The stochastic gradient descent (SGD) algorithm is widely used for parameter estimation, especially for huge data sets and online learning. While this recursive algorithm is popular for computation and memory efficiency, quantifying variability and randomness of the solutions has been rarely studied. This paper aims at conducting statistical inference of SGD-based estimates in an online setting. In particular, we propose a fully online estimator for the covariance matrix of averaged SGD iterates (ASGD) only using the iterates from SGD. We formally establish our online estimator's consistency and show that the convergence rate is comparable to offline counterparts. Based on the classic asymptotic normality results of ASGD, we construct asymptotically valid confidence intervals for model parameters. Upon receiving new observations, we can quickly update the covariance matrix estimate and the confidence intervals. This approach fits in an online setting and takes full advantage of SGD: efficiency in computation and memory.
We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed al…
Department of Computer Science, Stanford University（斯坦福大学计算机科学系）；Department of Computer Science, Columbia University（哥伦比亚大学计算机科学系）；Google Research, Mountain View（谷歌研究院（山景城））
We prove that the evolution of weight vectors in online gradient descent can encode arbitrary polynomial-space computations, even in very simple learning settings. Our results imply that, under weak complexity-theoretic assumptions, it is impossible to reason efficiently about the fine-grained behavior of online gradient descent.
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Borealis AI（北极光人工智能）
The choice of batch-size in a stochastic optimization algorithm plays a substantial role for both optimization and generalization. Increasing the batch-size used typically improves optimization but degrades generalization. To address the problem of improving generalization while maintaining optimal convergence in large-batch training, we propose to add covariance noise to the gradients. We demonstrate that the learning performance of our method is more accurately captured by the structure of the covariance matrix of the noise rather than by the variance of gradients. Moreover, over the convex-…
Johns Hopkins University（约翰斯·霍普金斯大学）；Missouri University of Science and Technology（密苏里科技大学）；Baidu Research（百度研究院）；Styling.AI Inc.（Styling.AI公司）；Peking University（北京大学）
The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and the covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the…
Nanyang Technological University（南洋理工大学）；Michigan State University（密歇根州立大学）
Kernel-based online learning has often shown state-of-the-art performance for many online learning tasks. It, however, suffers from a major shortcoming, that is, the unbounded number of support vectors, making it non-scalable and unsuitable for applications with large-scale datasets. In this work, we study the problem of bounded kernel-based online learning that aims to constrain the number of support vectors by a predefined budget. Although several algorithms have been proposed in literature, they are neither computationally efficient due to their intensive budget maintenance strategy nor eff…
Online learning naturally arises in many statistical and machine learning problems. The most widely used methods in online learning are stochastic first-order algorithms. Among this family of algorithms, there is a recently developed algorithm, Recursive One-Over-T SGD (ROOT-SGD). ROOT-SGD is advantageous in that it converges at a non-asymptotically fast rate, and its estimator further converges to a normal distribution. However, this normal distribution has unknown asymptotic covariance; thus cannot be directly applied to measure the uncertainty. To fill this gap, we develop two estimators fo…
National University of Defense Technology（国防科技大学）；Dongguan University of Technology（东莞理工学院）
We provide a new theoretical analysis framework to investigate online gradient descent in the dynamic environment. Comparing with the previous work, the new framework recovers the state-of-the-art dynamic regret, but does not require extra gradient queries for every iteration. Specifically, when functions are $α$ strongly convex and $β$ smooth, to achieve the state-of-the-art dynamic regret, the previous work requires $O(κ)$ with $κ= \fracβα$ queries of gradients at every iteration. But, our framework shows that the query complexity can be improved to be $O(1)$, which does not depend on $κ$. T…
We prove the familiar Lazy Online Gradient Descent algorithm is universal on polytope domains. That means it gets $O(1)$ pseudo-regret against i.i.d opponents, while simultaneously achieving the well-known $O(\sqrt N)$ worst-case regret bound. For comparison the bulk of the literature focuses on variants of the Hedge (exponential weights) algorithm on the simplex. These can in principle be lifted to general polytopes; however the process is computationally unfeasible for many important classes where the number of vertices grows quickly with the dimension. The lifting procedure also ignores any…
Department of Statistics, North Carolina State University（北卡罗来纳州立大学统计系）
Online decision making aims to learn the optimal decision rule by making personalized decisions and updating the decision rule recursively. It has become easier than before with the help of big data, but new challenges also come along. Since the decision rule should be updated once per step, an offline update which uses all the historical data is inefficient in computation and storage. To this end, we propose a completely online algorithm that can make decisions and update the decision rule online via stochastic gradient descent. It is not only efficient but also supports all kinds of parametr…
University of Toronto（多伦多大学）；Institute for Aerospace Studies（航空航天研究所）
This short note reviews so-called Natural Gradient Descent (NGD) for multivariate Gaussians. The Fisher Information Matrix (FIM) is derived for several different parameterizations of Gaussians. Careful attention is paid to the symmetric nature of the covariance matrix when calculating derivatives. We show that there are some advantages to choosing a parameterization comprising the mean and inverse covariance matrix and provide a simple NGD update that accounts for the symmetric (and sparse) nature of the inverse covariance matrix.
