We demonstrate the ability of hybrid regularization methods to automatically avoid the double descent phenomenon arising in the training of random feature models (RFM). The hallmark feature of the double descent phenomenon is a spike in the regularization gap at the interpolation threshold, i.e. when the number of features in the RFM equals the number of training samples. To close this gap, the hybrid method considered in our paper combines the respective strengths of the two most common forms of regularization: early stopping and weight decay. The scheme does not require hyperparameter tuning as it automatically selects the stopping iteration and weight decay hyperparameter by using generalized cross-validation (GCV). This also avoids the necessity of a dedicated validation set. While the benefits of hybrid methods have been well-documented for ill-posed inverse problems, our work presents the first use case in machine learning. To expose the need for regularization and motivate hybrid methods, we perform detailed numerical experiments inspired by image classification. In those examples, the hybrid scheme successfully avoids the double descent phenomenon and yields RFMs whose generalization is comparable with classical regularization approaches whose hyperparameters are tuned optimally using the test data. We provide our MATLAB codes for implementing the numerical experiments in this paper at https://github.com/EmoryMLIP/HybridRFM.
We analyze the training dynamics for deep linear networks using a new metric - layer imbalance - which defines the flatness of a solution. We demonstrate that different regularization methods, such as weight decay or noise data augmentation, behave in a similar way. Training has two distinct phases: 1) optimization and 2) regularization. First, during the optimization phase, the loss function monotonically decreases, and the trajectory goes toward a minima manifold. Then, during the regularization phase, the layer imbalance decreases, and the trajectory goes along the minima manifold toward a …
Rice University（莱斯大学）；Technical University of Munich（慕尼黑工业大学）
The risk of overparameterized models, in particular deep neural networks, is often double-descent shaped as a function of the model size. Recently, it was shown that the risk as a function of the early-stopping time can also be double-descent shaped, and this behavior can be explained as a super-position of bias-variance tradeoffs. In this paper, we show that the risk of explicit L2-regularized models can exhibit double descent behavior as a function of the regularization strength, both in theory and practice. We find that for linear regression, a double descent shaped risk is caused by a supe…
Gradient descent can be surprisingly good at optimizing deep neural networks without overfitting and without explicit regularization. We find that the discrete steps of gradient descent implicitly regularize models by penalizing gradient descent trajectories that have large loss gradients. We call this Implicit Gradient Regularization (IGR) and we use backward error analysis to calculate the size of this regularization. We confirm empirically that implicit gradient regularization biases gradient descent toward flat minima, where test errors are small and solutions are robust to noisy parameter…
Northeast Normal University（东北师范大学）；Beihua University（北华大学）
The study on the implicit regularization induced by gradient-based optimization is a longstanding pursuit. In the present paper, we characterize the implicit regularization of momentum gradient descent (MGD) with early stopping by comparing with the explicit $\ell_2$-regularization (ridge). In details, we study MGD in the continuous-time view, so-called momentum gradient flow (MGF), and show that its tendency is closer to ridge than the gradient descent (GD) [Ali et al., 2019] for least squares regression. Moreover, we prove that, under the calibration $t=\sqrt{2/λ}$, where $t$ is the time par…
In machine learning and statistical data analysis, we often run into objective function that is a summation: the number of terms in the summation possibly is equal to the sample size, which can be enormous. In such a setting, the stochastic mirror descent (SMD) algorithm is a numerically efficient method -- each iteration involving a very small subset of the data. The variance reduction version of SMD (VRSMD) can further improve SMD by inducing faster convergence. On the other hand, algorithms such as gradient descent and stochastic gradient descent have the implicit regularization property th…
We study discrete-time mirror descent applied to the unregularized empirical risk in matrix sensing. In both the general case of rectangular matrices and the particular case of positive semidefinite matrices, a simple potential-based analysis in terms of the Bregman divergence allows us to establish convergence of mirror descent -- with different choices of the mirror maps -- to a matrix that, among all global minimizers of the empirical risk, minimizes a quantity explicitly related to the nuclear norm, the Frobenius norm, and the von Neumann entropy. In both cases, this characterization impli…
