Gaussian processes provide a framework for nonlinear nonparametric Bayesian inference widely applicable across science and engineering. Unfortunately, their computational burden scales cubically with the training sample size, which in the case that samples arrive in perpetuity, approaches infinity. This issue necessitates approximations for use with streaming data, which to date mostly lack convergence guarantees. Thus, we develop the first online Gaussian process approximation that preserves convergence to the population posterior, i.e., asymptotic posterior consistency, while ameliorating its intractable complexity growth with the sample size. We propose an online compression scheme that, following each a posteriori update, fixes an error neighborhood with respect to the Hellinger metric centered at the current posterior, and greedily tosses out past kernel dictionary elements until its boundary is hit. We call the resulting method Parsimonious Online Gaussian Processes (POG). For diminishing error radius, exact asymptotic consistency is preserved (Theorem 1(i)) at the cost of unbounded memory in the limit. On the other hand, for constant error radius, POG converges to a neighborhood of the population posterior (Theorem 1(ii))but with finite memory at-worst determined by the metric entropy of the feature space (Theorem 2). Experimental results are presented on several nonlinear regression problems which illuminates the merits of this approach as compared with alternatives that fix the subspace dimension defining the history of past points.
Volvo Car Corporation（沃尔沃汽车公司）；Chalmers University of Technology（查尔姆斯理工大学）
In this paper, we study bottleneck identification in networks via extracting minimax paths. Many real-world networks have stochastic weights for which full knowledge is not available in advance. Therefore, we model this task as a combinatorial semi-bandit problem to which we apply a combinatorial version of Thompson Sampling and establish an upper bound on the corresponding Bayesian regret. Due to the computational intractability of the problem, we then devise an alternative problem formulation which approximates the original objective. Finally, we experimentally evaluate the performance of Th…
Online Passive-Aggressive (PA) learning is a class of online margin-based algorithms suitable for a wide range of real-time prediction tasks, including classification and regression. PA algorithms are formulated in terms of deterministic point-estimation problems governed by a set of user-defined hyperparameters: the approach fails to capture model/prediction uncertainty and makes their performance highly sensitive to hyperparameter configurations. In this paper, we introduce a novel PA learning framework for regression that overcomes the above limitations. We contribute a Bayesian state-space…
University of Cantabria（坎塔布里亚大学）；CIBER-BBN（西班牙生物医学、生物工程和纳米医学研究中心联盟）；University of Zaragoza（萨拉戈萨大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
We study the relationship between online Gaussian process (GP) regression and kernel least mean squares (KLMS) algorithms. While the latter have no capacity of storing the entire posterior distribution during online learning, we discover that their operation corresponds to the assumption of a fixed posterior covariance that follows a simple parametric model. Interestingly, several well-known KLMS algorithms correspond to specific cases of this model. The probabilistic perspective allows us to understand how each of them handles uncertainty, which could explain some of their performance differe…
Australian Artificial Intelligence Institute（澳大利亚人工智能研究所）；University of Technology Sydney（悉尼科技大学）
As a powerful Bayesian non-parameterized algorithm, the Gaussian process (GP) has performed a significant role in Bayesian optimization and signal processing. GPs have also advanced online decision-making systems because their posterior distribution has a closed-form solution. However, its training and inference process requires all historic data to be stored and the GP model to be trained from scratch. For those reasons, several online GP algorithms, such as O-SGPR and O-SVGP, have been specifically designed for streaming settings. In this paper, we present a new theoretical framework for onl…
Virginia Polytechnic Institute and State University（弗吉尼亚理工学院暨州立大学）；U.S. Army Research Laboratory（美国陆军研究实验室）
Gaussian processes (GPs) are a well-known nonparametric Bayesian inference technique, but they suffer from scalability problems for large sample sizes, and their performance can degrade for non-stationary or spatially heterogeneous data. In this work, we seek to overcome these issues through (i) employing variational free energy approximations of GPs operating in tandem with online expectation propagation steps; and (ii) introducing a local splitting step which instantiates a new GP whenever the posterior distribution changes significantly as quantified by the Wasserstein metric over posterior…
