Training a deep neural network with a small amount of data is a challenging problem as it is vulnerable to overfitting. However, one of the practical difficulties that we often face is to collect many samples. Transfer learning is a cost-effective solution to this problem. By using the source model trained with a large-scale dataset, the target model can alleviate the overfitting originated from the lack of training data. Resorting to the ability of generalization of the source model, several methods proposed to use the source knowledge during the whole training procedure. However, this is likely to restrict the potential of the target model and some transferred knowledge from the source can interfere with the training procedure. For improving the generalization performance of the target model with a few training samples, we proposed a regularization method called sample-based regularization (SBR), which does not rely on the source's knowledge during training. With SBR, we suggested a new training framework for transfer learning. Experimental results showed that our framework outperformed existing methods in various configurations.
A crucial aspect in reliable machine learning is to design a deployable system in generalizing new related but unobserved environments. Domain generalization aims to alleviate such a prediction gap between the observed and unseen environments. Previous approaches commonly incorporated learning invariant representation for achieving good empirical performance. In this paper, we reveal that merely learning invariant representation is vulnerable to the unseen environment. To this end, we derive novel theoretical analysis to control the unseen test environment error in the representation learning,…
Neural networks have achieved remarkable success in many cognitive tasks. However, when they are trained sequentially on multiple tasks without access to old data, their performance on early tasks tend to drop significantly. This problem is often referred to as catastrophic forgetting, a key challenge in continual learning of neural networks. The regularization-based approach is one of the primary classes of methods to alleviate catastrophic forgetting. In this paper, we provide a novel viewpoint of regularization-based continual learning by formulating it as a second-order Taylor approximatio…
Kakao Brain；Chung-Ang University（中央大学）；Upstage AI Research（Upstage人工智能研究院）；NAVER AI Lab（NAVER人工智能实验室）
Domain generalization (DG) aims to learn a generalized model to an unseen target domain using only limited source domains. Previous attempts to DG fail to learn domain-invariant representations only from the source domains due to the significant domain shifts between training and test domains. Instead, we re-formulate the DG objective using mutual information with the oracle model, a model generalized to any possible domain. We derive a tractable variational lower bound via approximating the oracle model by a pre-trained model, called Mutual Information Regularization with Oracle (MIRO). Our e…
Institute for Infocomm Research, A*STAR（A*STAR资讯通信研究院）；Centre for Frontier AI Research, A*STAR（A*STAR前沿人工智能研究中心）
Domain generalization methods aim to learn models robust to domain shift with data from a limited number of source domains and without access to target domain samples during training. Popular domain alignment methods for domain generalization seek to extract domain-invariant features by minimizing the discrepancy between feature distributions across all domains, disregarding inter-domain relationships. In this paper, we instead propose a novel representation learning methodology that selectively enforces prediction consistency between source domains estimated to be closely-related. Specificall…
Optimal L2 Regularization in High-dimensional Continual Linear Regression
Department of Data and Decisions Sciences, Technion（以色列理工学院数据与决策科学系）；School of Engineering, University of Edinburgh（爱丁堡大学工程学院）；Department of Computer Science, Technion（以色列理工学院计算机科学系）；Department of Electrical and Computing Engineering, Technion（以色列理工学院电气与计算机工程系）；Meta
Nanyang Technological University（南洋理工大学）；City University of Hong Kong（香港城市大学）；Karolinska Institute（卡罗林斯卡学院）；Karolinska University Hospital（卡罗林斯卡大学医院）
Recently, we have witnessed great progress in the field of medical imaging classification by adopting deep neural networks. However, the recent advanced models still require accessing sufficiently large and representative datasets for training, which is often unfeasible in clinically realistic environments. When trained on limited datasets, the deep neural network is lack of generalization capability, as the trained deep neural network on data within a certain distribution (e.g. the data captured by a certain device vendor or patient population) may not be able to generalize to the data with a…
We propose an effective regularization strategy (CW-TaLaR) for solving continual learning problems. It uses a penalizing term expressed by the Cramer-Wold distance between two probability distributions defined on a target layer of an underlying neural network that is shared by all tasks, and the simple architecture of the Cramer-Wold generator for modeling output data representation. Our strategy preserves target layer distribution while learning a new task but does not require remembering previous tasks' datasets. We perform experiments involving several common supervised frameworks, which pr…
Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）；Università degli Studi di Genova（热那亚大学）
We study the generalization properties of stochastic gradient methods for learning with convex loss functions and linearly parameterized functions. We show that, in the absence of penalizations or constraints, the stability and approximation properties of the algorithm can be controlled by tuning either the step-size or the number of passes over the data. In this view, these parameters can be seen to control a form of implicit regularization. Numerical results complement the theoretical findings.
