Functional Regularization for Representation Learning: A Unified Theoretical Perspective
Amazon Alexa AI Search（亚马逊Alexa AI搜索）；Department of Computer Sciences（计算机科学系）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Manhattan Beach, CA, USA（美国加利福尼亚州曼哈顿海滩（此处为地址，按要求不提取））
Unsupervised and self-supervised learning approaches have become a crucial tool to learn representations for downstream prediction tasks. While these approaches are widely used in practice and achieve impressive empirical gains, their theoretical understanding largely lags behind. Towards bridging this gap, we present a unifying perspective where several such approaches can be viewed as imposing a regularization on the representation via a learnable function using unlabeled data. We propose a discriminative theoretical framework for analyzing the sample complexity of these approaches, which generalizes the framework of (Balcan and Blum, 2010) to allow learnable regularization functions. Our sample complexity bounds show that, with carefully chosen hypothesis classes to exploit the structure in the data, these learnable regularization functions can prune the hypothesis space, and help reduce the amount of labeled data needed. We then provide two concrete examples of functional regularization, one using auto-encoders and the other using masked self-supervision, and apply our framework to quantify the reduction in the sample complexity bound of labeled data. We also provide complementary empirical results to support our analysis.
Chicago Booth School of Business（芝加哥大学布斯商学院）；University of Chicago（芝加哥大学）；George Mason University（乔治梅森大学）
Deep learning is a form of machine learning for nonlinear high dimensional pattern matching and prediction. By taking a Bayesian probabilistic perspective, we provide a number of insights into more efficient algorithms for optimisation and hyper-parameter tuning. Traditional high-dimensional data reduction techniques, such as principal component analysis (PCA), partial least squares (PLS), reduced rank regression (RRR), projection pursuit regression (PPR) are all shown to be shallow learners. Their deep learning counterparts exploit multiple deep layers of data reduction which provide predicti…
Purdue University（普渡大学）；Hong Kong University of Science and Technology（香港科技大学）；East China Normal University（华东师范大学）
Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well overparametrized neural networks can recover the true target function in the presence of random noises. We establish a lower bound on the $L_2$ estimation error with respect to the GD iterations, which is away from zero without a delicate scheme of early stopping. In turn, through a comprehensive analysis of $\ell_2$-regularized GD trajectories, we prove that for…
Stony Brook University（石溪大学）；Tencent AI Lab（腾讯人工智能实验室）
Multi-task learning (MTL) is a common paradigm that seeks to improve the generalization performance of task learning by training related tasks simultaneously. However, it is still a challenging problem to search the flexible and accurate architecture that can be shared among multiple tasks. In this paper, we propose a novel deep learning model called Task Adaptive Activation Network (TAAN) that can automatically learn the optimal network architecture for MTL. The main principle of TAAN is to derive flexible activation functions for different tasks from the data with other parameters of the net…
Bridging the Gap Between Target Networks and Functional Regularization
ServiceNow Research（ServiceNow研究院）；Mila（米拉人工智能研究所）；Université de Montréal（蒙特利尔大学）；DeepMind（深度思维）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Polytechnique Montréal（蒙特利尔理工学院）
The University of Chicago（芝加哥大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Motivated by the rising abundance of observational data with continuous treatments, we investigate the problem of estimating the average dose-response curve (ADRF). Available parametric methods are limited in their model space, and previous attempts in leveraging neural network to enhance model expressiveness relied on partitioning continuous treatment into blocks and using separate heads for each block; this however produces in practice discontinuous ADRFs. Therefore, the question of how to adapt the structure and training of neural network to estimate ADRFs remains open. This paper makes two…
In this paper we study the problem of convergence and generalization error bound of stochastic momentum for deep learning from the perspective of regularization. To do so, we first interpret momentum as solving an $\ell_2$-regularized minimization problem to learn the offsets between arbitrary two successive model parameters. We call this {\em time-delay momentum} because the model parameter is updated after a few iterations towards finding the minimizer. We then propose our learning algorithm, \ie stochastic gradient descent (SGD) with time-delay momentum. We show that our algorithm can be in…
We propose a new point of view for regularizing deep neural networks by using the norm of a reproducing kernel Hilbert space (RKHS). Even though this norm cannot be computed, it admits upper and lower approximations leading to various practical strategies. Specifically, this perspective (i) provides a common umbrella for many existing regularization principles, including spectral norm and gradient penalties, or adversarial training, (ii) leads to new effective regularization penalties, and (iii) suggests hybrid strategies combining lower and upper bounds to get better approximations of the RKH…
The adversarial vulnerability of deep neural networks (DNNs) has been actively investigated in the past several years. This paper investigates the scale-variant property of cross-entropy loss, which is the most commonly used loss function in classification tasks, and its impact on the effective margin and adversarial robustness of deep neural networks. Since the loss function is not invariant to logit scaling, increasing the effective weight norm will make the loss approach zero and its gradient vanish while the effective margin is not adequately maximized. On typical DNNs, we demonstrate that…
