In many online learning problems the computational bottleneck for gradient-based methods is the projection operation. For this reason, in many problems the most efficient algorithms are based on the Frank-Wolfe method, which replaces projections by linear optimization. In the general case, however, online projection-free methods require more iterations than projection-based methods: the best known regret bound scales as $T^{3/4}$. Despite significant work on various variants of the Frank-Wolfe method, this bound has remained unchanged for a decade. In this paper we give an efficient projection-free algorithm that guarantees $T^{2/3}$ regret for general online convex optimization with smooth cost functions and one linear optimization computation per iteration. As opposed to previous Frank-Wolfe approaches, our algorithm is derived using the Follow-the-Perturbed-Leader method and is analyzed using an online primal-dual framework.
Zhejiang University（浙江大学）；University of Pennsylvania（宾夕法尼亚大学）；Tencent AI Lab（腾讯人工智能实验室）；University of Western Ontario（西安大略大学）
This paper focuses on projection-free methods for solving smooth Online Convex Optimization (OCO) problems. Existing projection-free methods either achieve suboptimal regret bounds or have high per-iteration computational costs. To fill this gap, two efficient projection-free online methods called ORGFW and MORGFW are proposed for solving stochastic and adversarial OCO problems, respectively. By employing a recursive gradient estimator, our methods achieve optimal regret bounds (up to a logarithmic factor) while possessing low per-iteration computational costs. Experimental results demonstrate…
In this paper, we develop new efficient projection-free algorithms for Online Convex Optimization (OCO). Online Gradient Descent (OGD) is an example of a classical OCO algorithm that guarantees the optimal $O(\sqrt{T})$ regret bound. However, OGD and other projection-based OCO algorithms need to perform a Euclidean projection onto the feasible set $\mathcal{C}\subset \mathbb{R}^d$ whenever their iterates step outside $\mathcal{C}$. For various sets of interests, this projection step can be computationally costly, especially when the ambient dimension is large. This has motivated the developmen…
Technion - Israel Inst. of Tech.（以色列理工学院）；IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）
The computational bottleneck in applying online learning to massive data sets is usually the projection step. We present efficient online learning algorithms that eschew projections in favor of much more efficient linear optimization steps using the Frank-Wolfe technique. We obtain a range of regret bounds for online convex optimization, with better bounds for specific cases such as stochastic online smooth convex optimization. Besides the computational advantage, other desirable features of our algorithms are that they are parameter-free in the stochastic case and produce sparse decisions. We…
Technion - Israel Institute of Technology（以色列理工学院）
We present new efficient \textit{projection-free} algorithms for online convex optimization (OCO), where by projection-free we refer to algorithms that avoid computing orthogonal projections onto the feasible set, and instead relay on different and potentially much more efficient oracles. While most state-of-the-art projection-free algorithms are based on the \textit{follow-the-leader} framework, our algorithms are fundamentally different and are based on the \textit{online gradient descent} algorithm with a novel and efficient approach to computing so-called \textit{infeasible projections}. A…
University of Pennsylvania（宾夕法尼亚大学）；U.S. Army Research Laboratory（美国陆军研究实验室）
Despite their attractiveness, popular perception is that techniques for nonparametric function approximation do not scale to streaming data due to an intractable growth in the amount of storage they require. To solve this problem in a memory-affordable way, we propose an online technique based on functional stochastic gradient descent in tandem with supervised sparsification based on greedy function subspace projections. The method, called parsimonious online learning with kernels (POLK), provides a controllable tradeoff? between its solution accuracy and the amount of memory it requires. We d…
Yale University（耶鲁大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Pennsylvania（宾夕法尼亚大学）
How can we efficiently mitigate the overhead of gradient communications in distributed optimization? This problem is at the heart of training scalable machine learning models and has been mainly studied in the unconstrained setting. In this paper, we propose Quantized-Frank-Wolfe (QFW), the first projection-free and communication-efficient algorithm for solving constrained optimization problems at scale. We consider both convex and non-convex objective functions, expressed as a finite-sum or more generally a stochastic optimization problem, and provide strong theoretical guarantees on the conv…
Technion - Israel Institute of Technology（以色列理工学院）
We consider the setting of online convex optimization (OCO) with \textit{exp-concave} losses. The best regret bound known for this setting is $O(n\log{}T)$, where $n$ is the dimension and $T$ is the number of prediction rounds (treating all other quantities as constants and assuming $T$ is sufficiently large), and is attainable via the well-known Online Newton Step algorithm (ONS). However, ONS requires on each iteration to compute a projection (according to some matrix-induced norm) onto the feasible convex set, which is often computationally prohibitive in high-dimensional settings and when …
Technion - Israel Institute of Technology（以色列理工学院）
