Weiran Wang
We provide a simple and efficient algorithm for the projection operator for weighted $\ell_1$-norm regularization subject to a sum constraint, together with an elementary proof. The implementation of the proposed algorithm can be downloaded from the author's homepage.
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This paper presents an algorithm for efficient training of sparse linear models with elastic net regularization. Extending previous work on delayed updates, the new algorithm applies stochastic gradient updates to non-zero features only, bringing weights current as needed with closed-form updates. Closed-form delayed updates for the $\ell_1$, $\ell_{\infty}$, and rarely used $\ell_2$ regularizers have been described previously. This paper provides closed-form updates for the popular squared norm $\ell^2_2$ and elastic net regularizers. We provide dynamic programming algorithms that perform eac…
Feature selection in learning to rank has recently emerged as a crucial issue. Whereas several preprocessing approaches have been proposed, only a few works have been focused on integrating the feature selection into the learning process. In this work, we propose a general framework for feature selection in learning to rank using SVM with a sparse regularization term. We investigate both classical convex regularizations such as $\ell\_1$ or weighted $\ell\_1$ and non-convex regularization terms such as log penalty, Minimax Concave Penalty (MCP) or $\ell\_p$ pseudo norm with $p\textless{}1$. Tw…
In this paper, we made an extension to the convergence analysis of the dynamics of two-layered bias-free networks with one $ReLU$ output. We took into consideration two popular regularization terms: the $\ell_1$ and $\ell_2$ norm of the parameter vector $w$, and added it to the square loss function with coefficient $λ/2$. We proved that when $λ$ is small, the weight vector $w$ converges to the optimal solution $\hat{w}$ (with respect to the new loss function) with probability $\geq (1-\varepsilon)(1-A_d)/2$ under random initiations in a sphere centered at the origin, where $\varepsilon$ is a s…
Machine learning algorithms typically require abundant data under a stationary environment. However, environments are nonstationary in many real-world applications. Critical issues lie in how to effectively adapt models under an ever-changing environment. We propose a method for transferring knowledge from a source domain to a target domain via $\ell_1$ regularization. We incorporate $\ell_1$ regularization of differences between source parameters and target parameters, in addition to an ordinary $\ell_1$ regularization. Hence, our method yields sparsity for both the estimates themselves and c…
In logistic regression, it is often desirable to utilize regularization to promote sparse solutions, particularly for problems with a large number of features compared to available labels. In this paper, we present screening rules that safely remove features from logistic regression with $\ell_0-\ell_2$ regularization before solving the problem. The proposed safe screening rules are based on lower bounds from the Fenchel dual of strong conic relaxations of the logistic regression problem. Numerical experiments with real and synthetic data suggest that a high percentage of the features can be e…
With the capability of accurately representing a functional relationship between the inputs of a physical system's model and output quantities of interest, neural networks have become popular for surrogate modeling in scientific applications. However, as these networks are over-parameterized, their training often requires a large amount of data. To prevent overfitting and improve generalization error, regularization based on, e.g., $\ell_1$- and $\ell_2$-norms of the parameters is applied. Similarly, multiple connections of the network may be pruned to increase sparsity in the network paramete…
Regularization is a popular technique in machine learning for model estimation and avoiding overfitting. Prior studies have found that modern ordered regularization can be more effective in handling highly correlated, high-dimensional data than traditional regularization. The reason stems from the fact that the ordered regularization can reject irrelevant variables and yield an accurate estimation of the parameters. How to scale up the ordered regularization problems when facing the large-scale training data remains an unanswered question. This paper explores the problem of parameter estimatio…
Fixed Design Analysis of Regularization-Based Continual Learning
该研究在固定设计的双线性回归持续学习场景下，分析ℓ₂正则化CL算法的风险紧界，揭示其遗忘性与顽固性的理论权衡，说明调优的ℓ₂正则化可部分缓解异质任务的灾难性遗忘。
Effects of width-dependent model hyperparameters and $\ell_2$-regularization on the loss landscape of two-layer ReLU networks
研究两层ReLU网络中宽度依赖模型超参数和$\ell_2$正则化对损失景观的影响，推导超参数条件及解析解，通过实验发现AdamW可防参数坍缩，揭示了相关因素对损失景观几何结构的影响。
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…
