Qian Lin

Jianfa Lai, Manyun Xu, Rui Chen et al.

We perform a study on the generalization ability of the wide two-layer ReLU neural network on $\mathbb{R}$. We first establish some spectral properties of the neural tangent kernel (NTK): $a)$ $K_{d}$, the NTK defined on $\mathbb{R}^{d}$, is positive definite; $b)$ $λ_{i}(K_{1})$, the $i$-th largest eigenvalue of $K_{1}$, is proportional to $i^{-2}$. We then show that: $i)$ when the width $m\rightarrow\infty$, the neural network kernel (NNK) uniformly converges to the NTK; $ii)$ the minimax rate of regression over the RKHS associated to $K_{1}$ is $n^{-2/3}$; $iii)$ if one adopts the early stopping strategy in training a wide neural network, the resulting neural network achieves the minimax rate; $iv)$ if one trains the neural network till it overfits the data, the resulting neural network can not generalize well. Finally, we provide an explanation to reconcile our theory and the widely observed ``benign overfitting phenomenon''.

Haobo Zhang, Yicheng Li, Qian Lin

In the misspecified spectral algorithms problem, researchers usually assume the underground true function $f_ρ^{*} \in [\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ for some $s\in (0,1)$. The existing minimax optimal results require $\|f_ρ^{*}\|_{L^{\infty}}<\infty$ which implicitly requires $s > α_{0}$ where $α_{0}\in (0,1)$ is the embedding index, a constant depending on $\mathcal{H}$. Whether the spectral algorithms are optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that spectral algorithms are minimax optimal for any $α_{0}-\frac{1}β < s < 1$, where $β$ is the eigenvalue decay rate of $\mathcal{H}$. We also give several classes of RKHSs whose embedding index satisfies $ α_0 = \frac{1}β $. Thus, the spectral algorithms are minimax optimal for all $s\in (0,1)$ on these RKHSs.

Yicheng Li, Haobo Zhang, Qian Lin

The widely observed 'benign overfitting phenomenon' in the neural network literature raises the challenge to the 'bias-variance trade-off' doctrine in the statistical learning theory. Since the generalization ability of the 'lazy trained' over-parametrized neural network can be well approximated by that of the neural tangent kernel regression, the curve of the excess risk (namely, the learning curve) of kernel ridge regression attracts increasing attention recently. However, most recent arguments on the learning curve are heuristic and are based on the 'Gaussian design' assumption. In this paper, under mild and more realistic assumptions, we rigorously provide a full characterization of the learning curve: elaborating the effect and the interplay of the choice of the regularization parameter, the source condition and the noise. In particular, our results suggest that the 'benign overfitting phenomenon' exists in very wide neural networks only when the noise level is small.

Weihao Lu, Haobo Zhang, Yicheng Li et al.

We perform a study on kernel regression for large-dimensional data (where the sample size $n$ is polynomially depending on the dimension $d$ of the samples, i.e., $n\asymp d^γ$ for some $γ>0$ ). We first build a general tool to characterize the upper bound and the minimax lower bound of kernel regression for large dimensional data through the Mendelson complexity $\varepsilon_{n}^{2}$ and the metric entropy $\bar{\varepsilon}_{n}^{2}$ respectively. When the target function falls into the RKHS associated with a (general) inner product model defined on $\mathbb{S}^{d}$, we utilize the new tool to show that the minimax rate of the excess risk of kernel regression is $n^{-1/2}$ when $n\asymp d^γ$ for $γ=2, 4, 6, 8, \cdots$. We then further determine the optimal rate of the excess risk of kernel regression for all the $γ>0$ and find that the curve of optimal rate varying along $γ$ exhibits several new phenomena including the multiple descent behavior and the periodic plateau behavior. As an application, For the neural tangent kernel (NTK), we also provide a similar explicit description of the curve of optimal rate. As a direct corollary, we know these claims hold for wide neural networks as well.

Qian Lin, Bo Tang, Zifan Wu et al.

