Quantizing weights and activations of deep neural networks results in significant improvement in inference efficiency at the cost of lower accuracy. A source of the accuracy gap between full precision and quantized models is the quantization error. In this work, we focus on the binary quantization, in which values are mapped to -1 and 1. We provide a unified framework to analyze different scaling strategies. Inspired by the pareto-optimality of 2-bits versus 1-bit quantization, we introduce a novel 2-bits quantization with provably least squares error. Our quantization algorithms can be implemented efficiently on the hardware using bitwise operations. We present proofs to show that our proposed methods are optimal, and also provide empirical error analysis. We conduct experiments on the ImageNet dataset and show a reduced accuracy gap when using the proposed least squares quantization algorithms.
Scalar Quantization as Sparse Least Square Optimization
College of Electrical Engineering, Sichuan University（四川大学电气工程学院）；Department of Computer Science, Rutgers University -- New Brunswick（罗格斯大学新布朗斯维广场分校计算机科学系）；Engineering Practice Center, Chengdu University of Information Technology（成都信息科技大学工程实践中心）
Least squares is by far the simplest and most commonly applied computational method in many fields. In almost all applications, the least squares objective is rarely the true objective. We account for this discrepancy by parametrizing the least squares problem and automatically adjusting these parameters using an optimization algorithm. We apply our method, which we call least squares auto-tuning, to data fitting.
Here, we provide a supplementary material for Takayuki Osogami, "Uncorrected least-squares temporal difference with lambda-return," which appears in {\it Proceedings of the 34th AAAI Conference on Artificial Intelligence} (AAAI-20).
Recovering sparse signals from observed data is an important topic in signal/imaging processing, statistics and machine learning. Nonconvex penalized least squares have been attracted a lot of attentions since they enjoy nice statistical properties. Computationally, coordinate descent (CD) is a workhorse for minimizing the nonconvex penalized least squares criterion due to its simplicity and scalability. In this work, we prove the linear convergence rate to CD for solving MCP/SCAD penalized least squares problems.
This paper sheds light on the risk landscape of unsupervised least squares in the context of deep auto-encoding neural nets. We formally establish an equivalence between unsupervised least squares and principal manifolds. This link provides insight into the risk landscape of auto--encoding under the mean squared error, in particular all non-trivial critical points are saddlepoints. Finding saddlepoints is in itself difficult, overcomplete auto-encoding poses the additional challenge that the saddlepoints are degenerate. Within this context we discuss regularization of auto-encoders, in particu…
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Helsinki Institute for Information Technology（赫尔辛基信息技术研究所）
The kernel least mean squares (KLMS) algorithm is a computationally efficient nonlinear adaptive filtering method that "kernelizes" the celebrated (linear) least mean squares algorithm. We demonstrate that the least mean squares algorithm is closely related to the Kalman filtering, and thus, the KLMS can be interpreted as an approximate Bayesian filtering method. This allows us to systematically develop extensions of the KLMS by modifying the underlying state-space and observation models. The resulting extensions introduce many desirable properties such as "forgetting", and the ability to lear…
IBM T.J. Watson Research Center（IBM T.J. Watson研究中心）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
We study (constrained) least-squares regression as well as multiple response least-squares regression and ask the question of whether a subset of the data, a coreset, suffices to compute a good approximate solution to the regression. We give deterministic, low order polynomial-time algorithms to construct such coresets with approximation guarantees, together with lower bounds indicating that there is not much room for improvement upon our results.
In this technical report we analyse the performance of diffusion strategies applied to the Least-Mean-Square adaptive filter. We configure a network of cooperative agents running adaptive filters and discuss their behaviour when compared with a non-cooperative agent which represents the average of the network. The analysis provides conditions under which diversity in the filter parameters is beneficial in terms of convergence and stability. Simulations drive and support the analysis.
The least-squares support vector machine is a frequently used kernel method for non-linear regression and classification tasks. Here we discuss several approximation algorithms for the least-squares support vector machine classifier. The proposed methods are based on randomized block kernel matrices, and we show that they provide good accuracy and reliable scaling for multi-class classification problems with relatively large data sets. Also, we present several numerical experiments that illustrate the practical applicability of the proposed methods.
We study randomized sketching methods for approximately solving least-squares problem with a general convex constraint. The quality of a least-squares approximation can be assessed in different ways: either in terms of the value of the quadratic objective function (cost approximation), or in terms of some distance measure between the approximate minimizer and the true minimizer (solution approximation). Focusing on the latter criterion, our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least-squares problem; as a su…
A set of introductory notes on the subject of data classification using a linear classifier and least-squares cost function, and the negative effect of the presence of outliers on the decision boundary of the linear discriminant. We also show how a simple scaling could make the outlier less significant, thereby obtaining a much better decision boundary. We present some numerical results.
Massachusetts Institute of Technology（麻省理工学院）；UC Berkeley（加州大学伯克利分校）
We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dimension $d\geq 6$. Specifically, we establish that Least Squares is mimimax sub-optimal, and achieves a rate of $\tildeΘ_d(n^{-2/(d-1)})$ whereas the minimax rate is $Θ_d(n^{-4/(d+3)})$.
Sparse linear regression, which entails finding a sparse solution to an underdetermined system of linear equations, can formally be expressed as an $l_0$-constrained least-squares problem. The Orthogonal Least-Squares (OLS) algorithm sequentially selects the features (i.e., columns of the coefficient matrix) to greedily find an approximate sparse solution. In this paper, a generalization of Orthogonal Least-Squares which relies on a recursive relation between the components of the optimal solution to select L features at each step and solve the resulting overdetermined system of equations is p…
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine parameters of the deep neural network. There are various least-squares functionals for a partial differential equation. This paper focuses on the so-called first-order system least-squares (FOSLS) functional studied in [3], which is based on a first-order system of scalar second-orde…