We revisit the challenge of designing online algorithms for the bandit convex optimization problem (BCO) which are also scalable to high dimensional problems. Hence, we consider algorithms that are \textit{projection-free}, i.e., based on the conditional gradient method whose only access to the feasible decision set, is through a linear optimization oracle (as opposed to other methods which require potentially much more computationally-expensive subprocedures, such as computing Euclidean projections). We present the first such algorithm that attains $O(T^{3/4})$ expected regret using only $O(T…
Optimization algorithms such as projected Newton's method, FISTA, mirror descent, and its variants enjoy near-optimal regret bounds and convergence rates, but suffer from a computational bottleneck of computing ``projections'' in potentially each iteration (e.g., $O(T^{1/2})$ regret of online mirror descent). On the other hand, conditional gradient variants solve a linear optimization in each iteration, but result in suboptimal rates (e.g., $O(T^{3/4})$ regret of online Frank-Wolfe). Motivated by this trade-off in runtime v/s convergence rates, we consider iterative projections of close-by poi…
In constrained convex optimization, existing methods based on the ellipsoid or cutting plane method do not scale well with the dimension of the ambient space. Alternative approaches such as Projected Gradient Descent only provide a computational benefit for simple convex sets such as Euclidean balls, where Euclidean projections can be performed efficiently. For other sets, the cost of the projections can be too high. To circumvent these issues, alternative methods based on the famous Frank-Wolfe algorithm have been studied and used. Such methods use a Linear Optimization Oracle at each iterati…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）；University of Pennsylvania（宾夕法尼亚大学）
Online optimization has been a successful framework for solving large-scale problems under computational constraints and partial information. Current methods for online convex optimization require either a projection or exact gradient computation at each step, both of which can be prohibitively expensive for large-scale applications. At the same time, there is a growing trend of non-convex optimization in machine learning community and a need for online methods. Continuous DR-submodular functions, which exhibit a natural diminishing returns condition, have recently been proposed as a broad cla…
Universal Online Convex Optimization with $1$ Projection per Round
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）
To deal with complicated constraints via locally light computations in distributed online learning, a recent study has presented a projection-free algorithm called distributed online conditional gradient (D-OCG), and achieved an $O(T^{3/4})$ regret bound for convex losses, where $T$ is the number of total rounds. However, it requires $T$ communication rounds, and cannot utilize the strong convexity of losses. In this paper, we propose an improved variant of D-OCG, namely D-BOCG, which can attain the same $O(T^{3/4})$ regret bound with only $O(\sqrt{T})$ communication rounds for convex losses, …
Deep neural networks are commonly trained using stochastic non-convex optimization procedures, which are driven by gradient information estimated on fractions (batches) of the dataset. While it is commonly accepted that batch size is an important parameter for offline tuning, the benefits of online selection of batches remain poorly understood. We investigate online batch selection strategies for two state-of-the-art methods of stochastic gradient-based optimization, AdaDelta and Adam. As the loss function to be minimized for the whole dataset is an aggregation of loss functions of individual …
We study an online multi-task learning setting, in which instances of related tasks arrive sequentially, and are handled by task-specific online learners. We consider an algorithmic framework to model the relationship of these tasks via a set of convex constraints. To exploit this relationship, we design a novel algorithm -- COOL -- for coordinating the individual online learners: Our key idea is to coordinate their parameters via weighted projections onto a convex set. By adjusting the rate and accuracy of the projection, the COOL algorithm allows for a trade-off between the benefit of coordi…
Texas Tech University（德州理工大学）；North Dakota State University（北达科他州立大学）；New Jersey Institute of Technology（新泽西理工学院）
Stochastic gradient descent (SGD) and projected stochastic gradient descent (PSGD) are scalable algorithms to compute model parameters in unconstrained and constrained optimization problems. In comparison with SGD, PSGD forces its iterative values into the constrained parameter space via projection. From a statistical point of view, this paper studies the limiting distribution of PSGD-based estimate when the true parameters satisfy some linear-equality constraints. Our theoretical findings reveal the role of projection played in the uncertainty of the PSGD-based estimate. As a byproduct, we pr…
The aim of this paper is to design computationally-efficient and optimal algorithms for the online and stochastic exp-concave optimization settings. Typical algorithms for these settings, such as the Online Newton Step (ONS), can guarantee a $O(d\ln T)$ bound on their regret after $T$ rounds, where $d$ is the dimension of the feasible set. However, such algorithms perform so-called generalized projections whenever their iterates step outside the feasible set. Such generalized projections require $Ω(d^3)$ arithmetic operations even for simple sets such a Euclidean ball, making the total runtime…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
Minimizing a convex function over the spectrahedron, i.e., the set of all positive semidefinite matrices with unit trace, is an important optimization task with many applications in optimization, machine learning, and signal processing. It is also notoriously difficult to solve in large-scale since standard techniques require expensive matrix decompositions. An alternative, is the conditional gradient method (aka Frank-Wolfe algorithm) that regained much interest in recent years, mostly due to its application to this specific setting. The key benefit of the CG method is that it avoids expensiv…