The University of Sheffield（谢菲尔德大学）；University of Canterbury（坎特伯雷大学）；The University of Oxford（牛津大学）
Gaussian process regression is a machine learning approach which has been shown its power for estimation of unknown functions. However, Gaussian processes suffer from high computational complexity, as in a basic form they scale cubically with the number of observations. Several approaches based on inducing points were proposed to handle this problem in a static context. These methods though face challenges with real-time tasks and when the data is received sequentially over time. In this paper, a novel online algorithm for training sparse Gaussian process models is presented. It treats the mea…
COIN-GP: Cooperative Online Learning in Networked Distributed Systems with Partial Measurements via Gaussian Process Regression
Technical University of Munich（慕尼黑工业大学）；The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；Tongji University（同济大学）；Shanghai Yangzhi Rehabilitation Hospital（上海市养志康复医院）；University of Science and Technology of China（中国科学技术大学）；The Hong Kong Polytechnic University（香港理工大学）
Byzantine-resilient federated online learning for Gaussian process regression
School of Electrical Engineering and Computer Science, The Pennsylvania State University, University Park, PA 16802 USA（电气工程与计算机科学学院，宾夕法尼亚州立大学，大学公园，PA 16802 USA）；Virginia Tech Transportation Institute, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 USA（弗吉尼亚理工交通研究所，弗吉尼亚理工学院与州立大学，布莱克堡，VA 24061 USA）
We establish optimal rates for online regression for arbitrary classes of regression functions in terms of the sequential entropy introduced in (Rakhlin, Sridharan, Tewari, 2010). The optimal rates are shown to exhibit a phase transition analogous to the i.i.d./statistical learning case, studied in (Rakhlin, Sridharan, Tsybakov 2013). In the frequently encountered situation when sequential entropy and i.i.d. empirical entropy match, our results point to the interesting phenomenon that the rates for statistical learning with squared loss and online nonparametric regression are the same. In addi…
University of Potsdam（波茨坦大学）；Centre de Recherche INRIA de Paris（法国国家信息与自动化研究所巴黎研究中心）；IRT Saint Exupéry（圣埃克苏佩里技术研究院）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）
In this work we investigate the variation of the online kernelized ridge regression algorithm in the setting of $d-$dimensional adversarial nonparametric regression. We derive the regret upper bounds on the classes of Sobolev spaces $W_{p}^β(\mathcal{X})$, $p\geq 2, β>\frac{d}{p}$. The upper bounds are supported by the minimax regret analysis, which reveals that in the cases $β> \frac{d}{2}$ or $p=\infty$ these rates are (essentially) optimal. Finally, we compare the performance of the kernelized ridge regression forecaster to the known non-parametric forecasters in terms of the regret rates a…
We consider the problem of online linear regression on arbitrary deterministic sequences when the ambient dimension d can be much larger than the number of time rounds T. We introduce the notion of sparsity regret bound, which is a deterministic online counterpart of recent risk bounds derived in the stochastic setting under a sparsity scenario. We prove such regret bounds for an online-learning algorithm called SeqSEW and based on exponential weighting and data-driven truncation. In a second part we apply a parameter-free version of this algorithm to the stochastic setting (regression model w…
Online dimension reduction is a common method for high-dimensional streaming data processing. Online principal component analysis, online sliced inverse regression, online kernel principal component analysis and other methods have been studied in depth, but as far as we know, online supervised nonlinear dimension reduction methods have not been fully studied. In this article, an online kernel sliced inverse regression method is proposed. By introducing the approximate linear dependence condition and dictionary variable sets, we address the problem of increasing variable dimensions with the sam…
INRIA（法国国家信息与自动化研究所）；PSL Research University（巴黎文理研究大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；LJK（约瑟夫·傅立叶实验室）
Mixability has been shown to be a powerful tool to obtain algorithms with optimal regret. However, the resulting methods often suffer from high computational complexity which has reduced their practical applicability. For example, in the case of multiclass logistic regression, the aggregating forecaster (Foster et al. (2018)) achieves a regret of $O(\log(Bn))$ whereas Online Newton Step achieves $O(e^B\log(n))$ obtaining a double exponential gain in $B$ (a bound on the norm of comparative functions). However, this high statistical performance is at the price of a prohibitive computational comp…