MIT（麻省理工学院）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）；MosaicML（MosaicML公司）；Google Research（谷歌研究院）
Practitioners frequently observe that pruning improves model generalization. A long-standing hypothesis based on bias-variance trade-off attributes this generalization improvement to model size reduction. However, recent studies on over-parameterization characterize a new model size regime, in which larger models achieve better generalization. Pruning models in this over-parameterized regime leads to a contradiction -- while theory predicts that reducing model size harms generalization, pruning to a range of sparsities nonetheless improves it. Motivated by this contradiction, we re-examine pru…
Variational autoencoders optimize an objective that combines a reconstruction loss (the distortion) and a KL term (the rate). The rate is an upper bound on the mutual information, which is often interpreted as a regularizer that controls the degree of compression. We here examine whether inclusion of the rate also acts as an inductive bias that improves generalization. We perform rate-distortion analyses that control the strength of the rate term, the network capacity, and the difficulty of the generalization problem. Decreasing the strength of the rate paradoxically improves generalization in…
Overparameterized neural networks can be highly accurate on average on an i.i.d. test set yet consistently fail on atypical groups of the data (e.g., by learning spurious correlations that hold on average but not in such groups). Distributionally robust optimization (DRO) allows us to learn models that instead minimize the worst-case training loss over a set of pre-defined groups. However, we find that naively applying group DRO to overparameterized neural networks fails: these models can perfectly fit the training data, and any model with vanishing average training loss also already has vanis…
Tsinghua University（清华大学）；University of Minnesota（明尼苏达大学）
A crucial problem in neural networks is to select the most appropriate number of hidden neurons and obtain tight statistical risk bounds. In this work, we present a new perspective towards the bias-variance tradeoff in neural networks. As an alternative to selecting the number of neurons, we theoretically show that $L_1$ regularization can control the generalization error and sparsify the input dimension. In particular, with an appropriate $L_1$ regularization on the output layer, the network can produce a statistical risk that is near minimax optimal. Moreover, an appropriate $L_1$ regulariza…
Over-parameterized deep neural networks trained by simple first-order methods are known to be able to fit any labeling of data. Such over-fitting ability hinders generalization when mislabeled training examples are present. On the other hand, simple regularization methods like early-stopping can often achieve highly nontrivial performance on clean test data in these scenarios, a phenomenon not theoretically understood. This paper proposes and analyzes two simple and intuitive regularization methods: (i) regularization by the distance between the network parameters to initialization, and (ii) a…
Regularization is a well recognized powerful strategy to improve the performance of a learning machine and $l^q$ regularization schemes with $0<q<\infty$ are central in use. It is known that different $q$ leads to different properties of the deduced estimators, say, $l^2$ regularization leads to smooth estimators while $l^1$ regularization leads to sparse estimators. Then, how does the generalization capabilities of $l^q$ regularization learning vary with $q$? In this paper, we study this problem in the framework of statistical learning theory and show that implementing $l^q$ coefficient regul…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of …
Komaba Institute for Science, the University of Tokyo（东京大学驹场科学研究所）；University of Twente（特文特大学）
The classical statistical learning theory implies that fitting too many parameters leads to overfitting and poor performance. That modern deep neural networks generalize well despite a large number of parameters contradicts this finding and constitutes a major unsolved problem towards explaining the success of deep learning. While previous work focuses on the implicit regularization induced by stochastic gradient descent (SGD), we study here how the local geometry of the energy landscape around local minima affects the statistical properties of SGD with Gaussian gradient noise. We argue that u…
Generalization is essential for deep learning. In contrast to previous works claiming that Deep Neural Networks (DNNs) have an implicit regularization implemented by the stochastic gradient descent, we demonstrate explicitly Bayesian regularizations in a specific category of DNNs, i.e., Convolutional Neural Networks (CNNs). First, we introduce a novel probabilistic representation for the hidden layers of CNNs and demonstrate that CNNs correspond to Bayesian networks with the serial connection. Furthermore, we show that the hidden layers close to the input formulate prior distributions, thus CN…