CERC, Polytechnique Montréal（蒙特利尔理工学院计算与工程研究中心）；University of Pisa（比萨大学）；Huawei Montreal Research Centre（华为蒙特利尔研究中心）；Jacobs Technion-Cornell Institute（雅各布斯泰克尼翁-康奈尔研究所）；Cornell Tech（康奈尔理工学院）；Technion - IIT（以色列理工学院）
In Machine Learning, Artificial Neural Networks (ANNs) are a very powerful tool, broadly used in many applications. Often, the selected (deep) architectures include many layers, and therefore a large amount of parameters, which makes training, storage and inference expensive. This motivated a stream of research about compressing the original networks into smaller ones without excessively sacrificing performances. Among the many proposed compression approaches, one of the most popular is \emph{pruning}, whereby entire elements of the ANN (links, nodes, channels, \ldots) and the corresponding we…
Institute for Computational & Mathematical Engineering, Stanford University（斯坦福大学计算与数学工程研究所）；Management Science & Engineering, Stanford University（斯坦福大学管理科学与工程系）
This paper provides a unifying view of a wide range of problems of interest in machine learning by framing them as the minimization of functionals defined on the space of probability measures. In particular, we show that generative adversarial networks, variational inference, and actor-critic methods in reinforcement learning can all be seen through the lens of our framework. We then discuss a generic optimization algorithm for our formulation, called probability functional descent (PFD), and show how this algorithm recovers existing methods developed independently in the settings mentioned ea…
Many recent datasets contain a variety of different data modalities, for instance, image, question, and answer data in visual question answering (VQA). When training deep net classifiers on those multi-modal datasets, the modalities get exploited at different scales, i.e., some modalities can more easily contribute to the classification results than others. This is suboptimal because the classifier is inherently biased towards a subset of the modalities. To alleviate this shortcoming, we propose a novel regularization term based on the functional entropy. Intuitively, this term encourages to b…
A Random Matrix Perspective of Echo State Networks: From Precise Bias--Variance Characterization to Optimal Regularization
Huawei Noah’s Ark Lab, Huawei Technologies, Paris, France（华为诺亚实验室，华为技术，法国巴黎）；Chinese university of Hongkong, Shenzhen, China（香港中文大学（深圳））；Huazhong University of Science Technology (HUST), Wuhan, China（华中科技大学）
Multiparameter regularization and aggregation in the context of polynomial functional regression
Department of Radiology, Medical University of Innsbruck（放射科，因斯布鲁克医学大学）；Neuroimaging Research Core Facility, Medical University of Innsbruck（神经影像研究核心设施，因斯布鲁克医学大学）；Institute for Machine Learning, Johannes Kepler University（机器学习研究所，约瑟夫·施拉格大学）；Department of Neurology, Medical University of Innsbruck（神经病学系，因斯布鲁克医学大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（约瑟夫·拉东研究所，奥地利科学院）
Kidney DCE-MRI aims at both qualitative assessment of kidney anatomy and quantitative assessment of kidney function by estimating the tracer kinetic (TK) model parameters. Accurate estimation of TK model parameters requires an accurate measurement of the arterial input function (AIF) with high temporal resolution. Accelerated imaging is used to achieve high temporal resolution, which yields under-sampling artifacts in the reconstructed images. Compressed sensing (CS) methods offer a variety of reconstruction options. Most commonly, sparsity of temporal differences is encouraged for regularizat…
We consider functional outlier detection from a geometric perspective, specifically: for functional data sets drawn from a functional manifold which is defined by the data's modes of variation in amplitude and phase. Based on this manifold, we develop a conceptualization of functional outlier detection that is more widely applicable and realistic than previously proposed. Our theoretical and experimental analyses demonstrate several important advantages of this perspective: It considerably improves theoretical understanding and allows to describe and analyse complex functional outlier scenario…
Regularization vs. Relaxation: A conic optimization perspective of statistical variable selection
Department of Mathematics, Washington State University（数学系，华盛顿州立大学）；Department of Statistics, University of Connecticut（统计系，康涅狄格大学）；Department of Industrial and Systems Engineering, University of Wisconsin-Madison（工业与系统工程系，威斯康星大学麦迪逊分校）
Outlier-robust neural network training: variation regularization meets trimmed loss to prevent functional breakdown
Institute of Statistical Mathematics（统计数学研究所）；The Graduate University for Advanced Studies, SOKENDAI（高等研究大学、SOKENDAI）；RIKEN（理化学研究所）；Joint Support-Center for Data Science Research（数据科学联合支持中心）
Horizon Robotics（地平线机器人）；University at Buffalo（布法罗大学）；Tsinghua University（清华大学）
Knowledge distillation is an effective approach to leverage a well-trained network or an ensemble of them, named as the teacher, to guide the training of a student network. The outputs from the teacher network are used as soft labels for supervising the training of a new network. Recent studies \citep{muller2019does,yuan2020revisiting} revealed an intriguing property of the soft labels that making labels soft serves as a good regularization to the student network. From the perspective of statistical learning, regularization aims to reduce the variance, however how bias and variance change is n…
Center for Research on Intelligent Perception and Computing, CASIA（中国科学院自动化研究所智能感知与计算研究中心）；National Laboratory of Pattern Recognition, CASIA（中国科学院自动化研究所模式识别国家重点实验室）；Center for Excellence in Brain Science and Intelligence Technology, CAS（中国科学院脑科学与智能技术卓越创新中心）