Aiming at promoting the safe real-world deployment of Reinforcement Learning (RL), research on safe RL has made significant progress in recent years. However, most existing works in the literature still focus on the online setting where risky violations of the safety budget are likely to be incurred during training. Besides, in many real-world applications, the learned policy is required to respond to dynamically determined safety budgets (i.e., constraint threshold) in real time. In this paper, we target at the above real-time budget constraint problem under the offline setting, and propose Trajectory-based REal-time Budget Inference (TREBI) as a novel solution that models this problem from the perspective of trajectory distribution and solves it through diffusion model planning. Theoretically, we prove an error bound of the estimation on the episodic reward and cost under the offline setting and thus provide a performance guarantee for TREBI. Empirical results on a wide range of simulation tasks and a real-world large-scale advertising application demonstrate the capability of TREBI in solving real-time budget constraint problems under offline settings.

Yicheng Li, Haobo Zhang, Qian Lin

One of the most interesting problems in the recent renaissance of the studies in kernel regression might be whether the kernel interpolation can generalize well, since it may help us understand the `benign overfitting henomenon' reported in the literature on deep networks. In this paper, under mild conditions, we show that for any $\varepsilon>0$, the generalization error of kernel interpolation is lower bounded by $Ω(n^{-\varepsilon})$. In other words, the kernel interpolation generalizes poorly for a large class of kernels. As a direct corollary, we can show that overfitted wide neural networks defined on the sphere generalize poorly.

Qian Lin, Zongkai Liu, Danying Mo et al.

In recent years, significant progress has been made in multi-objective reinforcement learning (RL) research, which aims to balance multiple objectives by incorporating preferences for each objective. In most existing studies, specific preferences must be provided during deployment to indicate the desired policies explicitly. However, designing these preferences depends heavily on human prior knowledge, which is typically obtained through extensive observation of high-performing demonstrations with expected behaviors. In this work, we propose a simple yet effective offline adaptation framework for multi-objective RL problems without assuming handcrafted target preferences, but only given several demonstrations to implicitly indicate the preferences of expected policies. Additionally, we demonstrate that our framework can naturally be extended to meet constraints on safety-critical objectives by utilizing safe demonstrations, even when the safety thresholds are unknown. Empirical results on offline multi-objective and safe tasks demonstrate the capability of our framework to infer policies that align with real preferences while meeting the constraints implied by the provided demonstrations.

Zifan Wu, Bo Tang, Qian Lin et al.

Primal-dual safe RL methods commonly perform iterations between the primal update of the policy and the dual update of the Lagrange Multiplier. Such a training paradigm is highly susceptible to the error in cumulative cost estimation since this estimation serves as the key bond connecting the primal and dual update processes. We show that this problem causes significant underestimation of cost when using off-policy methods, leading to the failure to satisfy the safety constraint. To address this issue, we propose conservative policy optimization, which learns a policy in a constraint-satisfying area by considering the uncertainty in cost estimation. This improves constraint satisfaction but also potentially hinders reward maximization. We then introduce local policy convexification to help eliminate such suboptimality by gradually reducing the estimation uncertainty. We provide theoretical interpretations of the joint coupling effect of these two ingredients and further verify them by extensive experiments. Results on benchmark tasks show that our method not only achieves an asymptotic performance comparable to state-of-the-art on-policy methods while using much fewer samples, but also significantly reduces constraint violation during training. Our code is available at https://github.com/ZifanWu/CAL.

Yicheng Li, Haobo Zhang, Qian Lin

The saturation effect refers to the phenomenon that the kernel ridge regression (KRR) fails to achieve the information theoretical lower bound when the smoothness of the underground truth function exceeds certain level. The saturation effect has been widely observed in practices and a saturation lower bound of KRR has been conjectured for decades. In this paper, we provide a proof of this long-standing conjecture.

Zongkai Liu, Qian Lin, Chao Yu et al.