Baidu Research（百度研究院）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Regularization of Deep Neural Networks (DNNs) for the sake of improving their generalization capability is important and challenging. The development in this line benefits theoretical foundation of DNNs and promotes their usability in different areas of artificial intelligence. In this paper, we investigate the role of Rademacher complexity in improving generalization of DNNs and propose a novel regularizer rooted in Local Rademacher Complexity (LRC). While Rademacher complexity is well known as a distribution-free complexity measure of function class that help boost generalization of statisti…
From the statistical learning perspective, complexity control via explicit regularization is a necessity for improving the generalization of over-parameterized models. However, the impressive generalization performance of neural networks with only implicit regularization may be at odds with this conventional wisdom. In this work, we revisit the importance of explicit regularization for obtaining well-calibrated predictive uncertainty. Specifically, we introduce a probabilistic measure of calibration performance, which is lower bounded by the log-likelihood. We then explore explicit regularizat…
Distributionally robust optimization (DRO) has attracted attention in machine learning due to its connections to regularization, generalization, and robustness. Existing work has considered uncertainty sets based on phi-divergences and Wasserstein distances, each of which have drawbacks. In this paper, we study DRO with uncertainty sets measured via maximum mean discrepancy (MMD). We show that MMD DRO is roughly equivalent to regularization by the Hilbert norm and, as a byproduct, reveal deep connections to classic results in statistical learning. In particular, we obtain an alternative proof …
Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization
Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Prusa Research（普拉萨研究公司）；Institute for Robotics and Mechatronics, DLR-German Aerospace Center（德国航空航天中心机器人与机电研究所）
University of California, Los Angeles（加州大学洛杉矶分校）；The University of Hong Kong（香港大学）；Carnegie Mellon University（卡内基梅隆大学）
Adaptive gradient methods such as Adam have gained increasing popularity in deep learning optimization. However, it has been observed that compared with (stochastic) gradient descent, Adam can converge to a different solution with a significantly worse test error in many deep learning applications such as image classification, even with a fine-tuned regularization. In this paper, we provide a theoretical explanation for this phenomenon: we show that in the nonconvex setting of learning over-parameterized two-layer convolutional neural networks starting from the same random initialization, for …
University of Alberta（阿尔伯塔大学）；Google Research（谷歌研究院）；DeepMind Alberta（DeepMind 阿尔伯塔）
Deep reinforcement learning algorithms have shown an impressive ability to learn complex control policies in high-dimensional tasks. However, despite the ever-increasing performance on popular benchmarks, policies learned by deep reinforcement learning algorithms can struggle to generalize when evaluated in remarkably similar environments. In this paper we propose a protocol to evaluate generalization in reinforcement learning through different modes of Atari 2600 games. With that protocol we assess the generalization capabilities of DQN, one of the most traditional deep reinforcement learning…
Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We give a new separation result between the generalization performance of stochastic gradient descent (SGD) and of full-batch gradient descent (GD) in the fundamental stochastic convex optimization model. While for SGD it is well-known that $O(1/ε^2)$ iterations suffice for obtaining a solution with $ε$ excess expected risk, we show that with the same number of steps GD may overfit and emit a solution with $Ω(1)$ generalization error. Moreover, we show that in fact $Ω(1/ε^4)$ iterations are necessary for GD to match the generalization performance of SGD, which is also tight due to recent work …
Technion – Israel Institute of Technology（以色列理工学院）
In the Bayesian reinforcement learning (RL) setting, a prior distribution over the unknown problem parameters -- the rewards and transitions -- is assumed, and a policy that optimizes the (posterior) expected return is sought. A common approximation, which has been recently popularized as meta-RL, is to train the agent on a sample of $N$ problem instances from the prior, with the hope that for large enough $N$, good generalization behavior to an unseen test instance will be obtained. In this work, we study generalization in Bayesian RL under the probably approximately correct (PAC) framework, …