Leiden University（莱顿大学）；Poznań University of Technology（波兹南理工大学）
A standard introduction to online learning might place Online Gradient Descent at its center and then proceed to develop generalizations and extensions like Online Mirror Descent and second-order methods. Here we explore the alternative approach of putting Exponential Weights (EW) first. We show that many standard methods and their regret bounds then follow as a special case by plugging in suitable surrogate losses and playing the EW posterior mean. For instance, we easily recover Online Gradient Descent by using EW with a Gaussian prior on linearized losses, and, more generally, all instances…
Missouri University of Science and Technology（密苏里科技大学）；Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Baidu Inc.（百度公司）
We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the \textit{quasi-potential}. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case that the quasi-potential function is related to the noise covariance structure of SGD via a partial differential equation of Hamilton-Jacobi type. This relation helps us to show that anisotropic noise leads to faster escape than isotropic noise. We then consider the dynamics of SGD…
Mila - Université de Montréal（米拉 - 蒙特利尔大学）；Facebook AI Research（脸书人工智能研究院）；CIFAR（加拿大高等研究院）
Optimization algorithms that leverage gradient covariance information, such as variants of natural gradient descent (Amari, 1998), offer the prospect of yielding more effective descent directions. For models with many parameters, the covariance matrix they are based on becomes gigantic, making them inapplicable in their original form. This has motivated research into both simple diagonal approximations and more sophisticated factored approximations such as KFAC (Heskes, 2000; Martens & Grosse, 2015; Grosse & Martens, 2016). In the present work we draw inspiration from both to propose a novel a…
University of Alberta（阿尔伯塔大学）；Google DeepMind（谷歌DeepMind）
This paper investigates different vector step-size adaptation approaches for non-stationary online, continual prediction problems. Vanilla stochastic gradient descent can be considerably improved by scaling the update with a vector of appropriately chosen step-sizes. Many methods, including AdaGrad, RMSProp, and AMSGrad, keep statistics about the learning process to approximate a second order update---a vector approximation of the inverse Hessian. Another family of approaches use meta-gradient descent to adapt the step-size parameters to minimize prediction error. These meta-descent strategies…
University of Oxford（牛津大学）；University of British Columbia（不列颠哥伦比亚大学）
We introduce a general method for improving the convergence rate of gradient-based optimizers that is easy to implement and works well in practice. We demonstrate the effectiveness of the method in a range of optimization problems by applying it to stochastic gradient descent, stochastic gradient descent with Nesterov momentum, and Adam, showing that it significantly reduces the need for the manual tuning of the initial learning rate for these commonly used algorithms. Our method works by dynamically updating the learning rate during optimization using the gradient with respect to the learning…
University of California, Davis（加州大学戴维斯分校）；University of California, Los Angeles（加州大学洛杉矶分校）
We consider the contextual bandit problem, where a player sequentially makes decisions based on past observations to maximize the cumulative reward. Although many algorithms have been proposed for contextual bandit, most of them rely on finding the maximum likelihood estimator at each iteration, which requires $O(t)$ time at the $t$-th iteration and are memory inefficient. A natural way to resolve this problem is to apply online stochastic gradient descent (SGD) so that the per-step time and memory complexity can be reduced to constant with respect to $t$, but a contextual bandit policy based …
The noise in stochastic gradient descent (SGD) provides a crucial implicit regularization effect for training overparameterized models. Prior theoretical work largely focuses on spherical Gaussian noise, whereas empirical studies demonstrate the phenomenon that parameter-dependent noise -- induced by mini-batches or label perturbation -- is far more effective than Gaussian noise. This paper theoretically characterizes this phenomenon on a quadratically-parameterized model introduced by Vaskevicius et el. and Woodworth et el. We show that in an over-parameterized setting, SGD with label noise r…
University of British Columbia（不列颠哥伦比亚大学）；Sony Computer Science Laboratories Inc.（索尼计算机科学实验室）；RIKEN Center for Advanced Intelligence Project（理化学研究所革新智能综合研究中心）；Alberta Machine Intelligence Institute（阿尔伯塔机器智能研究所）
Natural-gradient descent (NGD) on structured parameter spaces (e.g., low-rank covariances) is computationally challenging due to difficult Fisher-matrix computations. We address this issue by using \emph{local-parameter coordinates} to obtain a flexible and efficient NGD method that works well for a wide-variety of structured parameterizations. We show four applications where our method (1) generalizes the exponential natural evolutionary strategy, (2) recovers existing Newton-like algorithms, (3) yields new structured second-order algorithms via matrix groups, and (4) gives new algorithms to …
