Although pretrained language models can be fine-tuned to produce state-of-the-art results for a very wide range of language understanding tasks, the dynamics of this process are not well understood, especially in the low data regime. Why can we use relatively vanilla gradient descent algorithms (e.g., without strong regularization) to tune a model with hundreds of millions of parameters on datasets with only hundreds or thousands of labeled examples? In this paper, we argue that analyzing fine-tuning through the lens of intrinsic dimension provides us with empirical and theoretical intuitions to explain this remarkable phenomenon. We empirically show that common pre-trained models have a very low intrinsic dimension; in other words, there exists a low dimension reparameterization that is as effective for fine-tuning as the full parameter space. For example, by optimizing only 200 trainable parameters randomly projected back into the full space, we can tune a RoBERTa model to achieve 90\% of the full parameter performance levels on MRPC. Furthermore, we empirically show that pre-training implicitly minimizes intrinsic dimension and, perhaps surprisingly, larger models tend to have lower intrinsic dimension after a fixed number of pre-training updates, at least in part explaining their extreme effectiveness. Lastly, we connect intrinsic dimensionality with low dimensional task representations and compression based generalization bounds to provide intrinsic-dimension-based generalization bounds that are independent of the full parameter count.
Rutgers University（罗格斯大学）；The University of Tokyo（东京大学）；Komaba Institute for Science, The University of Tokyo（东京大学驹场科学研究所）；Center for Advanced Intelligence Project, RIKEN（理化学研究所革新智能综合研究中心）
In this study, we prove that an intrinsic low dimensionality of covariates is the main factor that determines the performance of deep neural networks (DNNs). DNNs generally provide outstanding empirical performance. Hence, numerous studies have actively investigated the theoretical properties of DNNs to understand their underlying mechanisms. In particular, the behavior of DNNs in terms of high-dimensional data is one of the most critical questions. However, this issue has not been sufficiently investigated from the aspect of covariates, although high-dimensional data have practically low intr…
Fine-tuning Happens in Tiny Subspaces: Exploring Intrinsic Task-specific Subspaces of Pre-trained Language Models
University of Electronic Science and Technology of China（电子科技大学）；Shenzhen Institute for Advanced Study, UESTC（电子科技大学深圳高等研究院）；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）
University of Washington（华盛顿大学）；Harvard University（哈佛大学）；Université de Montréal（蒙特利尔大学）
Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower dimensionality. Learning to identify and extract task-relevant variables from this embedded manifold is crucial when dealing with high-dimensional problems. We find that neural networks are often very effective at solving this task and investigate why. To this end, we apply state-of-the-art techniques for intrinsic dimensionality estimation to show that neural network…
Measuring the Data analytically finds the intrinsic manifold in big data. First, Optimal Transport generates the tangent space at each data point from which the intrinsic dimension is revealed. Then, the Koopman Dimensionality Reduction procedure derives a nonlinear transformation from the data to the intrinsic manifold. Measuring the data procedure is presented here, backed up with encouraging results.
Ariel University（阿里埃勒大学）；Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Weizmann Institute of Science（魏茨曼科学研究所）
We study adaptive data-dependent dimensionality reduction in the context of supervised learning in general metric spaces. Our main statistical contribution is a generalization bound for Lipschitz functions in metric spaces that are doubling, or nearly doubling. On the algorithmic front, we describe an analogue of PCA for metric spaces: namely an efficient procedure that approximates the data's intrinsic dimension, which is often much lower than the ambient dimension. Our approach thus leverages the dual benefits of low dimensionality: (1) more efficient algorithms, e.g., for proximity search, …
Why can pre-trained language models (PLMs) learn universal representations and effectively adapt to broad NLP tasks differing a lot superficially? In this work, we empirically find evidence indicating that the adaptations of PLMs to various few-shot tasks can be reparameterized as optimizing only a few free parameters in a unified low-dimensional intrinsic task subspace, which may help us understand why PLMs could easily adapt to various NLP tasks with small-scale data. To find such a subspace and examine its universality, we propose an analysis pipeline called intrinsic prompt tuning (IPT). S…