Distributed computing offers a high degree of flexibility to accommodate modern learning constraints and the ever increasing size of datasets involved in massive data issues. Drawing inspiration from the theory of distributed computation models developed in the context of gradient-type optimization algorithms, we present a consensus-based asynchronous distributed approach for nonparametric online regression and analyze some of its asymptotic properties. Substantial numerical evidence involving up to 28 parallel processors is provided on synthetic datasets to assess the excellent performance of…
INRIA - Département d’Informatique de l’École Normale Supérieure（法国国家信息与自动化研究所 - 巴黎高等师范学院信息学系）；PSL Research University（巴黎文理研究大学）
We consider the setting of online logistic regression and consider the regret with respect to the 2-ball of radius B. It is known (see [Hazan et al., 2014]) that any proper algorithm which has logarithmic regret in the number of samples (denoted n) necessarily suffers an exponential multiplicative constant in B. In this work, we design an efficient improper algorithm that avoids this exponential constant while preserving a logarithmic regret. Indeed, [Foster et al., 2018] showed that the lower bound does not apply to improper algorithms and proposed a strategy based on exponential weights with…
We consider the problem of online linear regression on individual sequences. The goal in this paper is for the forecaster to output sequential predictions which are, after $T$ time rounds, almost as good as the ones output by the best linear predictor in a given $\ell^1$-ball in $\\R^d$. We consider both the cases where the dimension~$d$ is small and large relative to the time horizon $T$. We first present regret bounds with optimal dependencies on $d$, $T$, and on the sizes $U$, $X$ and $Y$ of the $\ell^1$-ball, the input data and the observations. The minimax regret is shown to exhibit a reg…
Initiative for the Theoretical Sciences, The Graduate Center, CUNY（纽约市立大学研究生中心理论科学倡议）
Avoiding overfitting is a central challenge in machine learning, yet many large neural networks readily achieve zero training loss. This puzzling contradiction necessitates new approaches to the study of overfitting. Here we quantify overfitting via residual information, defined as the bits in fitted models that encode noise in training data. Information efficient learning algorithms minimize residual information while maximizing the relevant bits, which are predictive of the unknown generative models. We solve this optimization to obtain the information content of optimal algorithms for a lin…
We consider Online Convex Optimization (OCO) in the setting where the costs are $m$-strongly convex and the online learner pays a switching cost for changing decisions between rounds. We show that the recently proposed Online Balanced Descent (OBD) algorithm is constant competitive in this setting, with competitive ratio $3 + O(1/m)$, irrespective of the ambient dimension. Additionally, we show that when the sequence of cost functions is $ε$-smooth, OBD has near-optimal dynamic regret and maintains strong per-round accuracy. We demonstrate the generality of our approach by showing that the OBD…
We present a framework for supervised subspace tracking, when there are two time series $x_t$ and $y_t$, one being the high-dimensional predictors and the other being the response variables and the subspace tracking needs to take into consideration of both sequences. It extends the classic online subspace tracking work which can be viewed as tracking of $x_t$ only. Our online sufficient dimensionality reduction (OSDR) is a meta-algorithm that can be applied to various cases including linear regression, logistic regression, multiple linear regression, multinomial logistic regression, support ve…
Virginia Tech（弗吉尼亚理工大学）；George Mason University（乔治梅森大学）；U. S. Army Corps of Engineers（美国陆军工程兵团）
The presence of data corruption in user-generated streaming data, such as social media, motivates a new fundamental problem that learns reliable regression coefficient when features are not accessible entirely at one time. Until now, several important challenges still cannot be handled concurrently: 1) corrupted data estimation when only partial features are accessible; 2) online feature selection when data contains adversarial corruption; and 3) scaling to a massive dataset. This paper proposes a novel RObust regression algorithm via Online Feature Selection (\textit{RoOFS}) that concurrently…
Stanford University（斯坦福大学）；University of Washington（华盛顿大学）
We consider the problem of online active learning to collect data for regression modeling. Specifically, we consider a decision maker with a limited experimentation budget who must efficiently learn an underlying linear population model. Our main contribution is a novel threshold-based algorithm for selection of most informative observations; we characterize its performance and fundamental lower bounds. We extend the algorithm and its guarantees to sparse linear regression in high-dimensional settings. Simulations suggest the algorithm is remarkably robust: it provides significant benefits ove…
