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 paves the way for "on the fly'' post-training quantization to various bit-widths. We show that by modeling quantization as a $\ell_\infty$-bounded perturbation, the first-order term in the loss expansion can be regularized using the $\ell_1$-norm of gradients. We experimentally validate the effectiveness of our regularization scheme on different architectures on CIFAR-10 and ImageNet datasets.
Robust quantization improves the tolerance of networks for various implementations, allowing reliable output in different bit-widths or fragmented low-precision arithmetic. In this work, we perform extensive analyses to identify the sources of quantization error and present three insights to robustify a network against quantization: reduction of error propagation, range clamping for error minimization, and inherited robustness against quantization. Based on these insights, we propose two novel methods called symmetry regularization (SymReg) and saturating nonlinearity (SatNL). Applying the pro…
Mitigating the Impact of Outlier Channels for Language Model Quantization with Activation Regularization
Massachusetts Institute of Technology（麻省理工学院）；MIT-IBM Watson AI Lab（MIT-IBM沃森人工智能实验室）；IBM Research（IBM研究院）；IST Austria（奥地利科学技术研究所）；Neural Magic, Inc.（Neural Magic公司）
For on-device automatic speech recognition (ASR), quantization aware training (QAT) is ubiquitous to achieve the trade-off between model predictive performance and efficiency. Among existing QAT methods, one major drawback is that the quantization centroids have to be predetermined and fixed. To overcome this limitation, we introduce a regularization-free, "soft-to-hard" compression mechanism with self-adjustable centroids in a mu-Law constrained space, resulting in a simpler yet more versatile quantization scheme, called General Quantizer (GQ). We apply GQ to ASR tasks using Recurrent Neural …
KTH Royal Institute of Technology（瑞典皇家理工学院）；Research Institutes of Sweden（瑞典研究机构）
Autoencoders and their variations provide unsupervised models for learning low-dimensional representations for downstream tasks. Without proper regularization, autoencoder models are susceptible to the overfitting problem and the so-called posterior collapse phenomenon. In this paper, we introduce a quantization-based regularizer in the bottleneck stage of autoencoder models to learn meaningful latent representations. We combine both perspectives of Vector Quantized-Variational AutoEncoders (VQ-VAE) and classical denoising regularization methods of neural networks. We interpret quantizers as r…
University of Appl. Sciences Mittweida（米特韦达应用科学大学）；University Paris Sorbonne-Pantheon（巴黎索邦-先贤祠大学）
We propose in this contribution a method for l one regularization in prototype based relevance learning vector quantization (LVQ) for sparse relevance profiles. Sparse relevance profiles in hyperspectral data analysis fade down those spectral bands which are not necessary for classification. In particular, we consider the sparsity in the relevance profile enforced by LASSO optimization. The latter one is obtained by a gradient learning scheme using a differentiable parametrized approximation of the $l_{1}$-norm, which has an upper error bound. We extend this regularization idea also to the mat…
University of California San Diego（加州大学圣地亚哥分校）；CNRS（法国国家科学研究中心）；Université de Bordeaux（波尔多大学）；Bordeaux INP（波尔多国立理工学院）
As deep neural networks make their ways into different domains, their compute efficiency is becoming a first-order constraint. Deep quantization, which reduces the bitwidth of the operations (below 8 bits), offers a unique opportunity as it can reduce both the storage and compute requirements of the network super-linearly. However, if not employed with diligence, this can lead to significant accuracy loss. Due to the strong inter-dependence between layers and exhibiting different characteristics across the same network, choosing an optimal bitwidth per layer granularity is not a straight forwa…
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…
Toshiba Corporation（东芝公司）；The Institute of Statistical Mathematics（统计数理研究所）
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…
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…
Deep quantization of neural networks (below eight bits) offers significant promise in reducing their compute and storage cost. Albeit alluring, without special techniques for training and optimization, deep quantization results in significant accuracy loss. To further mitigate this loss, we propose a novel sinusoidal regularization, called SinReQ1, for deep quantized training. SinReQ adds a periodic term to the original objective function of the underlying training algorithm. SinReQ exploits the periodicity, differentiability, and the desired convexity profile in sinusoidal functions to automa…
Pennsylvania State University（宾夕法尼亚州立大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
