We revisit the problem of the existence of the maximum likelihood estimate for multi-class logistic regression. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset. The notion of data separability is not needed, which is in contrast to the classical set up of multi-class logistic regression in which each data sample belongs to one class. We also provide a general and constructive estimate of the convergence rate to the maximum likelihood estimate when gradient descent is used as the optimizer. Our estimate involves bounding the condition number of the Hessian of the maximum likelihood function. The approaches used in this article rely on a simple operator-theoretic framework.
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We study the training of finite-width two-layer smoothed ReLU networks for binary classification using the logistic loss. We show that gradient descent drives the training loss to zero if the initial loss is small enough. When the data satisfies certain cluster and separation conditions and the network is wide enough, we show that one step of gradient descent reduces the loss sufficiently that the first result applies.
In this paper, we consider supervised learning problems such as logistic regression and study the stochastic gradient method with averaging, in the usual stochastic approximation setting where observations are used only once. We show that after $N$ iterations, with a constant step-size proportional to $1/R^2 \sqrt{N}$ where $N$ is the number of observations and $R$ is the maximum norm of the observations, the convergence rate is always of order $O(1/\sqrt{N})$, and improves to $O(R^2 / μN)$ where $μ$ is the lowest eigenvalue of the Hessian at the global optimum (when this eigenvalue is greater…
University of California, Berkeley（加州大学伯克利分校）；Google（谷歌）
We establish conditions under which gradient descent applied to fixed-width deep networks drives the logistic loss to zero, and prove bounds on the rate of convergence. Our analysis applies for smoothed approximations to the ReLU, such as Swish and the Huberized ReLU, proposed in previous applied work. We provide two sufficient conditions for convergence. The first is simply a bound on the loss at initialization. The second is a data separation condition used in prior analyses.
Exponential Convergence of (Stochastic) Gradient Descent for Separable Logistic Regression
Undergraduate Programme Indian Institute of Science, Bangalore（印度科学研究院布达佩斯分校本科项目）；Dept. of Computer Science and Automation Indian Institute of Science, Bangalore（计算机科学与自动化系印度科学研究院布达佩斯分校）；Inria, Ecole Normale Supérieure PSL Research University, Paris, France（法国巴黎PSL研究大学埃cole Normale Supérieure与Inria）
Research in statistical relational learning has produced a number of methods for learning relational models from large-scale network data. While these methods have been successfully applied in various domains, they have been developed under the unrealistic assumption of full data access. In practice, however, the data are often collected by crawling the network, due to proprietary access, limited resources, and privacy concerns. Recently, we showed that the parameter estimates for relational Bayes classifiers computed from network samples collected by existing network crawlers can be quite ina…
We study gradient descent (GD) with a constant stepsize for $\ell_2$-regularized logistic regression with linearly separable data. Classical theory suggests small stepsizes to ensure monotonic reduction of the optimization objective, achieving exponential convergence in $\widetilde{\mathcal{O}}(κ)$ steps with $κ$ being the condition number. Surprisingly, we show that this can be accelerated to $\widetilde{\mathcal{O}}(\sqrtκ)$ by simply using a large stepsize -- for which the objective evolves nonmonotonically. The acceleration brought by large stepsizes extends to minimizing the population ri…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Gradient descent, when applied to the task of logistic regression, outputs iterates which are biased to follow a unique ray defined by the data. The direction of this ray is the maximum margin predictor of a maximal linearly separable subset of the data; the gradient descent iterates converge to this ray in direction at the rate $\mathcal{O}(\ln\ln t / \ln t)$. The ray does not pass through the origin in general, and its offset is the bounded global optimum of the risk over the remaining data; gradient descent recovers this offset at a rate $\mathcal{O}((\ln t)^2 / \sqrt{t})$.
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；INRIA（法国国家信息与自动化研究所）；ENS（巴黎高等师范学院）；PSL Research University（PSL研究大学）
Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training and generalization behavior of infinitely wide two-layer neural networks with homogeneous activations. We show that the limits of the gradient flow on exponentially tailed losses can be fully characterized as a max-margin classifier in a certain non-Hilbertian space of functions. In presence of hidden low-dimensional structures, the resulting margin is indepen…
Tel Aviv University（特拉维夫大学）；The Open University of Israel（以色列开放大学）
In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose first a feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the nonasymptotic bounds for misclassification excess risk of the resulting classifier. We establish also their tightness by deriving the corresponding minimax lower bounds. In particular, we show that there exist two regimes corresponding to small and large number of classes. The bounds can be reduced under the additional low noise condition. To…
Gradient Descent on Logistic Regression: Do Large Step-Sizes Work with Data on the Sphere?