Offline Multi-Agent Reinforcement Learning (MARL) is an emerging field that aims to learn optimal multi-agent policies from pre-collected datasets. Compared to single-agent case, multi-agent setting involves a large joint state-action space and coupled behaviors of multiple agents, which bring extra complexity to offline policy optimization. In this work, we revisit the existing offline MARL methods and show that in certain scenarios they can be problematic, leading to uncoordinated behaviors and out-of-distribution (OOD) joint actions. To address these issues, we propose a new offline MARL algorithm, named In-Sample Sequential Policy Optimization (InSPO). InSPO sequentially updates each agent's policy in an in-sample manner, which not only avoids selecting OOD joint actions but also carefully considers teammates' updated policies to enhance coordination. Additionally, by thoroughly exploring low-probability actions in the behavior policy, InSPO can well address the issue of premature convergence to sub-optimal solutions. Theoretically, we prove InSPO guarantees monotonic policy improvement and converges to quantal response equilibrium (QRE). Experimental results demonstrate the effectiveness of our method compared to current state-of-the-art offline MARL methods.

Haobo Zhang, Weihao Lu, Qian Lin

The generalization ability of kernel interpolation in large dimensions (i.e., $n \asymp d^γ$ for some $γ>0$) might be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us understand the 'benign overfitting phenomenon' reported in the neural networks literature. Focusing on the inner product kernel on the sphere, we fully characterized the exact order of both the variance and bias of large-dimensional kernel interpolation under various source conditions $s\geq 0$. Consequently, we obtained the $(s,γ)$-phase diagram of large-dimensional kernel interpolation, i.e., we determined the regions in $(s,γ)$-plane where the kernel interpolation is minimax optimal, sub-optimal and inconsistent.

Yicheng Li, Weiye Gan, Zuoqiang Shi et al.

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

Shaoyuan Xu, Yang Cheng, Qian Lin et al.

Emotion has an important role in daily life, as it helps people better communicate with and understand each other more efficiently. Facial expressions can be classified into 7 categories: angry, disgust, fear, happy, neutral, sad and surprise. How to detect and recognize these seven emotions has become a popular topic in the past decade. In this paper, we develop an emotion recognition system that can apply emotion recognition on both still images and real-time videos by using deep learning. We build our own emotion recognition classification and regression system from scratch, which includes dataset collection, data preprocessing , model training and testing. Given a certain image or a real-time video, our system is able to show the classification and regression results for all of the 7 emotions. The proposed system is tested on 2 different datasets, and achieved an accuracy of over 80\%. Moreover, the result obtained from real-time testing proves the feasibility of implementing convolutional neural networks in real time to detect emotions accurately and efficiently.

Haobo Zhang, Yicheng Li, Weihao Lu et al.

Motivated by the studies of neural networks (e.g.,the neural tangent kernel theory), we perform a study on the large-dimensional behavior of kernel ridge regression (KRR) where the sample size $n \asymp d^γ$ for some $γ> 0$. Given an RKHS $\mathcal{H}$ associated with an inner product kernel defined on the sphere $\mathbb{S}^{d}$, we suppose that the true function $f_ρ^{*} \in [\mathcal{H}]^{s}$, the interpolation space of $\mathcal{H}$ with source condition $s>0$. We first determined the exact order (both upper and lower bound) of the generalization error of kernel ridge regression for the optimally chosen regularization parameter $λ$. We then further showed that when $0<s\le1$, KRR is minimax optimal; and when $s>1$, KRR is not minimax optimal (a.k.a. he saturation effect). Our results illustrate that the curves of rate varying along $γ$ exhibit the periodic plateau behavior and the multiple descent behavior and show how the curves evolve with $s>0$. Interestingly, our work provides a unified viewpoint of several recent works on kernel regression in the large-dimensional setting, which correspond to $s=0$ and $s=1$ respectively.

Weihao Lu, Haobo Zhang, Yicheng Li et al.