The most effective dimensionality reduction procedures produce interpretable features from the raw input space while also providing good performance for downstream supervised learning tasks. For many methods, this requires optimizing one or more hyperparameters for a specific task, which can limit generalizability. In this study we propose sparse compressed agglomeration (SparCA), a novel dimensionality reduction procedure that involves a multistep hierarchical feature grouping, compression, and feature selection process. We demonstrate the characteristics and performance of the SparCA method …
Scientific Computing and Imaging Institute（科学计算与成像研究所）；University of Utah（犹他大学）
The manifold assumption for high-dimensional data assumes that the data is generated by varying a set of parameters obtained from a low-dimensional latent space. Deep generative models (DGMs) are widely used to learn data representations in an unsupervised way. DGMs parameterize the underlying low-dimensional manifold in the data space using bottleneck architectures such as variational autoencoders (VAEs). The bottleneck dimension for VAEs is treated as a hyperparameter that depends on the dataset and is fixed at design time after extensive tuning. As the intrinsic dimensionality of most real-…
The intrinsic dimensionality refers to the ``true'' dimensionality of the data, as opposed to the dimensionality of the data representation. For example, when attributes are highly correlated, the intrinsic dimensionality can be much lower than the number of variables. Local intrinsic dimensionality refers to the observation that this property can vary for different parts of the data set; and intrinsic dimensionality can serve as a proxy for the local difficulty of the data set. Most popular methods for estimating the local intrinsic dimensionality are based on distances, and the rate at which…
CNRS-IRISA（法国国家科学研究中心-法国信息与自动化研究所）；Amazon Web Services（亚马逊云科技）；The University of Melbourne（墨尔本大学）；National Institute of Informatics（国立信息学研究所）；University of Novi Sad, Faculty of Sciences（诺维萨德大学理学院）；Bank of Thailand（泰国银行）
Accurate estimation of Intrinsic Dimensionality (ID) is of crucial importance in many data mining and machine learning tasks, including dimensionality reduction, outlier detection, similarity search and subspace clustering. However, since their convergence generally requires sample sizes (that is, neighborhood sizes) on the order of hundreds of points, existing ID estimation methods may have only limited usefulness for applications in which the data consists of many natural groups of small size. In this paper, we propose a local ID estimation strategy stable even for `tight' localities consist…
Curvature as a tool for evaluating dimensionality reduction and estimating intrinsic dimension
Center for Scalable Data Analytics and Artificial Intelligence (ScaDS,AI)（可扩展数据分析与人工智能中心）；Max Planck Institute for Human Cognitive and Brain Sciences（人类认知与脑科学马克斯·普朗克研究所）；Max Planck Institute for Mathematics in the Sciences（科学数学马克斯·普朗克研究所）；Santa Fe Institute for the Sciences of Complexity（复杂性科学圣菲研究所）
This paper investigates the vulnerability of the nearest neighbors search, which is a pivotal tool in data analysis and machine learning. The vulnerability is gauged as the relative amount of perturbation that an attacker needs to add onto a dataset point in order to modify its neighbor rank w.r.t. a query. The statistical distribution of this quantity is derived from simple assumptions. Experiments on six large scale datasets validate this model up to some outliers which are explained in term of violations of the assumptions.
In the last decades the estimation of the intrinsic dimensionality of a dataset has gained considerable importance. Despite the great deal of research work devoted to this task, most of the proposed solutions prove to be unreliable when the intrinsic dimensionality of the input dataset is high and the manifold where the points lie is nonlinearly embedded in a higher dimensional space. In this paper we propose a novel robust intrinsic dimensionality estimator that exploits the twofold complementary information conveyed both by the normalized nearest neighbor distances and by the angles computed…
Institut Curie（居里研究所）；PSL Research University（巴黎文理研究大学）；Mines ParisTech（巴黎高等矿业学院）；INSERM（法国国家健康与医学研究院）；Lobachevsky University（罗巴切夫斯基大学）
Intrinsic dimensionality (ID) is one of the most fundamental characteristics of multi-dimensional data point clouds. Knowing ID is crucial to choose the appropriate machine learning approach as well as to understand its behavior and validate it. ID can be computed globally for the whole data point distribution, or computed locally in different regions of the data space. In this paper, we introduce new local estimators of ID based on linear separability of multi-dimensional data point clouds, which is one of the manifestations of concentration of measure. We empirically study the properties of …