We analyze the effect of quantizing weights and activations of neural networks on their loss and derive a simple regularization scheme that improves robustness against post-training quantization. By training quantization-ready networks, our approach enables storing a single set of weights that can be quantized on-demand to different bit-widths as energy and memory requirements of the application change. Unlike quantization-aware training using the straight-through estimator that only targets a specific bit-width and requires access to training data and pipeline, our regularization-based method…
Compressed Sensing using $\ell_1$ regularization is among the most powerful and popular sparsification technique in many applications, but why has it not been used to obtain sparse deep learning model such as convolutional neural network (CNN)? This paper is aimed to provide an answer to this question and to show how to make it work. We first demonstrate that the commonly used stochastic gradient decent (SGD) and variants training algorithm is not an appropriate match with $\ell_1$ regularization and then replace it with a different training algorithm based on a regularized dual averaging (RDA…
This work investigates the use of mixed-norm regularization for sensor selection in Event-Related Potential (ERP) based Brain-Computer Interfaces (BCI). The classification problem is cast as a discriminative optimization framework where sensor selection is induced through the use of mixed-norms. This framework is extended to the multi-task learning situation where several similar classification tasks related to different subjects are learned simultaneously. In this case, multi-task learning helps in leveraging data scarcity issue yielding to more robust classifiers. For this purpose, we have i…
While $\ell_2$ regularization is widely used in training gradient boosted trees, popular individualized feature attribution methods for trees such as Saabas and TreeSHAP overlook the training procedure. We propose Prediction Decomposition Attribution (PreDecomp), a novel individualized feature attribution for gradient boosted trees when they are trained with $\ell_2$ regularization. Theoretical analysis shows that the inner product between PreDecomp and labels on in-sample data is essentially the total gain of a tree, and that it can faithfully recover additive models in the population case wh…
Elliptic Loss Regularization
提出一种基于椭圆算子的神经网络正则化技术，通过修改经验风险最小化目标使损失在数据域上满足椭圆算子，从而增强输入到损失映射的平滑性，有效应对分布偏移和组不平衡问题。
Sparse Deep Learning Models with the $\ell_1$ Regularization
本文研究 ℓ1 正则化参数对神经网络稀疏度的影响，推导单参数与多参数稀疏模型，并提出迭代选参算法以获得满足预定稀疏度且逼近精度良好的网络。
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…
Orthogonally weighted $\ell_{2,1}$ regularization for rank-aware joint sparse recovery: algorithm and analysis
本文提出一种名为正交加权 $\ell_{2,1}$ 的正则化方法解决联合稀疏恢复问题，该方法具备秩感知特性，能利用解矩阵的满秩特征，并提供了理论证明与高效收敛算法，数值实验验证了其有效性。
Multi-label learning is usually used to mine the correlation between features and labels, and feature selection can retain as much information as possible through a small number of features. $\ell_{2,1}$ regularization method can get sparse coefficient matrix, but it can not solve multicollinearity problem effectively. The model proposed in this paper can obtain the most relevant few features by solving the joint constrained optimization problems of $\ell_{2,1}$ and $\ell_{F}$ regularization.In manifold regularization, we implement random walk strategy based on joint information matrix, and ge…
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.
We consider a new family of regularizers, termed {\it weighted sorted $\ell_1$ norms} (WSL1), which generalizes the recently introduced {\it octagonal shrinkage and clustering algorithm for regression} (OSCAR) and also contains the $\ell_1$ and $\ell_{\infty}$ norms as particular instances. We focus on a special case of the WSL1, the {\sl decreasing WSL1} (DWSL1), where the elements of the argument vector are sorted in non-increasing order and the weights are also non-increasing. In this paper, after showing that the DWSL1 is indeed a norm, we derive two key tools for its use as a regularizer:…
We investigate the generalizability of deep learning based on the sensitivity to input perturbation. We hypothesize that the high sensitivity to the perturbation of data degrades the performance on it. To reduce the sensitivity to perturbation, we propose a simple and effective regularization method, referred to as spectral norm regularization, which penalizes the high spectral norm of weight matrices in neural networks. We provide supportive evidence for the abovementioned hypothesis by experimentally confirming that the models trained using spectral norm regularization exhibit better general…
Adversarially robust generalization theory via Jacobian regularization for deep neural networks
本研究证明雅可比正则化可作为对抗训练的近似上界，并建立其鲁棒泛化差距，实验表明降低雅可比范数可提升标准和鲁棒泛化。
Convex regularizers are often used for sparse learning. They are easy to optimize, but can lead to inferior prediction performance. The difference of $\ell_1$ and $\ell_2$ ($\ell_{1-2}$) regularizer has been recently proposed as a nonconvex regularizer. It yields better recovery than both $\ell_0$ and $\ell_1$ regularizers on compressed sensing. However, how to efficiently optimize its learning problem is still challenging. The main difficulty is that both the $\ell_1$ and $\ell_2$ norms in $\ell_{1-2}$ are not differentiable, and existing optimization algorithms cannot be applied. In this pap…