The saturation effects, which originally refer to the fact that kernel ridge regression (KRR) fails to achieve the information-theoretical lower bound when the regression function is over-smooth, have been observed for almost 20 years and were rigorously proved recently for kernel ridge regression and some other spectral algorithms over a fixed dimensional domain. The main focus of this paper is to explore the saturation effects for a large class of spectral algorithms (including the KRR, gradient descent, etc.) in large dimensional settings where $n \asymp d^γ$. More precisely, we first propose an improved minimax lower bound for the kernel regression problem in large dimensional settings and show that the gradient flow with early stopping strategy will result in an estimator achieving this lower bound (up to a logarithmic factor). Similar to the results in KRR, we can further determine the exact convergence rates (both upper and lower bounds) of a large class of (optimal tuned) spectral algorithms with different qualification $τ$'s. In particular, we find that these exact rate curves (varying along $γ$) exhibit the periodic plateau behavior and the polynomial approximation barrier. Consequently, we can fully depict the saturation effects of the spectral algorithms and reveal a new phenomenon in large dimensional settings (i.e., the saturation effect occurs in large dimensional setting as long as the source condition $s>τ$ while it occurs in fixed dimensional setting as long as $s>2τ$).

Dongming Huang, Zhifan Li, Yicheng Li et al.

We study spectral algorithms in the setting where kernels are learned from data. We introduce the effective span dimension (ESD), an alignment-sensitive complexity measure that depends jointly on the signal, spectrum, and noise level $σ^2$. The ESD is well-defined for arbitrary kernels and signals without requiring eigen-decay conditions or source conditions. We prove that for sequence models whose ESD is at most $K$, the minimax excess risk scales as $σ^2 K$. Furthermore, we analyze over-parameterized gradient flow and prove that it can reduce the ESD. This finding establishes a connection between adaptive feature learning and provable improvements in generalization of spectral algorithms. We demonstrate the generality of the ESD framework by extending it to linear models and RKHS regression, and we support the theory with numerical experiments. This framework provides a novel perspective on generalization beyond traditional fixed-kernel theories.

Weiye Gan, Yicheng Li, Qian Lin et al.

Spectral bias is a significant phenomenon in neural network training and can be explained by neural tangent kernel (NTK) theory. In this work, we develop the NTK theory for deep neural networks with physics-informed loss, providing insights into the convergence of NTK during initialization and training, and revealing its explicit structure. We find that, in most cases, the differential operators in the loss function do not induce a faster eigenvalue decay rate and stronger spectral bias. Some experimental results are also presented to verify the theory.

Haobo Zhang, Jianfa Lai, Yicheng Li et al.

A primary advantage of neural networks lies in their feature learning characteristics, which is challenging to theoretically analyze due to the complexity of their training dynamics. We propose a new paradigm for studying feature learning and the resulting benefits in generalizability. After reviewing the neural tangent kernel (NTK) theory and recent results in kernel regression, which address the generalization issue of sufficiently wide neural networks, we examine limitations and implications of the fixed kernel theory (as the NTK theory) and review recent theoretical advancements in feature learning. Moving beyond the fixed kernel/feature theory, we consider neural networks as adaptive feature models. Finally, we propose an over-parameterized Gaussian sequence model as a prototype model to study the feature learning characteristics of neural networks.

Jianfa Lai, Zixiong Yu, Songtao Tian et al.

In this paper, we study the generalization ability of the wide residual network on $\mathbb{S}^{d-1}$ with the ReLU activation function. We first show that as the width $m\rightarrow\infty$, the residual network kernel (RNK) uniformly converges to the residual neural tangent kernel (RNTK). This uniform convergence further guarantees that the generalization error of the residual network converges to that of the kernel regression with respect to the RNTK. As direct corollaries, we then show $i)$ the wide residual network with the early stopping strategy can achieve the minimax rate provided that the target regression function falls in the reproducing kernel Hilbert space (RKHS) associated with the RNTK; $ii)$ the wide residual network can not generalize well if it is trained till overfitting the data. We finally illustrate some experiments to reconcile the contradiction between our theoretical result and the widely observed ``benign overfitting phenomenon''

Haobo Zhang, Yicheng Li, Weihao Lu et al.

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_ρ^{*} \in [\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ for some $s\in (0,1)$. The existing minimax optimal results require $\|f_ρ^{*}\|_{L^{\infty}}<\infty$ which implicitly requires $s > α_{0}$ where $α_{0}\in (0,1)$ is the embedding index, a constant depending on $\mathcal{H}$. Whether the KRR is optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any $s\in (0,1)$ when the $\mathcal{H}$ is a Sobolev RKHS.

Yicheng Li, Zixiong Yu, Guhan Chen et al.