Multiclass logistic regression is a fundamental task in machine learning with applications in classification and boosting. Previous work (Foster et al., 2018) has highlighted the importance of improper predictors for achieving "fast rates" in the online multiclass logistic regression problem without suffering exponentially from secondary problem parameters, such as the norm of the predictors in the comparison class. While Foster et al. (2018) introduced a statistically optimal algorithm, it is in practice computationally intractable due to its run-time complexity being a large polynomial in th…
EDF R&D（法国电力研发中心）；HEC Paris（巴黎高等商学院）；CNRS（法国国家科学研究中心）；Institut de Mathématiques de Toulouse, Université Paul Sabatier（保罗·萨巴捷大学图卢兹数学研究所）
We consider the problem of online nonparametric regression with arbitrary deterministic sequences. Using ideas from the chaining technique, we design an algorithm that achieves a Dudley-type regret bound similar to the one obtained in a non-constructive fashion by Rakhlin and Sridharan (2014). Our regret bound is expressed in terms of the metric entropy in the sup norm, which yields optimal guarantees when the metric and sequential entropies are of the same order of magnitude. In particular our algorithm is the first one that achieves optimal rates for online regression over H{ö}lder balls. In…
University of São Paulo（圣保罗大学）；State University of Londrina（隆德里纳州立大学）
One of the current challenges in machine learning is how to deal with data coming at increasing rates in data streams. New predictive learning strategies are needed to cope with the high throughput data and concept drift. One of the data stream mining tasks where new learning strategies are needed is multi-target regression, due to its applicability in a high number of real world problems. While reliable and effective learning strategies have been proposed for batch multi-target regression, few have been proposed for multi-target online learning in data streams. Besides, most of the existing s…
This paper considers online convex optimization (OCO) problems - the paramount framework for online learning algorithm design. The loss function of learning task in OCO setting is based on streaming data so that OCO is a powerful tool to model large scale applications such as online recommender systems. Meanwhile, real-world data are usually of extreme high-dimensional due to modern feature engineering techniques so that the quadratic regression is impractical. Factorization Machine as well as its variants are efficient models for capturing feature interactions with low-rank matrix model but t…
A fundamental challenge in contextual bandits is to develop flexible, general-purpose algorithms with computational requirements no worse than classical supervised learning tasks such as classification and regression. Algorithms based on regression have shown promising empirical success, but theoretical guarantees have remained elusive except in special cases. We provide the first universal and optimal reduction from contextual bandits to online regression. We show how to transform any oracle for online regression with a given value function class into an algorithm for contextual bandits with …
University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）
This paper establishes minimax rates for online regression with arbitrary classes of functions and general losses. We show that below a certain threshold for the complexity of the function class, the minimax rates depend on both the curvature of the loss function and the sequential complexities of the class. Above this threshold, the curvature of the loss does not affect the rates. Furthermore, for the case of square loss, our results point to the interesting phenomenon: whenever sequential and i.i.d. empirical entropies match, the rates for statistical and online learning are the same. In add…
School of Physical and Mathematical Sciences（物理与数学科学学院）；Nanyang Technological University（南洋理工大学）
Active regression considers a linear regression problem where the learner receives a large number of data points but can only observe a small number of labels. Since online algorithms can deal with incremental training data and take advantage of low computational cost, we consider an online extension of the active regression problem: the learner receives data points one by one and immediately decides whether it should collect the corresponding labels. The goal is to efficiently maintain the regression of received data points with a small budget of label queries. We propose novel algorithms for…
Recurrent Neural Networks (RNNs) are widely used for online regression due to their ability to generalize nonlinear temporal dependencies. As an RNN model, Long-Short-Term-Memory Networks (LSTMs) are commonly preferred in practice, as these networks are capable of learning long-term dependencies while avoiding the vanishing gradient problem. However, due to their large number of parameters, training LSTMs requires considerably longer training time compared to simple RNNs (SRNNs). In this paper, we achieve the online regression performance of LSTMs with SRNNs efficiently. To this end, we introd…