The analysis of nonconvex matrix completion has recently attracted much attention in the community of machine learning thanks to its computational convenience. Existing analysis on this problem, however, usually relies on $\ell_{2,\infty}$ projection or regularization that involves unknown model parameters, although they are observed to be unnecessary in numerical simulations, see, e.g., Zheng and Lafferty [2016]. In this paper, we extend the analysis of the vanilla gradient descent for positive semidefinite matrix completion proposed in Ma et al. [2017] to the rectangular case, and more signi…
We describe novel subgradient methods for a broad class of matrix optimization problems involving nuclear norm regularization. Unlike existing approaches, our method executes very cheap iterations by combining low-rank stochastic subgradients with efficient incremental SVD updates, made possible by highly optimized and parallelizable dense linear algebra operations on small matrices. Our practical algorithms always maintain a low-rank factorization of iterates that can be conveniently held in memory and efficiently multiplied to generate predictions in matrix completion settings. Empirical com…
Universitat Pompeu Fabra（庞培法布拉大学）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利理工学院）；Università degli Studi di Genova（热那亚大学）
We propose and analyze a variant of the classic Polyak-Ruppert averaging scheme, broadly used in stochastic gradient methods. Rather than a uniform average of the iterates, we consider a weighted average, with weights decaying in a geometric fashion. In the context of linear least squares regression, we show that this averaging scheme has a the same regularizing effect, and indeed is asymptotically equivalent, to ridge regression. In particular, we derive finite-sample bounds for the proposed approach that match the best known results for regularized stochastic gradient methods.
We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.
Huazhong University of Science and Technology（华中科技大学）
Takagi-Sugeno-Kang (TSK) fuzzy systems are flexible and interpretable machine learning models; however, they may not be easily optimized when the data size is large, and/or the data dimensionality is high. This paper proposes a mini-batch gradient descent (MBGD) based algorithm to efficiently and effectively train TSK fuzzy classifiers. It integrates two novel techniques: 1) uniform regularization (UR), which forces the rules to have similar average contributions to the output, and hence to increase the generalization performance of the TSK classifier; and, 2) batch normalization (BN), which e…
Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）
Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss …
Pacific Northwest National Laboratory（太平洋西北国家实验室）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We present a generic framework for parallel coordinate descent (CD) algorithms that includes, as special cases, the original sequential algorithms Cyclic CD and Stochastic CD, as well as the recent parallel Shotgun algorithm. We introduce two novel parallel algorithms that are also special cases---Thread-Greedy CD and Coloring-Based CD---and give performance measurements for an OpenMP implementation of these.
For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization benefit is not explained by convergence bounds, since the learning rate which maximizes test accuracy is often larger than the learning rate which minimizes training loss. To interpret this phenomenon we prove that for SGD with random shuffling, the mean SGD iterate also stays close to the path of gradient flow if the learning rate is small and finite, but on a m…
Texas A&M University（德克萨斯农工大学）；New York University（纽约大学）
We study the implicit regularization of gradient descent towards structured sparsity via a novel neural reparameterization, which we call a diagonally grouped linear neural network. We show the following intriguing property of our reparameterization: gradient descent over the squared regression loss, without any explicit regularization, biases towards solutions with a group sparsity structure. In contrast to many existing works in understanding implicit regularization, we prove that our training trajectory cannot be simulated by mirror descent. We analyze the gradient dynamics of the correspon…
We study the problem of learning-to-learn: inferring a learning algorithm that works well on tasks sampled from an unknown distribution. As class of algorithms we consider Stochastic Gradient Descent on the true risk regularized by the square euclidean distance to a bias vector. We present an average excess risk bound for such a learning algorithm. This result quantifies the potential benefit of using a bias vector with respect to the unbiased case. We then address the problem of estimating the bias from a sequence of tasks. We propose a meta-algorithm which incrementally updates the bias, as …
We demonstrate that locally linear embedding (LLE) inherently admits some unwanted results when no regularization is used, even for cases in which regularization is not supposed to be needed in the original algorithm. The existence of one special type of result, which we call ``projection pattern'', is mathematically proved in the situation that an exact local linear relation is achieved in each neighborhood of the data. These special patterns as well as some other bizarre results that may occur in more general situations are shown by numerical examples on the Swiss roll with a hole embedded i…