Maximum Probability Framework, powered by Maximum Probability Theorem, is a recent theoretical development in artificial intelligence, aiming to formally define probabilistic models, guiding development of objective functions, and regularization of probabilistic models. MPT uses the probability distribution that the models assume on random variables to provide an upper bound on the probability of the model. We apply MPT to challenging out-of-distribution (OOD) detection problems in computer vision by incorporating MPT as a regularization scheme in the training of CNNs and their energy-based va…
Georgia Institute of Technology（佐治亚理工学院）；University of California, Berkeley（加州大学伯克利分校）
Data-driven machine learning models are being increasingly employed in several important inference problems in biology, chemistry, and physics which require learning over combinatorial spaces. Recent empirical evidence (see, e.g., [1], [2], [3]) suggests that regularizing the spectral representation of such models improves their generalization power when labeled data is scarce. However, despite these empirical studies, the theoretical underpinning of when and how spectral regularization enables improved generalization is poorly understood. In this paper, we focus on learning pseudo-Boolean fun…
ETH Zurich（苏黎世联邦理工学院）；Rice University（莱斯大学）；Technical University of Munich（慕尼黑工业大学）
Numerous recent works show that overparameterization implicitly reduces variance for min-norm interpolators and max-margin classifiers. These findings suggest that ridge regularization has vanishing benefits in high dimensions. We challenge this narrative by showing that, even in the absence of noise, avoiding interpolation through ridge regularization can significantly improve generalization. We prove this phenomenon for the robust risk of both linear regression and classification and hence provide the first theoretical result on robust overfitting.
Hong Kong Baptist University（香港浸会大学）；Southern University of Science and Technology（南方科技大学）
We study gradient-based regularization methods for neural networks. We mainly focus on two regularization methods: the total variation and the Tikhonov regularization. Applying these methods is equivalent to using neural networks to solve some partial differential equations, mostly in high dimensions in practical applications. In this work, we introduce a general framework to analyze the generalization error of regularized networks. The error estimate relies on two assumptions on the approximation error and the quadrature error. Moreover, we conduct some experiments on the image classification…
We present a data dependent generalization bound for a large class of regularized algorithms which implement structured sparsity constraints. The bound can be applied to standard squared-norm regularization, the Lasso, the group Lasso, some versions of the group Lasso with overlapping groups, multiple kernel learning and other regularization schemes. In all these cases competitive results are obtained. A novel feature of our bound is that it can be applied in an infinite dimensional setting such as the Lasso in a separable Hilbert space or multiple kernel learning with a countable number of ke…
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…
University of Tokyo（东京大学）；Tsinghua University（清华大学）；Kyoto University（京都大学）
We propose a novel regularization method, called \textit{volumization}, for neural networks. Inspired by physics, we define a physical volume for the weight parameters in neural networks, and we show that this method is an effective way of regularizing neural networks. Intuitively, this method interpolates between an $L_2$ and $L_\infty$ regularization. Therefore, weight decay and weight clipping become special cases of the proposed algorithm. We prove, on a toy example, that the essence of this method is a regularization technique to control bias-variance tradeoff. The method is shown to do w…
In this paper, we develop a simple yet effective screening rule strategy to improve the computational efficiency in solving structured optimization involving nonconvex $\ell_{q,p}$ regularization. Based on an iteratively reweighted $\ell_1$ (IRL1) framework, the proposed screening rule works like a preprocessing module that potentially removes the inactive groups before starting the subproblem solver, thereby reducing the computational time in total. This is mainly achieved by heuristically exploiting the dual subproblem information during each iteration.Moreover, we prove that our screening r…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Regularization is typically understood as improving generalization by altering the landscape of local extrema to which the model eventually converges. Deep neural networks (DNNs), however, challenge this view: We show that removing regularization after an initial transient period has little effect on generalization, even if the final loss landscape is the same as if there had been no regularization. In some cases, generalization even improves after interrupting regularization. Conversely, if regularization is applied only after the initial transient, it has no effect on the final solution, who…