Cauchy-Schwarz Divergence Information Bottleneck for Regression
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；UiT - The Arctic University of Norway（挪威北极大学）；Brookhaven National Laboratory（布鲁克海文国家实验室）；NORCE Norwegian Research Centre（NORCE挪威研究中心）；University of Copenhagen（哥本哈根大学）；University of Florida（佛罗里达大学）
Theoretical results show that Bayesian methods can achieve lower bounds on regret for online logistic regression. In practice, however, such techniques may not be feasible especially for very large feature sets. Various approximations that, for huge sparse feature sets, diminish the theoretical advantages, must be used. Often, they apply stochastic gradient methods with hyper-parameters that must be tuned on some surrogate loss, defeating theoretical advantages of Bayesian methods. The surrogate loss, defined to approximate the mixture, requires techniques as Monte Carlo sampling, increasing c…
Virginia Tech（弗吉尼亚理工大学）；George Mason University（乔治梅森大学）；U. S. Army Corps of Engineers（美国陆军工程兵团）
In today's era of big data, robust least-squares regression becomes a more challenging problem when considering the adversarial corruption along with explosive growth of datasets. Traditional robust methods can handle the noise but suffer from several challenges when applied in huge dataset including 1) computational infeasibility of handling an entire dataset at once, 2) existence of heterogeneously distributed corruption, and 3) difficulty in corruption estimation when data cannot be entirely loaded. This paper proposes online and distributed robust regression approaches, both of which can c…
Online customer reviews have become important for managers and executives in the hospitality and catering industry who wish to obtain a comprehensive understanding of their customers' demands and expectations. We propose a Regularized Text Logistic (RTL) regression model to perform text analytics and sentiment classification on unstructured text data, which automatically identifies a set of statistically significant and operationally insightful word features, and achieves satisfactory predictive classification accuracy. We apply the RTL model to two online review datasets, Restaurant and Hotel…
Poznań University of Technology（波兹南理工大学）；Centrum Wiskunde & Informatica（数学与计算机科学中心）；University of California at Berkeley（加州大学伯克利分校）
We consider the online version of the isotonic regression problem. Given a set of linearly ordered points (e.g., on the real line), the learner must predict labels sequentially at adversarially chosen positions and is evaluated by her total squared loss compared against the best isotonic (non-decreasing) function in hindsight. We survey several standard online learning algorithms and show that none of them achieve the optimal regret exponent; in fact, most of them (including Online Gradient Descent, Follow the Leader and Exponential Weights) incur linear regret. We then prove that the Exponent…
Penn State University（宾夕法尼亚州立大学）；UC Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Computational efficiency is an important consideration for deploying machine learning models for time series prediction in an online setting. Machine learning algorithms adjust model parameters automatically based on the data, but often require users to set additional parameters, known as hyperparameters. Hyperparameters can significantly impact prediction accuracy. Traffic measurements, typically collected online by sensors, are serially correlated. Moreover, the data distribution may change gradually. A typical adaptation strategy is periodically re-tuning the model hyperparameters, at the c…
The Technion - Israel Institute of Technology（以色列理工学院）
The goal of a learner, in standard online learning, is to have the cumulative loss not much larger compared with the best-performing function from some fixed class. Numerous algorithms were shown to have this gap arbitrarily close to zero, compared with the best function that is chosen off-line. Nevertheless, many real-world applications, such as adaptive filtering, are non-stationary in nature, and the best prediction function may drift over time. We introduce two novel algorithms for online regression, designed to work well in non-stationary environment. Our first algorithm performs adaptive…
In this work we revisit two classic high-dimensional online learning problems, namely linear regression and contextual bandits, from the perspective of adversarial robustness. Existing works in algorithmic robust statistics make strong distributional assumptions that ensure that the input data is evenly spread out or comes from a nice generative model. Is it possible to achieve strong robustness guarantees even without distributional assumptions altogether, where the sequence of tasks we are asked to solve is adaptively and adversarially chosen? We answer this question in the affirmative for b…