We establish a theoretical link between adversarial training and operator norm regularization for deep neural networks. Specifically, we prove that $\ell_p$-norm constrained projected gradient ascent based adversarial training with an $\ell_q$-norm loss on the logits of clean and perturbed inputs is equivalent to data-dependent (p, q) operator norm regularization. This fundamental connection confirms the long-standing argument that a network's sensitivity to adversarial examples is tied to its spectral properties and hints at novel ways to robustify and defend against adversarial attacks. We p…
In this paper, we study the Temporal Difference (TD) learning with linear value function approximation. It is well known that most TD learning algorithms are unstable with linear function approximation and off-policy learning. Recent development of Gradient TD (GTD) algorithms has addressed this problem successfully. However, the success of GTD algorithms requires a set of well chosen features, which are not always available. When the number of features is huge, the GTD algorithms might face the problem of overfitting and being computationally expensive. To cope with this difficulty, regulariz…
Implicit Regularization Leads to Benign Overfitting for Sparse Linear Regression
本文针对稀疏线性回归中最小ℓ₁和ℓ₂范数插值器均无法达到最优测试损失的问题，提出一种新的模型参数化方法，通过梯度下降产生结合ℓ₁与ℓ₂优点的隐式正则化效应，从而获得接近最优测试损失的插值器。
We pursue a line of research that seeks to regularize the spectral norm of the Jacobian of the input-output mapping for deep neural networks. While previous work rely on upper bounding techniques, we provide a scheme that targets the exact spectral norm. We showcase that our algorithm achieves an improved generalization performance compared to previous spectral regularization techniques while simultaneously maintaining a strong safeguard against natural and adversarial noise. Moreover, we further explore some previous reasoning concerning the strong adversarial protection that Jacobian regular…
We compute precise asymptotic expressions for the learning curves of least squares random feature (RF) models with either a separable strongly convex regularization or the $\ell_1$ regularization. We propose a novel multi-level application of the convex Gaussian min max theorem (CGMT) to overcome the traditional difficulty of finding computable expressions for random features models with correlated data. Our result takes the form of a computable 4-dimensional scalar optimization. In contrast to previous results, our approach does not require solving an often intractable proximal operator, whic…
State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no connections between points from different subspaces) under broad theoretical conditions, but the clusters may not be connected. $\ell_2$ and nuclear norm regularization often improve connectivity, but give a subspace-preserving affinity only for independent subspaces. Mixed $\ell_1$, …
We establish an equivalence between the $\ell_2$-regularized solution path for a convex loss function, and the solution of an ordinary differentiable equation (ODE). Importantly, this equivalence reveals that the solution path can be viewed as the flow of a hybrid of gradient descent and Newton method applying to the empirical loss, which is similar to a widely used optimization technique called trust region method. This provides an interesting algorithmic view of $\ell_2$ regularization, and is in contrast to the conventional view that the $\ell_2$ regularization solution path is similar to t…
Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are independent of the dimension of the input space, which may be infinite as in the case of reproducing kernel Hilbert spaces. A byproduct of the proof are bounds on the expected norm of sums of random positive semidefinite matrices with subexponential moments.
We apply concepts from manifold regularization to develop new regularization techniques for training locally stable deep neural networks. Our regularizers are based on a sparsification of the graph Laplacian which holds with high probability when the data is sparse in high dimensions, as is common in deep learning. Empirically, our networks exhibit stability in a diverse set of perturbation models, including $\ell_2$, $\ell_\infty$, and Wasserstein-based perturbations; in particular, we achieve 40% adversarial accuracy on CIFAR-10 against an adaptive PGD attack using $\ell_\infty$ perturbation…
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…
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…
Learning Spatially Adaptive $\ell_1$-Norms Weights for Convolutional Synthesis Regularization
提出一种展开算法，在卷积合成正则化框架中学习空间自适应参数图，通过展开FISTA估计深度参数化稀疏特征图权重，用于低场MRI重建，结果与模型驱动深度学习方法相当且可解释性强。
Recognition of hand gestures is one of the most fundamental tasks in human-robot interaction. Sparse representation based methods have been widely used due to their efficiency and low demands on the training data. Recently, nonconvex regularization techniques including the $\ell_{1-2}$ regularization have been proposed in the image processing community to promote sparsity while achieving efficient performance. In this paper, we propose a vision-based hand gesture recognition model based on the $\ell_{1-2}$ regularization, which is solved by the alternating direction method of multipliers (ADMM…