Inria Paris（巴黎国家信息与自动化研究所）；Inria Saclay Palaiseau（萨克莱研究所）；ELLIS Institute Tübingen MPI for Intelligent Systems Tübingen AI Center, Germany（图宾根人工智能研究所）；Department of Computer Science Cornell University（康奈尔大学计算机科学系）
MIT Sloan School of Management（麻省理工学院斯隆管理学院）；University of California, Berkeley（加州大学伯克利分校）
Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of n…
Mila；Université de Montreal（蒙特利尔大学）；University of British Columbia（不列颠哥伦比亚大学）；Samsung（三星）；SAIT AI Lab（SAIT AI实验室）
We consider the problem of upper bounding the expected log-likelihood sub-optimality of the maximum likelihood estimate (MLE), or a conjugate maximum a posteriori (MAP) for an exponential family, in a non-asymptotic way. Surprisingly, we found no general solution to this problem in the literature. In particular, current theories do not hold for a Gaussian or in the interesting few samples regime. After exhibiting various facets of the problem, we show we can interpret the MAP as running stochastic mirror descent (SMD) on the log-likelihood. However, modern convergence results do not apply for …
Feature Selection and Regularization in Multi-Class Classification: An Empirical Study of One-vs-Rest Logistic Regression with Gradient Descent Optimization and L1 Sparsity Constraints
Department of Computer Science and Software Engineering, Auburn University, Alabama, USA（计算机科学与软件工程系，阿伯茨费尔德大学，美国阿拉巴马州）；Department of Information and Communication Technology, Bangladesh University of Professionals, Dhaka, Bangladesh（信息与通信技术系，孟加拉国专业大学，达卡）
Finite-Sample Performance of Gradient Descent in Logistic Regression with Gaussian Design
Department of Mathematics, University of Maryland, College Park（数学系，马里兰大学，College Park分校）；Halıcıoğlu Data Science Institute, UC San Diego（Halıcıoğlu数据科学研究所，加州大学圣地亚哥分校）
In this paper, we introduce the concept of sparse bilinear logistic regression for decision problems involving explanatory variables that are two-dimensional matrices. Such problems are common in computer vision, brain-computer interfaces, style/content factorization, and parallel factor analysis. The underlying optimization problem is bi-convex; we study its solution and develop an efficient algorithm based on block coordinate descent. We provide a theoretical guarantee for global convergence and estimate the asymptotical convergence rate using the Kurdyka-Łojasiewicz inequality. A range of e…
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）
In this paper, we study the implicit regularization of the gradient descent algorithm in homogeneous neural networks, including fully-connected and convolutional neural networks with ReLU or LeakyReLU activations. In particular, we study the gradient descent or gradient flow (i.e., gradient descent with infinitesimal step size) optimizing the logistic loss or cross-entropy loss of any homogeneous model (possibly non-smooth), and show that if the training loss decreases below a certain threshold, then we can define a smoothed version of the normalized margin which increases over time. We also f…
Université Paris-Saclay（巴黎-萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能与替代能源委员会）；University of Genova（热那亚大学）
Acceleration of first order methods is mainly obtained via inertial techniques à la Nesterov, or via nonlinear extrapolation. The latter has known a recent surge of interest, with successful applications to gradient and proximal gradient techniques. On multiple Machine Learning problems, coordinate descent achieves performance significantly superior to full-gradient methods. Speeding up coordinate descent in practice is not easy: inertially accelerated versions of coordinate descent are theoretically accelerated, but might not always lead to practical speed-ups. We propose an accelerated versi…
UC Berkeley（加州大学伯克利分校）；National University of Defense Technology（国防科技大学）；Stanford University（斯坦福大学）；National University of Singapore（新加坡国立大学）
We consider a wide range of regularized stochastic minimization problems with two regularization terms, one of which is composed with a linear function. This optimization model abstracts a number of important applications in artificial intelligence and machine learning, such as fused Lasso, fused logistic regression, and a class of graph-guided regularized minimization. The computational challenges of this model are in two folds. On one hand, the closed-form solution of the proximal mapping associated with the composed regularization term or the expected objective function is not available. On…
University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）；Department of Electrical and Computer Engineering（电气与计算机工程系）
Extensive empirical evidence reveals that, for a wide range of different learning methods and datasets, the risk curve exhibits a double-descent (DD) trend as a function of the model size. In a recent paper [Zeyu,Kammoun,Thrampoulidis,2019] the authors studied binary linear classification models and showed that the test error of gradient descent (GD) with logistic loss undergoes a DD. In this paper, we complement these results by extending them to GD with square loss. We show that the DD phenomenon persists, but we also identify several differences compared to logistic loss. This emphasizes th…