In many problems in machine learning and operations research, we need to optimize a function whose input is a random variable or a probability density function, i.e. to solve optimization problems in an infinite dimensional space. On the other hand, online learning has the advantage of dealing with streaming examples, and better model a changing environ- ment. In this paper, we extend the celebrated online gradient descent algorithm to Hilbert spaces (function spaces), and analyze the convergence guarantee of the algorithm. Finally, we demonstrate that our algorithms can be useful in several i…
Multi-way data analysis has become an essential tool for capturing underlying structures in higher-order datasets stored in tensor $\mathcal{X} \in \mathbb{R} ^{I_1 \times \dots \times I_N} $. $CANDECOMP/PARAFAC$ (CP) decomposition has been extensively studied and applied to approximate $\mathcal{X}$ by $N$ loading matrices $A^{(1)}, \dots, A^{(N)}$ where $N$ represents the order of the tensor. We propose a new efficient CP decomposition solver named NeCPD for non-convex problem in multi-way online data based on stochastic gradient descent (SGD) algorithm. SGD is very useful in online setting …
Learning RBMs using standard algorithms such as CD(k) involves gradient descent on the negative log-likelihood. One of the terms in the gradient, which involves expectation w.r.t. the model distribution, is intractable and is obtained through an MCMC estimate. In this work we show that the Hessian of the log-likelihood can be written in terms of covariances of hidden and visible units and hence, all elements of the Hessian can also be estimated using the same MCMC samples with small extra computational costs. Since inverting the Hessian may be computationally expensive, we propose an algorithm…
In batch learning, stability together with existence and uniqueness of the solution corresponds to well-posedness of Empirical Risk Minimization (ERM) methods; recently, it was proved that CV_loo stability is necessary and sufficient for generalization and consistency of ERM. In this note, we introduce CV_on stability, which plays a similar note in online learning. We show that stochastic gradient descent (SDG) with the usual hypotheses is CVon stable and we then discuss the implications of CV_on stability for convergence of SGD.
Institute of High Performance Computing（高性能计算研究所）；California Institute of Technology（加州理工学院）
We study Smoothed Online Convex Optimization, a version of online convex optimization where the learner incurs a penalty for changing her actions between rounds. Given a $Ω(\sqrt{d})$ lower bound on the competitive ratio of any online algorithm, where $d$ is the dimension of the action space, we ask under what conditions this bound can be beaten. We introduce a novel algorithmic framework for this problem, Online Balanced Descent (OBD), which works by iteratively projecting the previous point onto a carefully chosen level set of the current cost function so as to balance the switching costs an…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；The Technion（以色列理工学院）；Università degli Studi di Milano（米兰大学）
Online learning algorithms are fast, memory-efficient, easy to implement, and applicable to many prediction problems, including classification, regression, and ranking. Several online algorithms were proposed in the past few decades, some based on additive updates, like the Perceptron, and some on multiplicative updates, like Winnow. A unifying perspective on the design and the analysis of online algorithms is provided by online mirror descent, a general prediction strategy from which most first-order algorithms can be obtained as special cases. We generalize online mirror descent to time-vary…
Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other…
Sorbonne Université（索邦大学）；Institut des Systèmes Intelligents et de Robotique（智能与机器人系统研究院）
In this paper, we provide an overview of first-order and second-order variants of the gradient descent method that are commonly used in machine learning. We propose a general framework in which 6 of these variants can be interpreted as different instances of the same approach. They are the vanilla gradient descent, the classical and generalized Gauss-Newton methods, the natural gradient descent method, the gradient covariance matrix approach, and Newton's method. Besides interpreting these methods within a single framework, we explain their specificities and show under which conditions some of…
University of Milan（米兰大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；University College London（伦敦大学学院）
We introduce and analyze MT-OMD, a multitask generalization of Online Mirror Descent (OMD) which operates by sharing updates between tasks. We prove that the regret of MT-OMD is of order $\sqrt{1 + σ^2(N-1)}\sqrt{T}$, where $σ^2$ is the task variance according to the geometry induced by the regularizer, $N$ is the number of tasks, and $T$ is the time horizon. Whenever tasks are similar, that is $σ^2 \le 1$, our method improves upon the $\sqrt{NT}$ bound obtained by running independent OMDs on each task. We further provide a matching lower bound, and show that our multitask extensions of Online…