In this paper, we provide a strategy to determine the eigenvalue decay rate (EDR) of a large class of kernel functions defined on a general domain rather than $\mathbb S^{d}$. This class of kernel functions include but are not limited to the neural tangent kernel associated with neural networks with different depths and various activation functions. After proving that the dynamics of training the wide neural networks uniformly approximated that of the neural tangent kernel regression on general domains, we can further illustrate the minimax optimality of the wide neural network provided that the underground truth function $f\in [\mathcal H_{\mathrm{NTK}}]^{s}$, an interpolation space associated with the RKHS $\mathcal{H}_{\mathrm{NTK}}$ of NTK. We also showed that the overfitted neural network can not generalize well. We believe our approach for determining the EDR of kernels might be also of independent interests.

Qian Lin, Hwee Tou Ng

Identifying breakdowns in ongoing dialogues helps to improve communication effectiveness. Most prior work on this topic relies on human annotated data and data augmentation to learn a classification model. While quality labeled dialogue data requires human annotation and is usually expensive to obtain, unlabeled data is easier to collect from various sources. In this paper, we propose a novel semi-supervised teacher-student learning framework to tackle this task. We introduce two teachers which are trained on labeled data and perturbed labeled data respectively. We leverage unlabeled data to improve classification in student training where we employ two teachers to refine the labeling of unlabeled data through teacher-student learning in a bootstrapping manner. Through our proposed training approach, the student can achieve improvements over single-teacher performance. Experimental results on the Dialogue Breakdown Detection Challenge dataset DBDC5 and Learning to Identify Follow-Up Questions dataset LIF show that our approach outperforms all previous published approaches as well as other supervised and semi-supervised baseline methods.

Xiaoyu Xiang, Qian Lin, Jan P. Allebach

Due to the limits of bandwidth and storage space, digital images are usually down-scaled and compressed when transmitted over networks, resulting in loss of details and jarring artifacts that can lower the performance of high-level visual tasks. In this paper, we aim to generate an artifact-free high-resolution image from a low-resolution one compressed with an arbitrary quality factor by exploring joint compression artifacts reduction (CAR) and super-resolution (SR) tasks. First, we propose a context-aware joint CAR and SR neural network (CAJNN) that integrates both local and non-local features to solve CAR and SR in one-stage. Finally, a deep reconstruction network is adopted to predict high quality and high-resolution images. Evaluation on CAR and SR benchmark datasets shows that our CAJNN model outperforms previous methods and also takes 26.2% shorter runtime. Based on this model, we explore addressing two critical challenges in high-level computer vision: optical character recognition of low-resolution texts, and extremely tiny face detection. We demonstrate that CAJNN can serve as an effective image preprocessing method and improve the accuracy for real-scene text recognition (from 85.30% to 85.75%) and the average precision for tiny face detection (from 0.317 to 0.611).

Xiaoyu Xiang, Yang Cheng, Shaoyuan Xu et al.

The evolving algorithms for 2D facial landmark detection empower people to recognize faces, analyze facial expressions, etc. However, existing methods still encounter problems of unstable facial landmarks when applied to videos. Because previous research shows that the instability of facial landmarks is caused by the inconsistency of labeling quality among the public datasets, we want to have a better understanding of the influence of annotation noise in them. In this paper, we make the following contributions: 1) we propose two metrics that quantitatively measure the stability of detected facial landmarks, 2) we model the annotation noise in an existing public dataset, 3) we investigate the influence of different types of noise in training face alignment neural networks, and propose corresponding solutions. Our results demonstrate improvements in both accuracy and stability of detected facial landmarks.

Augusto C. Valente, Cristina Wada, Deangela Neves et al.

Efficient automated print defect mapping is valuable to the printing industry since such defects directly influence customer-perceived printer quality and manually mapping them is cost-ineffective. Conventional methods consist of complicated and hand-crafted feature engineering techniques, usually targeting only one type of defect. In this paper, we propose the first end-to-end framework to map print defects at pixel level, adopting an approach based on semantic segmentation. Our framework uses Convolutional Neural Networks, specifically DeepLab-v3+, and achieves promising results in the identification of defects in printed images. We use synthetic training data by simulating two types of print defects and a print-scan effect with image processing and computer graphic techniques. Compared with conventional methods, our framework is versatile, allowing two inference strategies, one being near real-time and providing coarser results, and the other focusing on offline processing with more fine-grained detection. Our model is evaluated on a dataset of real printed images.