Improving generalization is one of the main challenges for training deep neural networks on classification tasks. In particular, a number of techniques have been proposed, aiming to boost the performance on unseen data: from standard data augmentation techniques to the $\ell_2$ regularization, dropout, batch normalization, entropy-driven SGD and many more.\\ In this work we propose an elegant, simple and principled approach: post-synaptic potential regularization (PSP). We tested this regularization on a number of different state-of-the-art scenarios. Empirical results show that PSP achieves a…
Microsoft Research（微软研究院）；Ochanomizu University（御茶水女子大学）；University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
We address the problem of enhancing model robustness through regularization. Specifically, we focus on methods that regularize the model posterior difference between clean and noisy inputs. Theoretically, we provide a connection of two recent methods, Jacobian Regularization and Virtual Adversarial Training, under this framework. Additionally, we generalize the posterior differential regularization to the family of $f$-divergences and characterize the overall regularization framework in terms of Jacobian matrix. Empirically, we systematically compare those regularizations and standard BERT tra…
University of California, Los Angeles（加州大学洛杉矶分校）；University of Cambridge（剑桥大学）；The Alan Turing Institute（艾伦图灵研究所）
Regularization improves generalization of supervised models to out-of-sample data. Prior works have shown that prediction in the causal direction (effect from cause) results in lower testing error than the anti-causal direction. However, existing regularization methods are agnostic of causality. We introduce Causal Structure Learning (CASTLE) regularization and propose to regularize a neural network by jointly learning the causal relationships between variables. CASTLE learns the causal directed acyclical graph (DAG) as an adjacency matrix embedded in the neural network's input layers, thereby…
Space Applications Centre（空间应用中心）；Indian Space Research Organization（印度空间研究组织）
Despite numerous attempts sought to provide empirical evidence of adversarial regularization outperforming sole supervision, the theoretical understanding of such phenomena remains elusive. In this study, we aim to resolve whether adversarial regularization indeed performs better than sole supervision at a fundamental level. To bring this insight into fruition, we study vanishing gradient issue, asymptotic iteration complexity, gradient flow and provable convergence in the context of sole supervision and adversarial regularization. The key ingredient is a theoretical justification supported by…
Using weight decay to penalize the L2 norms of weights in neural networks has been a standard training practice to regularize the complexity of networks. In this paper, we show that a family of regularizers, including weight decay, is ineffective at penalizing the intrinsic norms of weights for networks with positively homogeneous activation functions, such as linear, ReLU and max-pooling functions. As a result of homogeneity, functions specified by the networks are invariant to the shifting of weight scales between layers. The ineffective regularizers are sensitive to such shifting and thus p…
Toyota Technological Institute at Chicago（芝加哥丰田技术学院）
In an attempt to better understand generalization in deep learning, we study several possible explanations. We show that implicit regularization induced by the optimization method is playing a key role in generalization and success of deep learning models. Motivated by this view, we study how different complexity measures can ensure generalization and explain how optimization algorithms can implicitly regularize complexity measures. We empirically investigate the ability of these measures to explain different observed phenomena in deep learning. We further study the invariances in neural netwo…
Advanced data augmentation strategies have widely been studied to improve the generalization ability of deep learning models. Regional dropout is one of the popular solutions that guides the model to focus on less discriminative parts by randomly removing image regions, resulting in improved regularization. However, such information removal is undesirable. On the other hand, recent strategies suggest to randomly cut and mix patches and their labels among training images, to enjoy the advantages of regional dropout without having any pointless pixel in the augmented images. We argue that such r…