Joint sparsity offers powerful structural cues for feature selection, especially for variables that are expected to demonstrate a "grouped" behavior. Such behavior is commonly modeled via group-lasso, multitask lasso, and related methods where feature selection is effected via mixed-norms. Several mixed-norm based sparse models have received substantial attention, and for some cases efficient algorithms are also available. Surprisingly, several constrained sparse models seem to be lacking scalable algorithms. We address this deficiency by presenting batch and online (stochastic-gradient) optim…
Tsinghua University（清华大学）；South China University of Technology（华南理工大学）
Factorization Machine (FM) is a supervised learning approach with a powerful capability of feature engineering. It yields state-of-the-art performance in various batch learning tasks where all the training data is made available prior to the training. However, in real-world applications where the data arrives sequentially in a streaming manner, the high cost of re-training with batch learning algorithms has posed formidable challenges in the online learning scenario. The initial challenge is that no prior formulations of FM could fulfill the requirements in Online Convex Optimization (OCO) -- …
University of Pittsburgh（匹兹堡大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；MIIT Key Laboratory of Pattern Analysis & Machine Intelligence（工业和信息化部模式分析与机器智能重点实验室）；JD Finance America Corporation（京东金融美国公司）
In this paper, we propose a faster stochastic alternating direction method of multipliers (ADMM) for nonconvex optimization by using a new stochastic path-integrated differential estimator (SPIDER), called as SPIDER-ADMM. Moreover, we prove that the SPIDER-ADMM achieves a record-breaking incremental first-order oracle (IFO) complexity of $\mathcal{O}(n+n^{1/2}ε^{-1})$ for finding an $ε$-approximate stationary point, which improves the deterministic ADMM by a factor $\mathcal{O}(n^{1/2})$, where $n$ denotes the sample size. As one of major contribution of this paper, we provide a new theoretica…
In order to find Nash-equilibria for two-player zero-sum games where each player plays combinatorial objects like spanning trees, matchings etc, we consider two online learning algorithms: the online mirror descent (OMD) algorithm and the multiplicative weights update (MWU) algorithm. The OMD algorithm requires the computation of a certain Bregman projection, that has closed form solutions for simple convex sets like the Euclidean ball or the simplex. However, for general polyhedra one often needs to exploit the general machinery of convex optimization. We give a novel primal-style algorithm f…
University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Georgia Institute of Technology（佐治亚理工学院）
We revisit the Frank-Wolfe (FW) optimization under strongly convex constraint sets. We provide a faster convergence rate for FW without line search, showing that a previously overlooked variant of FW is indeed faster than the standard variant. With line search, we show that FW can converge to the global optimum, even for smooth functions that are not convex, but are quasi-convex and locally-Lipschitz. We also show that, for the general case of (smooth) non-convex functions, FW with line search converges with high probability to a stationary point at a rate of $O\left(\frac{1}{t}\right)$, as lo…
This paper presents a proposal of a faster Wasserstein $k$-means algorithm for histogram data by reducing Wasserstein distance computations and exploiting sparse simplex projection. We shrink data samples, centroids, and the ground cost matrix, which leads to considerable reduction of the computations used to solve optimal transport problems without loss of clustering quality. Furthermore, we dynamically reduced the computational complexity by removing lower-valued data samples and harnessing sparse simplex projection while keeping the degradation of clustering quality lower. We designate this…
This paper studies online optimization from a high-level unified theoretical perspective. We not only generalize both Optimistic-DA and Optimistic-MD in normed vector space, but also unify their analysis methods for dynamic regret. Regret bounds are the tightest possible due to the introduction of $ϕ$-convex. As instantiations, regret bounds of normalized exponentiated subgradient and greedy/lazy projection are better than the currently known optimal results. By replacing losses of online game with monotone operators, and extending the definition of regret, namely regret$^n$, we extend online …
U.S. Army Research Laboratory（美国陆军研究实验室）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）
An open challenge in supervised learning is \emph{conceptual drift}: a data point begins as classified according to one label, but over time the notion of that label changes. Beyond linear autoregressive models, transfer and meta learning address drift, but require data that is representative of disparate domains at the outset of training. To relax this requirement, we propose a memory-efficient \emph{online} universal function approximator based on compressed kernel methods. Our approach hinges upon viewing non-stationary learning as online convex optimization with dynamic comparators, for wh…
Michigan State University（密歇根州立大学）；GE Global Research（通用电气全球研究中心）
In this paper we propose a general framework to characterize and solve the stochastic optimization problems with multiple objectives underlying many real world learning applications. We first propose a projection based algorithm which attains an $O(T^{-1/3})$ convergence rate. Then, by leveraging on the theory of Lagrangian in constrained optimization, we devise a novel primal-dual stochastic approximation algorithm which attains the optimal convergence rate of $O(T^{-1/2})$ for general Lipschitz continuous objectives.
