Revisiting Random Weight Perturbation for Efficiently Improving Generalization

Improving the generalization ability of modern deep neural networks (DNNs) is a fundamental challenge in machine learning. Two branches of methods have been proposed to seek flat minima and improve generalization: one led by sharpness-aware minimization (SAM) minimizes the worst-case neighborhood loss through adversarial weight perturbation (AWP), and the other minimizes the expected Bayes objective with random weight perturbation (RWP). While RWP offers advantages in computation and is closely linked to AWP on a mathematical basis, its empirical performance has consistently lagged behind that of AWP. In this paper, we revisit the use of RWP for improving generalization and propose improvements from two perspectives: i) the trade-off between generalization and convergence and ii) the random perturbation generation. Through extensive experimental evaluations, we demonstrate that our enhanced RWP methods achieve greater efficiency in enhancing generalization, particularly in large-scale problems, while also offering comparable or even superior performance to SAM. The code is released at https://github.com/nblt/mARWP.

Sparsity via Sparse Group $k$-max Regularization

For the linear inverse problem with sparsity constraints, the $l_0$ regularized problem is NP-hard, and existing approaches either utilize greedy algorithms to find almost-optimal solutions or to approximate the $l_0$ regularization with its convex counterparts. In this paper, we propose a novel and concise regularization, namely the sparse group $k$-max regularization, which can not only simultaneously enhance the group-wise and in-group sparsity, but also casts no additional restraints on the magnitude of variables in each group, which is especially important for variables at different scales, so that it approximate the $l_0$ norm more closely. We also establish an iterative soft thresholding algorithm with local optimality conditions and complexity analysis provided. Through numerical experiments on both synthetic and real-world datasets, we verify the effectiveness and flexibility of the proposed method.

Kernel PCA for Out-of-Distribution Detection

Out-of-Distribution (OoD) detection is vital for the reliability of Deep Neural Networks (DNNs). Existing works have shown the insufficiency of Principal Component Analysis (PCA) straightforwardly applied on the features of DNNs in detecting OoD data from In-Distribution (InD) data. The failure of PCA suggests that the network features residing in OoD and InD are not well separated by simply proceeding in a linear subspace, which instead can be resolved through proper nonlinear mappings. In this work, we leverage the framework of Kernel PCA (KPCA) for OoD detection, seeking subspaces where OoD and InD features are allocated with significantly different patterns. We devise two feature mappings that induce non-linear kernels in KPCA to advocate the separability between InD and OoD data in the subspace spanned by the principal components. Given any test sample, the reconstruction error in such subspace is then used to efficiently obtain the detection result with $\mathcal{O}(1)$ time complexity in inference. Extensive empirical results on multiple OoD data sets and network structures verify the superiority of our KPCA-based detector in efficiency and efficacy with state-of-the-art OoD detection performances.

Self-Attention through Kernel-Eigen Pair Sparse Variational Gaussian Processes

While the great capability of Transformers significantly boosts prediction accuracy, it could also yield overconfident predictions and require calibrated uncertainty estimation, which can be commonly tackled by Gaussian processes (GPs). Existing works apply GPs with symmetric kernels under variational inference to the attention kernel; however, omitting the fact that attention kernels are in essence asymmetric. Moreover, the complexity of deriving the GP posteriors remains high for large-scale data. In this work, we propose Kernel-Eigen Pair Sparse Variational Gaussian Processes (KEP-SVGP) for building uncertainty-aware self-attention where the asymmetry of attention kernels is tackled by Kernel SVD (KSVD) and a reduced complexity is acquired. Through KEP-SVGP, i) the SVGP pair induced by the two sets of singular vectors from KSVD w.r.t. the attention kernel fully characterizes the asymmetry; ii) using only a small set of adjoint eigenfunctions from KSVD, the derivation of SVGP posteriors can be based on the inversion of a diagonal matrix containing singular values, contributing to a reduction in time complexity; iii) an evidence lower bound is derived so that variational parameters can be optimized towards this objective. Experiments verify our excellent performances and efficiency on in-distribution, distribution-shift and out-of-distribution benchmarks.

Revisiting Deep Ensemble for Out-of-Distribution Detection: A Loss Landscape Perspective

Existing Out-of-Distribution (OoD) detection methods address to detect OoD samples from In-Distribution data (InD) mainly by exploring differences in features, logits and gradients in Deep Neural Networks (DNNs). We in this work propose a new perspective upon loss landscape and mode ensemble to investigate OoD detection. In the optimization of DNNs, there exist many local optima in the parameter space, or namely modes. Interestingly, we observe that these independent modes, which all reach low-loss regions with InD data (training and test data), yet yield significantly different loss landscapes with OoD data. Such an observation provides a novel view to investigate the OoD detection from the loss landscape and further suggests significantly fluctuating OoD detection performance across these modes. For instance, FPR values of the RankFeat method can range from 46.58% to 84.70% among 5 modes, showing uncertain detection performance evaluations across independent modes. Motivated by such diversities on OoD loss landscape across modes, we revisit the deep ensemble method for OoD detection through mode ensemble, leading to improved performance and benefiting the OoD detector with reduced variances. Extensive experiments covering varied OoD detectors and network structures illustrate high variances across modes and also validate the superiority of mode ensemble in boosting OoD detection. We hope this work could attract attention in the view of independent modes in the OoD loss landscape and more reliable evaluations on OoD detectors.