$\ell_1$ regularization has been used for logistic regression to circumvent the overfitting and use the estimated sparse coefficient for feature selection. However, the challenge of such a regularization is that the $\ell_1$ norm is not differentiable, making the standard algorithms for convex optimization not applicable to this problem. This paper presents a simple projection neural network for $\ell_1$-regularized logistics regression. In contrast to many available solvers in the literature, the proposed neural network does not require any extra auxiliary variable nor any smooth approximatio…
Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on w…
Estimating Implicit Regularization in Deep Learning
本文提出通过梯度匹配方法估计深度学习中的隐式正则化，验证了其在处理复杂网络时的有效性，并展示了在Dropout等方法中的应用。
Adaptive Norm-Based Regularization for Neural Networks
本文提出两种自适应范数正则化策略，通过引入输入特征协方差结构改进传统岭和Lasso惩罚，提升高维数据下的预测性能和复杂度控制。
Multi-task learning leverages structural similarities between multiple tasks to learn despite very few samples. Motivated by the recent success of neural networks applied to data-scarce tasks, we consider a linear low-dimensional shared representation model. Despite an extensive literature, existing theoretical results either guarantee weak estimation rates or require a large number of samples per task. This work provides the first estimation error bound for the trace norm regularized estimator when the number of samples per task is small. The advantages of trace norm regularization for learni…
On the Compression of Neural Networks Using $\ell_0$-Norm Regularization and Weight Pruning
本文提出结合新型 ℓ0 范数正则化、权重剪枝、ℓ2 正则化与微调的神经网络压缩方案，以训练稀疏性并获得小型高效网络。
The de-facto standard approach of promoting sparsity by means of $\ell_1$-regularization becomes ineffective in the presence of simplex constraints, i.e.,~the target is known to have non-negative entries summing up to a given constant. The situation is analogous for the use of nuclear norm regularization for low-rank recovery of Hermitian positive semidefinite matrices with given trace. In the present paper, we discuss several strategies to deal with this situation, from simple to more complex. As a starting point, we consider empirical risk minimization (ERM). It follows from existing theory …
Domain Generalization via Nuclear Norm Regularization
本文提出基于所学特征核范数的正则化方法用于领域泛化，通过抑制环境特征影响来学习领域不变特征，并在多种方法与 DomainBed 基准上验证了其一致的有效性。
Deep neural networks (DNNs) have become increasingly important due to their excellent empirical performance on a wide range of problems. However, regularization is generally achieved by indirect means, largely due to the complex set of functions defined by a network and the difficulty in measuring function complexity. There exists no method in the literature for additive regularization based on a norm of the function, as is classically considered in statistical learning theory. In this work, we propose sampling-based approximations to weighted function norms as regularizers for deep neural net…
A Unified Fractional Regularization Framework for Sparse Recovery
提出基于 ℓ1/ℓp^q 模型的统一分数阶正则化框架，通过参数 p 和 q 泛化多种稀疏正则化器，理论刻画了与 ℓ1-αℓp 模型的一阶驻点等价性，并建立了基于 RIP 的充分恢复条件，开发了 MM 算法并证明收敛性，实验表明在稀疏恢复和 MRI 重建中优于现有方法。
In this paper, we discuss the statistical properties of the $\ell_q$ optimization methods $(0<q\leq 1)$, including the $\ell_q$ minimization method and the $\ell_q$ regularization method, for estimating a sparse parameter from noisy observations in high-dimensional linear regression with either a deterministic or random design. For this purpose, we introduce a general $q$-restricted eigenvalue condition (REC) and provide its sufficient conditions in terms of several widely-used regularity conditions such as sparse eigenvalue condition, restricted isometry property, and mutual incoherence prope…
Low-rank tensor models have been applied in accelerating dynamic magnetic resonance imaging (dMRI). Recently, a new tensor nuclear norm based on t-SVD has been proposed and applied to tensor completion. Inspired by the different properties of the tensor nuclear norm (TNN) and the Casorati matrix nuclear norm (MNN), we introduce a combined TNN and Casorati MNN regularizations framework to reconstruct dMRI, which we term as TMNN. The proposed method simultaneously exploits the spatial structure and the temporal correlation of the dynamic MR data. The optimization problem can be efficiently solve…
Minimum norm interpolation by perceptra: Explicit regularization and implicit bias
本研究分析浅层ReLU网络插值行为，证明在特定权重衰减下经验风险最小化器收敛至最小范数插值，并数值探究优化算法的隐式偏差。
An L-BFGS-B approach for linear and nonlinear system identification under $\ell_1$ and group-Lasso regularization
提出基于 L-BFGS-B 的线性与非线性状态空间辨识方法，支持 ℓ1、group-Lasso 及稳定性惩罚，在机器人 MIMO 基准上验证，并提供 jax-sysid 实现。
Fast Robust Kernel Regression through Sign Gradient Descent with Early Stopping
本文提出KRR的等价目标函数，利用 $\ell_\infty$ 和 $\ell_1$ 惩罚项实现鲁棒和稀疏核回归，并将其分别与符号梯度下降和坐标下降的早停解建立联系，提出高效计算方法，在五个数据集上速度快一到两个数量级且不损失精度。
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