In this paper, we focus on the question of the extent to which online learning can benefit from distributed computing. We focus on the setting in which $N$ agents online-learn cooperatively, where each agent only has access to its own data. We propose a generic data-distributed online learning meta-algorithm. We then introduce the Distributed Weighted Majority and Distributed Online Mirror Descent algorithms, as special cases. We show, using both theoretical analysis and experiments, that compared to a single agent: given the same computation time, these distributed algorithms achieve smaller …
Google AI Princeton（谷歌AI普林斯顿）；Princeton University（普林斯顿大学）
We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, but theory f…
Purdue University（普渡大学）；University of Minnesota（明尼苏达大学）
Past research has indicated that the covariance of the Stochastic Gradient Descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the Multiplicative Stochastic Gradient Descent (M-SGD) algorithm as introduced in previous work, which has a much more general noise class than the SGD algorithm done …
Microsoft Research Asia（微软亚洲研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Stochastic gradient descent (SGD) and its variants are mainstream methods to train deep neural networks. Since neural networks are non-convex, more and more works study the dynamic behavior of SGD and the impact to its generalization, especially the escaping efficiency from local minima. However, these works take the over-simplified assumption that the covariance of the noise in SGD is (or can be upper bounded by) constant, although it is actually state-dependent. In this work, we conduct a formal study on the dynamic behavior of SGD with state-dependent noise. Specifically, we show that the c…
We analyze and evaluate an online gradient descent algorithm with adaptive per-coordinate adjustment of learning rates. Our algorithm can be thought of as an online version of batch gradient descent with a diagonal preconditioner. This approach leads to regret bounds that are stronger than those of standard online gradient descent for general online convex optimization problems. Experimentally, we show that our algorithm is competitive with state-of-the-art algorithms for large scale machine learning problems.
In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate $O\left(\frac{1}{\varepsilon^2}\right)$ improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of Hazan and Kale [2012] with convergence rate $O\left(\frac{1}{\varepsilon^4}\right)$.
UC Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）；Microsoft Research New England（微软新英格兰研究院）
Matrix completion, where we wish to recover a low rank matrix by observing a few entries from it, is a widely studied problem in both theory and practice with wide applications. Most of the provable algorithms so far on this problem have been restricted to the offline setting where they provide an estimate of the unknown matrix using all observations simultaneously. However, in many applications, the online version, where we observe one entry at a time and dynamically update our estimate, is more appealing. While existing algorithms are efficient for the offline setting, they could be highly i…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；The University of Chicago（芝加哥大学）
We propose a flexible yet interpretable model for high-dimensional data with time-varying second order statistics, motivated and applied to functional neuroimaging data. Motivated by the neuroscience literature, we factorize the covariances into sparse spatial and smooth temporal components. While this factorization results in both parsimony and domain interpretability, the resulting estimation problem is nonconvex. To this end, we design a two-stage optimization scheme with a carefully tailored spectral initialization, combined with iteratively refined alternating projected gradient descent. …
California Institute of Technology（加州理工学院）；Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）
We study online convex optimization in a setting where the learner seeks to minimize the sum of a per-round hitting cost and a movement cost which is incurred when changing decisions between rounds. We prove a new lower bound on the competitive ratio of any online algorithm in the setting where the costs are $m$-strongly convex and the movement costs are the squared $\ell_2$ norm. This lower bound shows that no algorithm can achieve a competitive ratio that is $o(m^{-1/2})$ as $m$ tends to zero. No existing algorithms have competitive ratios matching this bound, and we show that the state-of-t…
University of Science and Technology of China（中国科学技术大学）