Low-Rank Multitask Learning based on Tensorized SVMs and LSSVMs

Multitask learning (MTL) leverages task-relatedness to enhance performance. With the emergence of multimodal data, tasks can now be referenced by multiple indices. In this paper, we employ high-order tensors, with each mode corresponding to a task index, to naturally represent tasks referenced by multiple indices and preserve their structural relations. Based on this representation, we propose a general framework of low-rank MTL methods with tensorized support vector machines (SVMs) and least square support vector machines (LSSVMs), where the CP factorization is deployed over the coefficient tensor. Our approach allows to model the task relation through a linear combination of shared factors weighted by task-specific factors and is generalized to both classification and regression problems. Through the alternating optimization scheme and the Lagrangian function, each subproblem is transformed into a convex problem, formulated as a quadratic programming or linear system in the dual form. In contrast to previous MTL frameworks, our decision function in the dual induces a weighted kernel function with a task-coupling term characterized by the similarities of the task-specific factors, better revealing the explicit relations across tasks in MTL. Experimental results validate the effectiveness and superiority of our proposed methods compared to existing state-of-the-art approaches in MTL. The code of implementation will be available at https://github.com/liujiani0216/TSVM-MTL.

Nonlinear SVD with Asymmetric Kernels: feature learning and asymmetric Nyström method

Asymmetric data naturally exist in real life, such as directed graphs. Different from the common kernel methods requiring Mercer kernels, this paper tackles the asymmetric kernel-based learning problem. We describe a nonlinear extension of the matrix Singular Value Decomposition through asymmetric kernels, namely KSVD. First, we construct two nonlinear feature mappings w.r.t. rows and columns of the given data matrix. The proposed optimization problem maximizes the variance of each mapping projected onto the subspace spanned by the other, subject to a mutual orthogonality constraint. Through Lagrangian duality, we show that it can be solved by the left and right singular vectors in the feature space induced by the asymmetric kernel. Moreover, we start from the integral equations with a pair of adjoint eigenfunctions corresponding to the singular vectors on an asymmetrical kernel, and extend the Nystr\"om method to asymmetric cases through the finite sample approximation, which can be applied to speedup the training in KSVD. Experiments show that asymmetric KSVD learns features outperforming Mercer-kernel based methods that resort to symmetrization, and also verify the effectiveness of the asymmetric Nystr\"om method.

Primal-Attention: Self-attention through Asymmetric Kernel SVD in Primal Representation

Recently, a new line of works has emerged to understand and improve self-attention in Transformers by treating it as a kernel machine. However, existing works apply the methods for symmetric kernels to the asymmetric self-attention, resulting in a nontrivial gap between the analytical understanding and numerical implementation. In this paper, we provide a new perspective to represent and optimize self-attention through asymmetric Kernel Singular Value Decomposition (KSVD), which is also motivated by the low-rank property of self-attention normally observed in deep layers. Through asymmetric KSVD, $i$) a primal-dual representation of self-attention is formulated, where the optimization objective is cast to maximize the projection variances in the attention outputs; $ii$) a novel attention mechanism, i.e., Primal-Attention, is proposed via the primal representation of KSVD, avoiding explicit computation of the kernel matrix in the dual; $iii$) with KKT conditions, we prove that the stationary solution to the KSVD optimization in Primal-Attention yields a zero-value objective. In this manner, KSVD optimization can be implemented by simply minimizing a regularization loss, so that low-rank property is promoted without extra decomposition. Numerical experiments show state-of-the-art performance of our Primal-Attention with improved efficiency. Moreover, we demonstrate that the deployed KSVD optimization regularizes Primal-Attention with a sharper singular value decay than that of the canonical self-attention, further verifying the great potential of our method. To the best of our knowledge, this is the first work that provides a primal-dual representation for the asymmetric kernel in self-attention and successfully applies it to modeling and optimization.

Tensorized LSSVMs for Multitask Regression

Multitask learning (MTL) can utilize the relatedness between multiple tasks for performance improvement. The advent of multimodal data allows tasks to be referenced by multiple indices. High-order tensors are capable of providing efficient representations for such tasks, while preserving structural task-relations. In this paper, a new MTL method is proposed by leveraging low-rank tensor analysis and constructing tensorized Least Squares Support Vector Machines, namely the tLSSVM-MTL, where multilinear modelling and its nonlinear extensions can be flexibly exerted. We employ a high-order tensor for all the weights with each mode relating to an index and factorize it with CP decomposition, assigning a shared factor for all tasks and retaining task-specific latent factors along each index. Then an alternating algorithm is derived for the nonconvex optimization, where each resulting subproblem is solved by a linear system. Experimental results demonstrate promising performances of our tLSSVM-MTL.

Deep Kernel Principal Component Analysis for Multi-level Feature Learning

Principal Component Analysis (PCA) and its nonlinear extension Kernel PCA (KPCA) are widely used across science and industry for data analysis and dimensionality reduction. Modern deep learning tools have achieved great empirical success, but a framework for deep principal component analysis is still lacking. Here we develop a deep kernel PCA methodology (DKPCA) to extract multiple levels of the most informative components of the data. Our scheme can effectively identify new hierarchical variables, called deep principal components, capturing the main characteristics of high-dimensional data through a simple and interpretable numerical optimization. We couple the principal components of multiple KPCA levels, theoretically showing that DKPCA creates both forward and backward dependency across levels, which has not been explored in kernel methods and yet is crucial to extract more informative features. Various experimental evaluations on multiple data types show that DKPCA finds more efficient and disentangled representations with higher explained variance in fewer principal components, compared to the shallow KPCA. We demonstrate that our method allows for effective hierarchical data exploration, with the ability to separate the key generative factors of the input data both for large datasets and when few training samples are available. Overall, DKPCA can facilitate the extraction of useful patterns from high-dimensional data by learning more informative features organized in different levels, giving diversified aspects to explore the variation factors in the data, while maintaining a simple mathematical formulation.