The vulnerability of machine learning models to adversarial perturbations has motivated a significant amount of research under the broad umbrella of adversarial machine learning. Sophisticated attacks may cause learning algorithms to learn decision functions or make decisions with poor predictive performance. In this context, there is a growing body of literature that uses local intrinsic dimensionality (LID), a local metric that describes the minimum number of latent variables required to describe each data point, for detecting adversarial samples and subsequently mitigating their effects. Th…
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Halıcıǧlu Data Science Institute, University of California San Diego（加利福尼亚大学圣迭戈分校Halıcıǧlu数据科学研究所）；Simula Research Laboratory（Simula研究实验室）
We study the approximation of two-layer compositions $f(x) = g(ϕ(x))$ via deep networks with ReLU activation, where $ϕ$ is a geometrically intuitive, dimensionality reducing feature map. We focus on two intuitive and practically relevant choices for $ϕ$: the projection onto a low-dimensional embedded submanifold and a distance to a collection of low-dimensional sets. We achieve near optimal approximation rates, which depend only on the complexity of the dimensionality reducing map $ϕ$ rather than the ambient dimension. Since $ϕ$ encapsulates all nonlinear features that are material to the func…
The University of Melbourne（墨尔本大学）；National Institute of Informatics（国立信息学研究所）
Generative Adversarial Networks (GANs) are an elegant mechanism for data generation. However, a key challenge when using GANs is how to best measure their ability to generate realistic data. In this paper, we demonstrate that an intrinsic dimensional characterization of the data space learned by a GAN model leads to an effective evaluation metric for GAN quality. In particular, we propose a new evaluation measure, CrossLID, that assesses the local intrinsic dimensionality (LID) of real-world data with respect to neighborhoods found in GAN-generated samples. Intuitively, CrossLID measures the d…
The University of Tokyo（东京大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Kyushu Institute of Technology（九州工业大学）
Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-dependent smoothness and includes several function classes that have been investigated thus far. We demonstrate that the approximation error and estimation error of deep learning only depend on the average value of the smoothness parameters in all directions. Consequently, the curse o…
Most exact methods for k-nearest neighbour search suffer from the curse of dimensionality; that is, their query times exhibit exponential dependence on either the ambient or the intrinsic dimensionality. Dynamic Continuous Indexing (DCI) offers a promising way of circumventing the curse and successfully reduces the dependence of query time on intrinsic dimensionality from exponential to sublinear. In this paper, we propose a variant of DCI, which we call Prioritized DCI, and show a remarkable improvement in the dependence of query time on intrinsic dimensionality. In particular, a linear incre…
National Chung Hsing University（国立中兴大学）；IBM Research（IBM研究院）
Understanding and characterizing the subspaces of adversarial examples aid in studying the robustness of deep neural networks (DNNs) to adversarial perturbations. Very recently, Ma et al. (ICLR 2018) proposed to use local intrinsic dimensionality (LID) in layer-wise hidden representations of DNNs to study adversarial subspaces. It was demonstrated that LID can be used to characterize the adversarial subspaces associated with different attack methods, e.g., the Carlini and Wagner's (C&W) attack and the fast gradient sign attack. In this paper, we use MNIST and CIFAR-10 to conduct two new sets o…
The University of Melbourne（墨尔本大学）；University of California, Berkeley（加利福尼亚大学伯克利分校）；Tsinghua University（清华大学）；University of Michigan（密歇根大学）；National Institute of Informatics（国立信息学研究所）
Deep Neural Networks (DNNs) have recently been shown to be vulnerable against adversarial examples, which are carefully crafted instances that can mislead DNNs to make errors during prediction. To better understand such attacks, a characterization is needed of the properties of regions (the so-called 'adversarial subspaces') in which adversarial examples lie. We tackle this challenge by characterizing the dimensional properties of adversarial regions, via the use of Local Intrinsic Dimensionality (LID). LID assesses the space-filling capability of the region surrounding a reference example, ba…
Dimensionality-Aware Outlier Detection: Theoretical and Experimental Analysis
The University of Newcastle（纽卡斯尔大学）；The University of Melbourne（墨尔本大学）；University of Southern Denmark（南丹麦大学）；New Jersey Institute of Technology（新泽西理工学院）；University of Novi Sad（诺威萨大学）
The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched and the effect of random linear projections on measures such as intrinsic dimensionality still is an ongoing effort. In this paper, we investigate the less explored depths of linear projections onto explicit subspaces of varying dimensionality and the expectations of variance that ensue. The result is a new family of bounds for Euclidean distances and inner pr…