Follow-the-Regularized-Lead (FTRL) and Online Mirror Descent (OMD) are regret minimization algorithms for Online Convex Optimization (OCO), they are mathematically elegant but less practical in solving Extensive-Form Games (EFGs). Counterfactual Regret Minimization (CFR) is a technique for approximating Nash equilibria in EFGs. CFR and its variants have a fast convergence rate in practice, but their theoretical results are not satisfactory. In recent years, researchers have been trying to link CFRs with OCO algorithms, which may provide new theoretical results and inspire new algorithms. Howev…
Columbia University（哥伦比亚大学）；Rutgers University（罗格斯大学）；Seoul National University（首尔国立大学）；McMaster University（麦克马斯特大学）
We develop a new method of online inference for a vector of parameters estimated by the Polyak-Ruppert averaging procedure of stochastic gradient descent (SGD) algorithms. We leverage insights from time series regression in econometrics and construct asymptotically pivotal statistics via random scaling. Our approach is fully operational with online data and is rigorously underpinned by a functional central limit theorem. Our proposed inference method has a couple of key advantages over the existing methods. First, the test statistic is computed in an online fashion with only SGD iterates and t…
In Iterative Learning Control (ILC), a sequence of feedforward control actions is generated at each iteration on the basis of partial model knowledge and past measurements with the goal of steering the system toward a desired reference trajectory. This is framed here as an online learning task, where the decision-maker takes sequential decisions by solving a sequence of optimization problems having only partial knowledge of the cost functions. Having established this connection, the performance of an online gradient-descent based scheme using inexact gradient information is analyzed in the set…
Online mirror descent (OMD) and dual averaging (DA) -- two fundamental algorithms for online convex optimization -- are known to have very similar (and sometimes identical) performance guarantees when used with a fixed learning rate. Under dynamic learning rates, however, OMD is provably inferior to DA and suffers a linear regret, even in common settings such as prediction with expert advice. We modify the OMD algorithm through a simple technique that we call stabilization. We give essentially the same abstract regret bound for OMD with stabilization and for DA by modifying the classical OMD c…
Harmonized Gradient Descent for Class Imbalanced Data Stream Online Learning
Peng Cheng Laboratory（鹏城实验室）；School of Automation, Chongqing University（重庆大学自动化学院）；School of Information Science and Engineering, East China University of Science and Technology（东华大学信息科学与工程学院）
Online Covariance Matrix Estimation in Sketched Newton Methods
Department of Statistics, The University of Chicago（芝加哥大学统计系）；Mathematics and Computer Science Division, Argonne National Laboratory（阿贡国家实验室数学与计算机科学部）；School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院工业与系统工程系）
Online Covariance Estimation in Nonsmooth Stochastic Approximation
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（H. Milton Stewart工业与系统工程学院，佐治亚理工学院）；Department of Statistics, Texas A&M University（统计学系，德克萨斯大学安德森分校）；Department of Statistics, University of California, Davis（统计学系，加州大学戴维斯分校）；Wharton Department of Statistics and Data Science, University of Pennsylvania（沃顿统计与数据科学系，宾夕法尼亚大学）；Department of Mathematics, University of Washington（数学系，华盛顿大学）
National Technical University of Athens（雅典国立技术大学）；Singapore University of Technology and Design（新加坡科技设计大学）
We consider a natural model of online preference aggregation, where sets of preferred items $R_1, R_2, \ldots, R_t$ along with a demand for $k_t$ items in each $R_t$, appear online. Without prior knowledge of $(R_t, k_t)$, the learner maintains a ranking $π_t$ aiming that at least $k_t$ items from $R_t$ appear high in $π_t$. This is a fundamental problem in preference aggregation with applications to, e.g., ordering product or news items in web pages based on user scrolling and click patterns. The widely studied Generalized Min-Sum-Set-Cover (GMSSC) problem serves as a formal model for the set…
Microsoft Research（微软研究院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）
This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares. This result is obtained by analyzing SGD as a stochastic process and by sharply characterizing the stationary covariance matrix of this process. The finite rate optimality characterization captures the constant factors and addresses model mis-specification.
Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
This work addresses decentralized online optimization in non-stationary environments. A network of agents aim to track the minimizer of a global time-varying convex function. The minimizer evolves according to a known dynamics corrupted by an unknown, unstructured noise. At each time, the global function can be cast as a sum of a finite number of local functions, each of which is assigned to one agent in the network. Moreover, the local functions become available to agents sequentially, and agents do not have a prior knowledge of the future cost functions. Therefore, agents must communicate wi…