Massachusetts Institute of Technology（麻省理工学院）；California Institute of Technology（加州理工学院）
Despite perfectly interpolating the training data, deep neural networks (DNNs) can often generalize fairly well, in part due to the "implicit regularization" induced by the learning algorithm. Nonetheless, various forms of regularization, such as "explicit regularization" (via weight decay), are often used to avoid overfitting, especially when the data is corrupted. There are several challenges with explicit regularization, most notably unclear convergence properties. Inspired by convergence properties of stochastic mirror descent (SMD) algorithms, we propose a new method for training DNNs wit…
Overfitting is one of the most common problems when training deep neural networks on comparatively small datasets. Here, we demonstrate that neural network activation sparsity is a reliable indicator for overfitting which we utilize to propose novel targeted sparsity visualization and regularization strategies. Based on these strategies we are able to understand and counteract overfitting caused by activation sparsity and filter correlation in a targeted layer-by-layer manner. Our results demonstrate that targeted sparsity regularization can efficiently be used to regularize well-known dataset…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Microsoft Research（微软研究院）
Recent work across many machine learning disciplines has highlighted that standard descent methods, even without explicit regularization, do not merely minimize the training error, but also exhibit an implicit bias. This bias is typically towards a certain regularized solution, and relies upon the details of the learning process, for instance the use of the cross-entropy loss. In this work, we show that for empirical risk minimization over linear predictors with arbitrary convex, strictly decreasing losses, if the risk does not attain its infimum, then the gradient-descent path and the algorit…
We propose \textit{Meta-Regularization}, a novel approach for the adaptive choice of the learning rate in first-order gradient descent methods. Our approach modifies the objective function by adding a regularization term on the learning rate, and casts the joint updating process of parameters and learning rates into a maxmin problem. Given any regularization term, our approach facilitates the generation of practical algorithms. When \textit{Meta-Regularization} takes the $φ$-divergence as a regularizer, the resulting algorithms exhibit comparable theoretical convergence performance with other …
Harvard University（哈佛大学）；Microsoft Research（微软研究院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
Recent empirical and theoretical studies have shown that many learning algorithms -- from linear regression to neural networks -- can have test performance that is non-monotonic in quantities such the sample size and model size. This striking phenomenon, often referred to as "double descent", has raised questions of if we need to re-think our current understanding of generalization. In this work, we study whether the double-descent phenomenon can be avoided by using optimal regularization. Theoretically, we prove that for certain linear regression models with isotropic data distribution, optim…
Huazhong University of Science and Technology（华中科技大学）
Takagi-Sugeno-Kang (TSK) fuzzy systems are very useful machine learning models for regression problems. However, to our knowledge, there has not existed an efficient and effective training algorithm that ensures their generalization performance, and also enables them to deal with big data. Inspired by the connections between TSK fuzzy systems and neural networks, we extend three powerful neural network optimization techniques, i.e., mini-batch gradient descent, regularization, and AdaBound, to TSK fuzzy systems, and also propose three novel techniques (DropRule, DropMF, and DropMembership) spe…
Random Smoothing Regularization in Kernel Gradient Descent Learning
Fudan University（复旦大学）；Purdue University（普渡大学）；The Hong Kong University of Science and Technology（香港科技大学）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））
Generalized linear model with $L_1$ and $L_2$ regularization is a widely used technique for solving classification, class probability estimation and regression problems. With the numbers of both features and examples growing rapidly in the fields like text mining and clickstream data analysis parallelization and the use of cluster architectures becomes important. We present a novel algorithm for fitting regularized generalized linear models in the distributed environment. The algorithm splits data between nodes by features, uses coordinate descent on each node and line search to merge results …
Solving logistic regression with L1-regularization in distributed settings is an important problem. This problem arises when training dataset is very large and cannot fit the memory of a single machine. We present d-GLMNET, a new algorithm solving logistic regression with L1-regularization in the distributed settings. We empirically show that it is superior over distributed online learning via truncated gradient.