Stochastic Gradient Descent (SGD) is a central tool in machine learning. We prove that SGD converges to zero loss, even with a fixed (non-vanishing) learning rate - in the special case of homogeneous linear classifiers with smooth monotone loss functions, optimized on linearly separable data. Previous works assumed either a vanishing learning rate, iterate averaging, or loss assumptions that do not hold for monotone loss functions used for classification, such as the logistic loss. We prove our result on a fixed dataset, both for sampling with or without replacement. Furthermore, for logistic …
We provide a detailed study on the implicit bias of gradient descent when optimizing loss functions with strictly monotone tails, such as the logistic loss, over separable datasets. We look at two basic questions: (a) what are the conditions on the tail of the loss function under which gradient descent converges in the direction of the $L_2$ maximum-margin separator? (b) how does the rate of margin convergence depend on the tail of the loss function and the choice of the step size? We show that for a large family of super-polynomial tailed losses, gradient descent iterates on linear networks o…
University of California, Santa Barbara（加州大学圣巴巴拉分校）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We consider a model for logistic regression where only a subset of features of size $p$ is used for training a linear classifier over $n$ training samples. The classifier is obtained by running gradient descent (GD) on logistic loss. For this model, we investigate the dependence of the classification error on the overparameterization ratio $κ=p/n$. First, building on known deterministic results on the implicit bias of GD, we uncover a phase-transition phenomenon for the case of Gaussian features: the classification error of GD is the same as that of the maximum-likelihood (ML) solution when $κ…
Center for Language & Speech Processing（语言与语音处理中心）；The Johns Hopkins University（约翰斯·霍普金斯大学）
We propose a novel parameter estimation procedure that works efficiently for conditional random fields (CRF). This algorithm is an extension to the maximum likelihood estimation (MLE), using loss functions defined by Bregman divergences which measure the proximity between the model expectation and the empirical mean of the feature vectors. This leads to a flexible training framework from which multiple update strategies can be derived using natural gradient descent (NGD). We carefully choose the convex function inducing the Bregman divergence so that the types of updates are reduced, while mak…
We present a strikingly simple proof that two rules are sufficient to automate gradient descent: 1) don't increase the stepsize too fast and 2) don't overstep the local curvature. No need for functional values, no line search, no information about the function except for the gradients. By following these rules, you get a method adaptive to the local geometry, with convergence guarantees depending only on the smoothness in a neighborhood of a solution. Given that the problem is convex, our method converges even if the global smoothness constant is infinity. As an illustration, it can minimize a…
In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding sparse logistic regression problem, and study its local optimality conditions and the choice of the regularization parameter to exclude trivial solutions. Despite the nonconvexity, a …
We examine a new form of smooth approximation to the zero one loss in which learning is performed using a reformulation of the widely used logistic function. Our approach is based on using the posterior mean of a novel generalized Beta-Bernoulli formulation. This leads to a generalized logistic function that approximates the zero one loss, but retains a probabilistic formulation conferring a number of useful properties. The approach is easily generalized to kernel logistic regression and easily integrated into methods for structured prediction. We present experiments in which we learn such mod…
University of California, Los Angeles（加州大学洛杉矶分校）；University of British Columbia（不列颠哥伦比亚大学）；Emory University（埃默里大学）
We present ADMM-Softmax, an alternating direction method of multipliers (ADMM) for solving multinomial logistic regression (MLR) problems. Our method is geared toward supervised classification tasks with many examples and features. It decouples the nonlinear optimization problem in MLR into three steps that can be solved efficiently. In particular, each iteration of ADMM-Softmax consists of a linear least-squares problem, a set of independent small-scale smooth, convex problems, and a trivial dual variable update. Solution of the least-squares problem can be be accelerated by pre-computing a f…
Arizona State University（亚利桑那州立大学）；Universidad Nacional Autónoma de México（墨西哥国立自治大学）