Daniel Mas Montserrat, Jianhang Chen, Qian Lin et al.

Applications that interact with the real world such as augmented reality or robot manipulation require a good understanding of the location and pose of the surrounding objects. In this paper, we present a new approach to estimate the 6 Degree of Freedom (DoF) or 6D pose of objects from a single RGB image. Our approach can be paired with an object detection and segmentation method to estimate, refine and track the pose of the objects by matching the input image with rendered images.

Biao Zhang, Deyi Xiong, Jinsong Su et al.

In this paper, we propose an additionsubtraction twin-gated recurrent network (ATR) to simplify neural machine translation. The recurrent units of ATR are heavily simplified to have the smallest number of weight matrices among units of all existing gated RNNs. With the simple addition and subtraction operation, we introduce a twin-gated mechanism to build input and forget gates which are highly correlated. Despite this simplification, the essential non-linearities and capability of modeling long-distance dependencies are preserved. Additionally, the proposed ATR is more transparent than LSTM/GRU due to the simplification. Forward self-attention can be easily established in ATR, which makes the proposed network interpretable. Experiments on WMT14 translation tasks demonstrate that ATR-based neural machine translation can yield competitive performance on English- German and English-French language pairs in terms of both translation quality and speed. Further experiments on NIST Chinese-English translation, natural language inference and Chinese word segmentation verify the generality and applicability of ATR on different natural language processing tasks.

Jing Yang, Biao Zhang, Yue Qin et al.

Although neural machine translation(NMT) yields promising translation performance, it unfortunately suffers from over- and under-translation is- sues [Tu et al., 2016], of which studies have become research hotspots in NMT. At present, these studies mainly apply the dominant automatic evaluation metrics, such as BLEU, to evaluate the overall translation quality with respect to both adequacy and uency. However, they are unable to accurately measure the ability of NMT systems in dealing with the above-mentioned issues. In this paper, we propose two quantitative metrics, the Otem and Utem, to automatically evaluate the system perfor- mance in terms of over- and under-translation respectively. Both metrics are based on the proportion of mismatched n-grams between gold ref- erence and system translation. We evaluate both metrics by comparing their scores with human evaluations, where the values of Pearson Cor- relation Coefficient reveal their strong correlation. Moreover, in-depth analyses on various translation systems indicate some inconsistency be- tween BLEU and our proposed metrics, highlighting the necessity and significance of our metrics.

Matey Neykov, Qian Lin, Jun S. Liu

In this paper we study the support recovery problem for single index models $Y=f(\boldsymbol{X}^{\intercal} \boldsymbolβ,\varepsilon)$, where $f$ is an unknown link function, $\boldsymbol{X}\sim N_p(0,\mathbb{I}_{p})$ and $\boldsymbolβ$ is an $s$-sparse unit vector such that $\boldsymbolβ_{i}\in \{\pm\frac{1}{\sqrt{s}},0\}$. In particular, we look into the performance of two computationally inexpensive algorithms: (a) the diagonal thresholding sliced inverse regression (DT-SIR) introduced by Lin et al. (2015); and (b) a semi-definite programming (SDP) approach inspired by Amini & Wainwright (2008). When $s=O(p^{1-δ})$ for some $δ>0$, we demonstrate that both procedures can succeed in recovering the support of $\boldsymbolβ$ as long as the rescaled sample size $κ=\frac{n}{s\log(p-s)}$ is larger than a certain critical threshold. On the other hand, when $κ$ is smaller than a critical value, any algorithm fails to recover the support with probability at least $\frac{1}{2}$ asymptotically. In other words, we demonstrate that both DT-SIR and the SDP approach are optimal (up to a scalar) for recovering the support of $\boldsymbolβ$ in terms of sample size. We provide extensive simulations, as well as a real dataset application to help verify our theoretical observations.