We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$ \mathbb{E} ~\left\| Ax_{k+1}-b\right\|^2_{2} \leq \left(1 + \frac{c_{A}}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} - \frac{2}{\|A\|_F^2} \left\|A^T A (x_k - x)\right\|^2_{2}.$$ This is a curious inequality: the last term has one more matrix applied to the residual $u_k - u$ than the remaining terms: if $x_k - x$ is mainly comprised of large singular vecto…
Texas A&M University（德克萨斯A&M大学）；AWS AI；New York University（纽约大学）
In this paper, we study the implicit bias of gradient descent for sparse regression. We extend results on regression with quadratic parametrization, which amounts to depth-2 diagonal linear networks, to more general depth-N networks, under more realistic settings of noise and correlated designs. We show that early stopping is crucial for gradient descent to converge to a sparse model, a phenomenon that we call implicit sparse regularization. This result is in sharp contrast to known results for noiseless and uncorrelated-design cases. We characterize the impact of depth and early stopping and …
University of Cambridge（剑桥大学）；Eötvös Loránd University（罗兰大学）
In gradient descent, changing how we parametrize the model can lead to drastically different optimization trajectories, giving rise to a surprising range of meaningful inductive biases: identifying sparse classifiers or reconstructing low-rank matrices without explicit regularization. This implicit regularization has been hypothesised to be a contributing factor to good generalization in deep learning. However, natural gradient descent is approximately invariant to reparameterization, it always follows the same trajectory and finds the same optimum. The question naturally arises: What happens …
Carnegie Mellon University（卡内基梅隆大学）；Princeton University（普林斯顿大学）；University of Southern California（南加州大学）
We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using …
Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often require proper regularization (e.g. trimming, regularized cost, projection) in order to guarantee fast convergence. For vanilla procedures such as gradient descent, however, prior theory either recommends highly conservative learning rates to avoid overshooting, or completely lacks performance guarantees. This paper uncovers a striking phenomenon in nonconvex op…
Data Science Institute（数据科学研究所）；Imperial College（帝国理工学院）
The role of $L^2$ regularization, in the specific case of deep neural networks rather than more traditional machine learning models, is still not fully elucidated. We hypothesize that this complex interplay is due to the combination of overparameterization and high dimensional phenomena that take place during training and make it unamenable to standard convex optimization methods. Using insights from statistical physics and random fields theory, we introduce a parameter factoring in both the level of the loss function and its remaining nonconvexity: the \emph{complexity}. We proceed to show th…
University of Oxford（牛津大学）；Instadeep Ltd（Instadeep有限公司）；University of Southern California（南加州大学）
We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervis…
ETH Zürich（苏黎世联邦理工学院）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
Injecting noise within gradient descent has several desirable features, such as smoothing and regularizing properties. In this paper, we investigate the effects of injecting noise before computing a gradient step. We demonstrate that small perturbations can induce explicit regularization for simple models based on the L1-norm, group L1-norms, or nuclear norms. However, when applied to overparametrized neural networks with large widths, we show that the same perturbations can cause variance explosion. To overcome this, we propose using independent layer-wise perturbations, which provably allow …
It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $β$ play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?". To answer this question, first, we show that the discrete H.B momentum update (GD+M) follows a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. Then, we show that this implicit regularizer for (GD+M) is stronger than that of (GD) by factor …
Nanjing University（南京大学）；the University of Iowa（爱荷华大学）；Michigan State University（密歇根州立大学）
In this paper, we utilize stochastic optimization to reduce the space complexity of convex composite optimization with a nuclear norm regularizer, where the variable is a matrix of size $m \times n$. By constructing a low-rank estimate of the gradient, we propose an iterative algorithm based on stochastic proximal gradient descent (SPGD), and take the last iterate of SPGD as the final solution. The main advantage of the proposed algorithm is that its space complexity is $O(m+n)$, in contrast, most of previous algorithms have a $O(mn)$ space complexity. Theoretical analysis shows that it achiev…
Recently there has been a surge of interest in understanding implicit regularization properties of iterative gradient-based optimization algorithms. In this paper, we study the statistical guarantees on the excess risk achieved by early-stopped unconstrained mirror descent algorithms applied to the unregularized empirical risk with the squared loss for linear models and kernel methods. By completing an inequality that characterizes convexity for the squared loss, we identify an intrinsic link between offset Rademacher complexities and potential-based convergence analysis of mirror descent meth…