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…
Department of Indutrial Engineering and Operations Research, University of California, Berkeley（加州大学伯克利分校工业工程与运筹学系）
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…
We propose a simple and easy to implement neural network compression algorithm that achieves results competitive with more complicated state-of-the-art methods. The key idea is to modify the original optimization problem by adding K independent Gaussian priors (corresponding to the k-means objective) over the network parameters to achieve parameter quantization, as well as an L1 penalty to achieve pruning. Unlike many existing quantization-based methods, our method uses hard clustering assignments of network parameters, which adds minimal change or overhead to standard network training. We als…
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…
Network quantization, which aims to reduce the bit-lengths of the network weights and activations, has emerged as one of the key ingredients to reduce the size of neural networks for their deployments to resource-limited devices. In order to overcome the nature of transforming continuous activations and weights to discrete ones, recent study called Relaxed Quantization (RQ) [Louizos et al. 2019] successfully employ the popular Gumbel-Softmax that allows this transformation with efficient gradient-based optimization. However, RQ with this Gumbel-Softmax relaxation still suffers from bias-varian…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）
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…
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…
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…
Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
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.
Astro: Activation-guided Structured Regularization for Outlier-Robust LLM Post-Training Quantization
Beijing Institute of Technology（北京理工大学）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；Hebei Province Key Laboratory of Big Data Science and Intelligent Technology（河北省大数据科学与智能技术重点实验室）
This paper describes an entropy regularization term for vector quantization (VQ) based on the analysis of persistent homology of the VQ embeddings. Higher embedding entropy positively correlates with higher codebook utilization, mitigating overfit towards the identity and codebook collapse in VQ-based autoencoders [1]. We show that homology-constrained regularization is an effective way to increase entropy of the VQ process (approximated to input entropy) while preserving the approximated topology in the quantized latent space, averaged over mini batches. This work further explores some patter…
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.
Johns Hopkins University（约翰斯·霍普金斯大学）；Beijing University of Posts and Telecommunications（北京邮电大学）
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…
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…
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…
Institut de Recherche en Informatique de Toulouse（图卢兹计算机科学研究所）；CNRS（法国国家科学研究中心）；Université de Toulouse（图卢兹大学）；Nomao SA（Nomao公司）；Laboratoire Lagrange（拉格朗日实验室）；Université de Nice-Sophia Antipolis（尼斯-索菲亚·安提波利斯大学）；INSA de Rouen（鲁昂国立应用科学学院）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Institut Universitaire de Formation de Maîtres（大学教师培训学院）
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…
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…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）；AITRICS
Network quantization, which aims to reduce the bit-lengths of the network weights and activations, has emerged for their deployments to resource-limited devices. Although recent studies have successfully discretized a full-precision network, they still incur large quantization errors after training, thus giving rise to a significant performance gap between a full-precision network and its quantized counterpart. In this work, we propose a novel quantization method for neural networks, Cluster-Promoting Quantization (CPQ) that finds the optimal quantization grids while naturally encouraging the …
Instituto de Telecomunicações（电信研究所）；Instituto Superior Técnico, University of Lisbon（里斯本大学高等技术学院）
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:…
Hong Kong University of Science and Technology（香港科技大学）
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 consider learning deep neural networks (DNNs) that consist of low-precision weights and activations for efficient inference of fixed-point operations. In training low-precision networks, gradient descent in the backward pass is performed with high-precision weights while quantized low-precision weights and activations are used in the forward pass to calculate the loss function for training. Thus, the gradient descent becomes suboptimal, and accuracy loss follows. In order to reduce the mismatch in the forward and backward passes, we utilize mean squared quantization error (MSQE) regularizat…