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…
Purdue University（普渡大学）；University of Chicago（芝加哥大学）
Structured output prediction is an important machine learning problem both in theory and practice, and the max-margin Markov network (\mcn) is an effective approach. All state-of-the-art algorithms for optimizing \mcn\ objectives take at least $O(1/ε)$ number of iterations to find an $ε$ accurate solution. Recent results in structured optimization suggest that faster rates are possible by exploiting the structure of the objective function. Towards this end \citet{Nesterov05} proposed an excessive gap reduction technique based on Euclidean projections which converges in $O(1/\sqrtε)$ iterations…
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 …
Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
The online problem of computing the top eigenvector is fundamental to machine learning. In both adversarial and stochastic settings, previous results (such as matrix multiplicative weight update, follow the regularized leader, follow the compressed leader, block power method) either achieve optimal regret but run slow, or run fast at the expense of loosing a $\sqrt{d}$ factor in total regret where $d$ is the matrix dimension. We propose a $\textit{follow-the-compressed-leader (FTCL)}$ framework which achieves optimal regret without sacrificing the running time. Our idea is to "compress" the ma…
Online-learning literature has focused on designing algorithms that ensure sub-linear growth of the cumulative long-term constraint violations. The drawback of this guarantee is that strictly feasible actions may cancel out constraint violations on other time slots. For this reason, we introduce a new performance measure called $\hCFit$, whose particular instance is the cumulative positive part of the constraint violations. We propose a class of non-causal algorithms for online-decision making, which guarantees, in slowly changing environments, sub-linear growth of this quantity despite noisy …
Technion - Israel Institute of Technology（以色列理工学院）
Projection-free optimization algorithms, which are mostly based on the classical Frank-Wolfe method, have gained significant interest in the machine learning community in recent years due to their ability to handle convex constraints that are popular in many applications, but for which computing projections is often computationally impractical in high-dimensional settings, and hence prohibit the use of most standard projection-based methods. In particular, a significant research effort was put on projection-free methods for online learning. In this paper we revisit the Online Frank-Wolfe (OFW)…
CSIE Department, National Taiwan University（台湾大学计算机科学与信息工程学系（CSIE Department, National Taiwan University）/ 台湾大学（National Taiwan University））
We study multi-label classification (MLC) with three important real-world issues: online updating, label space dimensional reduction (LSDR), and cost-sensitivity. Current MLC algorithms have not been designed to address these three issues simultaneously. In this paper, we propose a novel algorithm, cost-sensitive dynamic principal projection (CS-DPP) that resolves all three issues. The foundation of CS-DPP is an online LSDR framework derived from a leading LSDR algorithm. In particular, CS-DPP is equipped with an efficient online dimension reducer motivated by matrix stochastic gradient, and e…
Indraprastha Institute of Information Technology, Delhi（德里英迪拉普拉斯信息技术学院）
In this paper, we train a recurrent neural network to learn dynamics of a chaotic road environment and to project the future of the environment on an image. Future projection can be used to anticipate an unseen environment for example, in autonomous driving. Road environment is highly dynamic and complex due to the interaction among traffic participants such as vehicles and pedestrians. Even in this complex environment, a human driver is efficacious to safely drive on chaotic roads irrespective of the number of traffic participants. The proliferation of deep learning research has shown the eff…
Technion - Israel Institute of Technology（以色列理工学院）
This paper considers a convex composite optimization problem with affine constraints, which includes problems that take the form of minimizing a smooth convex objective function over the intersection of (simple) convex sets, or regularized with multiple (simple) functions. Motivated by high-dimensional applications in which exact projection/proximal computations are not tractable, we propose a \textit{projection-free} augmented Lagrangian-based method, in which primal updates are carried out using a \textit{weak proximal oracle} (WPO). In an earlier work, WPO was shown to be more powerful than…
Youtu Lab, Tencent（腾讯优图实验室）；Department of Computer Science and Engineering, The Chinese University of Hong Kong（香港中文大学计算机科学与工程学系）；Shenzhen Research Institute, The Chinese University of Hong Kong（香港中文大学深圳研究院）；Meitu (China)（美图（中国））；Department of Computing, Hang Seng University of Hong Kong（香港恒生大学计算机系）
Data-dependent hashing methods have demonstrated good performance in various machine learning applications to learn a low-dimensional representation from the original data. However, they still suffer from several obstacles: First, most of existing hashing methods are trained in a batch mode, yielding inefficiency for training streaming data. Second, the computational cost and the memory consumption increase extraordinarily in the big data setting, which perplexes the training procedure. Third, the lack of labeled data hinders the improvement of the model performance. To address these difficult…