KU Leuven；University of California, Berkeley（加州大学伯克利分校）
In this paper, we provide a precise characterization of generalization properties of high dimensional kernel ridge regression across the under- and over-parameterized regimes, depending on whether the number of training data n exceeds the feature dimension d. By establishing a bias-variance decomposition of the expected excess risk, we show that, while the bias is (almost) independent of d and monotonically decreases with n, the variance depends on n, d and can be unimodal or monotonically decreasing under different regularization schemes. Our refined analysis goes beyond the double descent th…
Chung-Ang University（中央大学）；University of Copenhagen（哥本哈根大学）
Regularization is essential for avoiding over-fitting to training data in network optimization, leading to better generalization of the trained networks. The label noise provides a strong implicit regularization by replacing the target ground truth labels of training examples by uniform random labels. However, it can cause undesirable misleading gradients due to the large loss associated with incorrect labels. We propose a first-order optimization method (Label-Noised Trim-SGD) that uses the label noise with the example trimming in order to remove the outliers based on the loss. The proposed a…
California Institute of Technology（加州理工学院）；Department of Computing and Mathematical Sciences（计算与数学科学系）；Department of Electrical Engineering（电气工程系）
Stochastic descent methods (of the gradient and mirror varieties) have become increasingly popular in optimization. In fact, it is now widely recognized that the success of deep learning is not only due to the special deep architecture of the models, but also due to the behavior of the stochastic descent methods used, which play a key role in reaching "good" solutions that generalize well to unseen data. In an attempt to shed some light on why this is the case, we revisit some minimax properties of stochastic gradient descent (SGD) for the square loss of linear models---originally developed in…
The volume of the distribution of weight sets associated with a loss value may be the source of implicit regularization from overparameterization due to the phenomenon of contracting volume with increasing dimensions for geometric figures demonstrated by hyperspheres. We introduce the geometric regularization conjecture and extract to an explanation for the double descent phenomenon by considering a similar property resulting from shrinking intrinsic dimensionality of the distribution of potential weight set updates available along training path, where if that distribution retracts across a vo…
University of California, Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）
Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We provide the first exact non-asymptotic expressions for double descent of the minimum norm linear e…
Most modern learning problems are highly overparameterized, meaning that there are many more parameters than the number of training data points, and as a result, the training loss may have infinitely many global minima (parameter vectors that perfectly interpolate the training data). Therefore, it is important to understand which interpolating solutions we converge to, how they depend on the initialization point and the learning algorithm, and whether they lead to different generalization performances. In this paper, we study these questions for the family of stochastic mirror descent (SMD) al…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Deep neural networks with remarkably strong generalization performances are usually over-parameterized. Despite explicit regularization strategies are used for practitioners to avoid over-fitting, the impacts are often small. Some theoretical studies have analyzed the implicit regularization effect of stochastic gradient descent (SGD) on simple machine learning models with certain assumptions. However, how it behaves practically in state-of-the-art models and real-world datasets is still unknown. To bridge this gap, we study the role of SGD implicit regularization in deep learning systems. We …
Università degli Studi di Genova（热那亚大学）；Machine Learning Genoa Center（热那亚机器学习中心）；Massachusetts Institute of Technology（麻省理工学院）；Istituto Italiano di Tecnologia（意大利技术研究院）
We study learning properties of accelerated gradient descent methods for linear least-squares in Hilbert spaces. We analyze the implicit regularization properties of Nesterov acceleration and a variant of heavy-ball in terms of corresponding learning error bounds. Our results show that acceleration can provides faster bias decay than gradient descent, but also suffers of a more unstable behavior. As a result acceleration cannot be in general expected to improve learning accuracy with respect to gradient descent, but rather to achieve the same accuracy with reduced computations. Our theoretical…