We analyze the optimization landscape of a recently introduced tunable class of loss functions called $α$-loss, $α\in (0,\infty]$, in the logistic model. This family encapsulates the exponential loss ($α= 1/2$), the log-loss ($α= 1$), and the 0-1 loss ($α= \infty$) and contains compelling properties that enable the practitioner to discern among a host of operating conditions relevant to emerging learning methods. Specifically, we study the evolution of the optimization landscape of $α$-loss with respect to $α$ using tools drawn from the study of strictly-locally-quasi-convex functions in addit…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This work studies the behavior of shallow ReLU networks trained with the logistic loss via gradient descent on binary classification data where the underlying data distribution is general, and the (optimal) Bayes risk is not necessarily zero. In this setting, it is shown that gradient descent with early stopping achieves population risk arbitrarily close to optimal in terms of not just logistic and misclassification losses, but also in terms of calibration, meaning the sigmoid mapping of its outputs approximates the true underlying conditional distribution arbitrarily finely. Moreover, the nec…
A Flexible Empirical Bayes Approach to Generalized Linear Models, with Applications to Sparse Logistic Regression
Department of Statistics The University of Chicago（统计学系芝加哥大学）；Department of Statistics University of California, Irvine（统计学系加州大学伊文斯顿分校）；Department of Statistics and Department of Human Genetics The University of Chicago（统计学系和人类遗传学系芝加哥大学）
This paper presents an original approach for jointly fitting survival times and classifying samples into subgroups. The Coxlogit model is a generalized linear model with a common set of selected features for both tasks. Survival times and class labels are here assumed to be conditioned by a common risk score which depends on those features. Learning is then naturally expressed as maximizing the joint probability of subgroup labels and the ordering of survival events, conditioned to a common weight vector. The model is estimated by minimizing a regularized log-likelihood through a coordinate de…
Jheronimus Academy of Data Science（杰罗尼莫斯数据科学学院）；Eindhoven University（埃因霍温大学）；Ferdowsi University of Mashhad（马什哈德 Ferdowsi 大学）
$\ell_1$ 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…
Johns Hopkins University（约翰斯·霍普金斯大学）；University of California, Berkeley（加州大学伯克利分校）
Stochastic gradient descent (SGD) is one of the most widely used optimization methods for solving various machine learning problems. SGD solves an optimization problem by iteratively sampling a few data points from the input data, computing gradients for the selected data points, and updating the solution. However, in a parallel setting, SGD requires interprocess communication at every iteration. We introduce a new communication-avoiding technique for solving the logistic regression problem using SGD. This technique re-organizes the SGD computations into a form that communicates every $s$ iter…
National Taiwan University（国立台湾大学）；Hon Hai (Foxconn) Quantum Computing Centre（鸿海（富士康）量子计算中心）
In maximum-likelihood quantum state tomography, both the sample size and dimension grow exponentially with the number of qubits. It is therefore desirable to develop a stochastic first-order method, just like stochastic gradient descent for modern machine learning, to compute the maximum-likelihood estimate. To this end, we propose an algorithm called stochastic mirror descent with the Burg entropy. Its expected optimization error vanishes at a $O ( \sqrt{ ( 1 / t ) d \log t } )$ rate, where $d$ and $t$ denote the dimension and number of iterations, respectively. Its per-iteration time complex…
University of Cambridge（剑桥大学）；Eötvös Loránd University（罗兰大学）
In gradient descent, changing how we parametrize the model can lead to drastically different optimization trajectories, giving rise to a surprising range of meaningful inductive biases: identifying sparse classifiers or reconstructing low-rank matrices without explicit regularization. This implicit regularization has been hypothesised to be a contributing factor to good generalization in deep learning. However, natural gradient descent is approximately invariant to reparameterization, it always follows the same trajectory and finds the same optimum. The question naturally arises: What happens …
Finite-sample performance of the maximum likelihood estimator in logistic regression
Department of Statistics, University of California, Berkeley, USA（加州大学伯克利分校统计学系）；Department of Statistics, CREST/ENSAE Paris, Palaiseau, France（巴黎中央银行与应用经济学院统计学系）
Solving logistic regression with L1-regularization in distributed settings is an important problem. This problem arises when training dataset is very large and cannot fit the memory of a single machine. We present d-GLMNET, a new algorithm solving logistic regression with L1-regularization in the distributed settings. We empirically show that it is superior over distributed online learning via truncated gradient.
PixelCNNs are a recently proposed class of powerful generative models with tractable likelihood. Here we discuss our implementation of PixelCNNs which we make available at https://github.com/openai/pixel-cnn. Our implementation contains a number of modifications to the original model that both simplify its structure and improve its performance. 1) We use a discretized logistic mixture likelihood on the pixels, rather than a 256-way softmax, which we find to speed up training. 2) We condition on whole pixels, rather than R/G/B sub-pixels, simplifying the model structure. 3) We use downsampling …