Self-paced learning (SPL) mimics the cognitive mechanism of humans and animals that gradually learns from easy to hard samples. One key issue in SPL is to obtain better weighting strategy that is determined by minimizer function. Existing methods usually pursue this by artificially designing the explicit form of SPL regularizer. In this paper, we focus on the minimizer function, and study a group of new regularizer, named self-paced implicit regularizer that is deduced from robust loss function. Based on the convex conjugacy theory, the minimizer function for self-paced implicit regularizer ca…
University of Pennsylvania（宾夕法尼亚大学）；Princeton University（普林斯顿大学）；Columbia University（哥伦比亚大学）
Sorted l1 regularization has been incorporated into many methods for solving high-dimensional statistical estimation problems, including the SLOPE estimator in linear regression. In this paper, we study how this relatively new regularization technique improves variable selection by characterizing the optimal SLOPE trade-off between the false discovery proportion (FDP) and true positive proportion (TPP) or, equivalently, between measures of type I error and power. Assuming a regime of linear sparsity and working under Gaussian random designs, we obtain an upper bound on the optimal trade-off fo…
We propose a simple architecture for deep reinforcement learning by embedding inputs into a learned Fourier basis and show that it improves the sample efficiency of both state-based and image-based RL. We perform infinite-width analysis of our architecture using the Neural Tangent Kernel and theoretically show that tuning the initial variance of the Fourier basis is equivalent to functional regularization of the learned deep network. That is, these learned Fourier features allow for adjusting the degree to which networks underfit or overfit different frequencies in the training data, and hence…
University of Helsinki（赫尔辛基大学）；University of Cambridge（剑桥大学）；Friedrich-Alexander Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Lappeenranta University of Technology（拉彭兰塔理工大学）；University of Genoa（热那亚大学）
We consider a statistical inverse learning problem, where the task is to estimate a function $f$ based on noisy point evaluations of $Af$, where $A$ is a linear operator. The function $Af$ is evaluated at i.i.d. random design points $u_n$, $n=1,...,N$ generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and $p$-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov no…
Ca’ Foscari University（威尼斯福斯卡里大学）；University of New South Wales（新南威尔士大学）；Computer Vision Center（计算机视觉中心）
In this paper, we investigate the continual learning of Vision Transformers (ViT) for the challenging exemplar-free scenario, with special focus on how to efficiently distill the knowledge of its crucial self-attention mechanism (SAM). Our work takes an initial step towards a surgical investigation of SAM for designing coherent continual learning methods in ViTs. We first carry out an evaluation of established continual learning regularization techniques. We then examine the effect of regularization when applied to two key enablers of SAM: (a) the contextualized embedding layers, for their abi…
Technion - Israel Institute of Technology（以色列理工学院）
Contemporary predictive models are hard to interpret as their deep nets exploit numerous complex relations between input elements. This work suggests a theoretical framework for model interpretability by measuring the contribution of relevant features to the functional entropy of the network with respect to the input. We rely on the log-Sobolev inequality that bounds the functional entropy by the functional Fisher information with respect to the covariance of the data. This provides a principled way to measure the amount of information contribution of a subset of features to the decision funct…
German Research Center for AI (DFKI)（德国人工智能研究中心（DFKI））；Centre for Cognitive Science（认知科学中心）
Recent methods for imitation learning directly learn a $Q$-function using an implicit reward formulation rather than an explicit reward function. However, these methods generally require implicit reward regularization to improve stability and often mistreat absorbing states. Previous works show that a squared norm regularization on the implicit reward function is effective, but do not provide a theoretical analysis of the resulting properties of the algorithms. In this work, we show that using this regularizer under a mixture distribution of the policy and the expert provides a particularly il…
Artificial Intelligence Department, Faculty of Computer Engineering, University of Isfahan（伊斯法罕大学计算机工程学院人工智能系）；Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab（格勒诺布尔阿尔卑斯大学（Univ. Grenoble Alpes）、法国国家科学研究中心（CNRS）、格勒诺布尔综合理工学院（Grenoble INP）、GIPSA实验室（GIPSA-lab））
Multimodal manifold modeling methods extend the spectral geometry-aware data analysis to learning from several related and complementary modalities. Most of these methods work based on two major assumptions: 1) there are the same number of homogeneous data samples in each modality, and 2) at least partial correspondences between modalities are given in advance as prior knowledge. This work proposes two new multimodal modeling methods. The first method establishes a general analyzing framework to deal with the multimodal information problem for heterogeneous data without any specific prior know…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
A family of regularization functionals is said to admit a linear representer theorem if every member of the family admits minimizers that lie in a fixed finite dimensional subspace. A recent characterization states that a general class of regularization functionals with differentiable regularizer admits a linear representer theorem if and only if the regularization term is a non-decreasing function of the norm. In this report, we improve over such result by replacing the differentiability assumption with lower semi-continuity and deriving a proof that is independent of the dimensionality of th…