We propose an axiomatic approach to the concept of an intrinsic dimension of a dataset, based on a viewpoint of geometry of high-dimensional structures. Our first axiom postulates that high values of dimension be indicative of the presence of the curse of dimensionality (in a certain precise mathematical sense). The second axiom requires the dimension to depend smoothly on a distance between datasets (so that the dimension of a dataset and that of an approximating principal manifold would be close to each other). The third axiom is a normalization condition: the dimension of the Euclidean $n$-…
LSEC（科学与工程计算国家重点实验室）；Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
Estimating intrinsic dimensionality of data is a classic problem in pattern recognition and statistics. Principal Component Analysis (PCA) is a powerful tool in discovering dimensionality of data sets with a linear structure; it, however, becomes ineffective when data have a nonlinear structure. In this paper, we propose a new PCA-based method to estimate intrinsic dimension of data with nonlinear structures. Our method works by first finding a minimal cover of the data set, then performing PCA locally on each subset in the cover and finally giving the estimation result by checking up the data…
Intrinsic Dimension Estimation for Robust Detection of AI-Generated Texts
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；AI Foundation and Algorithm Lab（人工智能基金会与算法实验室）；Artificial Intelligence Research Institute (AIRI)（人工智能研究所）；CNRS（法国国家科学研究中心）；Université Paris Cité（巴黎城市大学）；Steklov Institute of Mathematics（斯捷克洛夫数学研究所）
Electrical and Computer Engineering Program, King Abdullah University of Science and Technology（计算机与电气工程系，沙特国王阿卜杜拉科技大学）；Center of Excellence for NEOM Research at KAUST, King Abdullah University of Science and Technology（沙特国王阿卜杜拉科技大学NEOM研究卓越中心）
Unlike its intercept, a linear classifier's weight vector cannot be tuned by a simple grid search. Hence, this paper proposes weight vector tuning of a generic binary linear classifier through the parameterization of a decomposition of the discriminant by a scalar which controls the trade-off between conflicting informative and noisy terms. By varying this parameter, the original weight vector is modified in a meaningful way. Applying this method to a number of linear classifiers under a variety of data dimensionality and sample size settings reveals that the classification performance loss du…
In this paper, we develop a local rank correlation measure which quantifies the performance of dimension reduction methods. The local rank correlation is easily interpretable, and robust against the extreme skewness of nearest neighbor distributions in high dimensions. Some benchmark datasets are studied. We find that the local rank correlation closely corresponds to our visual interpretation of the quality of the output. In addition, we demonstrate that the local rank correlation is useful in estimating the intrinsic dimensionality of the original data, and in selecting a suitable value of tu…
University of Warsaw（华沙大学）；Polish National Institute for Machine Learning（波兰国家机器学习研究所）；Applica；Jagiellonian University（雅盖隆大学）；University of Oxford（牛津大学）
Most of the existing methods for estimating the local intrinsic dimension of a data distribution do not scale well to high-dimensional data. Many of them rely on a non-parametric nearest neighbors approach which suffers from the curse of dimensionality. We attempt to address that challenge by proposing a novel approach to the problem: Local Intrinsic Dimension estimation using approximate Likelihood (LIDL). Our method relies on an arbitrary density estimation method as its subroutine and hence tries to sidestep the dimensionality challenge by making use of the recent progress in parametric neu…
Nonlinear dimensionality reduction methods are a popular tool for data scientists and researchers to visualize complex, high dimensional data. However, while these methods continue to improve and grow in number, it is often difficult to evaluate the quality of a visualization due to a variety of factors such as lack of information about the intrinsic dimension of the data and additional tuning required for many evaluation metrics. In this paper, we seek to provide a systematic comparison of dimensionality reduction quality metrics using datasets where we know the ground truth manifold. We util…
Surprisingly High Redundancy in Electronic Structure Data Across Materials Explained by Low Intrinsic Dimensionality
Department of Mechanical and Aerospace Engineering, Michigan Technological University, Houghton, MI, USA.（机械与航空航天工程系，密歇根技术大学，霍顿，MI，美国）；Hopkins Extreme Materials Institute, Johns Hopkins University, Baltimore, MD, USA.（霍普金斯极端材料研究所，约翰霍普金斯大学，巴尔的摩，MD，美国）；Department of Materials Science and Engineering, University of California, Los Angeles, CA, USA.（材料科学与工程系，加州大学洛杉矶分校，CA，美国）；Center for Artificial Intelligence, Michigan Technological University, Houghton, MI, USA.（人工智能中心，密歇根技术大学，霍顿，MI，美国）
