据实验室）
We introduce a pruning algorithm that provably sparsifies the parameters of a trained model in a way that approximately preserves the model's predictive accuracy. Our algorithm uses a small batch of input points to construct a data-informed importance sampling distribution over the network's parameters, and adaptively mixes a sampling-based and deterministic pruning procedure to discard redundant weights. Our pruning method is simultaneously computationally efficient, provably accurate, and broadly applicable to various network architectures and data distributions. Our empirical comparisons sh…
University of Oslo（奥斯陆大学）；University of Copenhagen（哥本哈根大学）
Neural networks have achieved dramatic improvements in recent years and depict the state-of-the-art methods for many real-world tasks nowadays. One drawback is, however, that many of these models are overparameterized, which makes them both computationally and memory intensive. Furthermore, overparameterization can also lead to undesired overfitting side-effects. Inspired by recently proposed magnitude-based pruning schemes and the Wald test from the field of statistics, we introduce a novel magnitude and uncertainty (M&U) pruning criterion that helps to lessen such shortcomings. One important…
We propose a novel, structured pruning algorithm for neural networks -- the iterative, Sparse Structured Pruning algorithm, dubbed as i-SpaSP. Inspired by ideas from sparse signal recovery, i-SpaSP operates by iteratively identifying a larger set of important parameter groups (e.g., filters or neurons) within a network that contribute most to the residual between pruned and dense network output, then thresholding these groups based on a smaller, pre-defined pruning ratio. For both two-layer and multi-layer network architectures with ReLU activations, we show the error induced by pruning with i…
Learning on Transformers is Provable Low-Rank and Sparse: A One-layer Analysis
Rensselaer Polytechnic Institute（伦斯勒理工学院）；New Jersey Institute of Technology（新泽西理工学院）；Michigan State University（密歇根州立大学）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology（麻省理工学院计算机科学与人工智能实验室）
Neural network pruning is a popular technique used to reduce the inference costs of modern, potentially overparameterized, networks. Starting from a pre-trained network, the process is as follows: remove redundant parameters, retrain, and repeat while maintaining the same test accuracy. The result is a model that is a fraction of the size of the original with comparable predictive performance (test accuracy). Here, we reassess and evaluate whether the use of test accuracy alone in the terminating condition is sufficient to ensure that the resulting model performs well across a wide spectrum of…
University of Haifa（海法大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Carnegie Mellon University（卡内基梅隆大学）
Previous work showed empirically that large neural networks can be significantly reduced in size while preserving their accuracy. Model compression became a central research topic, as it is crucial for deployment of neural networks on devices with limited computational and memory resources. The majority of the compression methods are based on heuristics and offer no worst-case guarantees on the trade-off between the compression rate and the approximation error for an arbitrarily new sample. We propose the first efficient, data-independent neural pruning algorithm with a provable trade-off betw…
School of Mathematics and Natural Sciences, University of Wuppertal（伍珀塔尔大学数学与自然科学学院）
Overparameterization and overfitting are common concerns when designing and training deep neural networks, that are often counteracted by pruning and regularization strategies. However, these strategies remain secondary to most learning approaches and suffer from time and computational intensive procedures. We suggest a multiobjective perspective on the training of neural networks by treating its prediction accuracy and the network complexity as two individual objective functions in a biobjective optimization problem. As a showcase example, we use the cross entropy as a measure of the predicti…
Model compression is crucial for deployment of neural networks on devices with limited computational and memory resources. Many different methods show comparable accuracy of the compressed model and similar compression rates. However, the majority of the compression methods are based on heuristics and offer no worst-case guarantees on the trade-off between the compression rate and the approximation error for an arbitrarily new sample. We propose the first efficient structured pruning algorithm with a provable trade-off between its compression rate and the approximation error for any future tes…
Pruning is one of the predominant approaches for compressing deep neural networks (DNNs). Lately, coresets (provable data summarizations) were leveraged for pruning DNNs, adding the advantage of theoretical guarantees on the trade-off between the compression rate and the approximation error. However, coresets in this domain were either data-dependent or generated under restrictive assumptions on both the model's weights and inputs. In real-world scenarios, such assumptions are rarely satisfied, limiting the applicability of coresets. To this end, we suggest a novel and robust framework for com…
University of Oxford（牛津大学）；University of Georgia（佐治亚大学）；University of Liverpool（利物浦大学）
Despite the improved accuracy of deep neural networks, the discovery of adversarial examples has raised serious safety concerns. In this paper, we study two variants of pointwise robustness, the maximum safe radius problem, which for a given input sample computes the minimum distance to an adversarial example, and the feature robustness problem, which aims to quantify the robustness of individual features to adversarial perturbations. We demonstrate that, under the assumption of Lipschitz continuity, both problems can be approximated using finite optimisation by discretising the input space, a…
In continual learning (CL), the goal is to design models that can learn a sequence of tasks without catastrophic forgetting. While there is a rich set of techniques for CL, relatively little understanding exists on how representations built by previous tasks benefit new tasks that are added to the network. To address this, we study the problem of continual representation learning (CRL) where we learn an evolving representation as new tasks arrive. Focusing on zero-forgetting methods where tasks are embedded in subnetworks (e.g., PackNet), we first provide experiments demonstrating CRL can sign…
