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 and hitting costs. We demonstrate the generality of the OBD framework by showing how, with different choices of "balance," OBD can improve upon state-of-the-art performance guarantees for both competitive ratio and regret, in particular, OBD is the first algorithm to achieve a dimension-free competitive ratio, $3 + O(1/α)$, for locally polyhedral costs, where $α$ measures the "steepness" of the costs. We also prove bounds on the dynamic regret of OBD when the balance is performed in the dual space that are dimension-free and imply that OBD has sublinear static regret.
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.
To improve the off-sample generalization of classical procedures minimizing the empirical risk under potentially heavy-tailed data, new robust learning algorithms have been proposed in recent years, with generalized median-of-means strategies being particularly salient. These procedures enjoy performance guarantees in the form of sharp risk bounds under weak moment assumptions on the underlying loss, but typically suffer from a large computational overhead and substantial bias when the data happens to be sub-Gaussian, limiting their utility. In this work, we propose a novel robust gradient des…
Jacobs University Bremen（不来梅雅各布大学）；Katholische Universität Eichstätt-Ingolstadt（艾希施泰特-因戈尔施塔特天主教大学）
We present a Compressive Sensing algorithm for reconstructing binary signals from its linear measurements. The proposed algorithm minimizes a non-convex cost function expressed as a weighted sum of smoothed $\ell_0$ norms which takes into account the binariness of signals. We show that for binary signals the proposed algorithm outperforms other existing algorithms in recovery rate while requiring a short run time.
Online simultaneous inference for quantiles via smoothed stochastic gradient descent
Department of Mathematics and Statistics（数学与统计学系）；Washington University in St.Louis（圣路易斯华盛顿大学）；Chair of Statistics（统计学教授职位）；Humboldt-Universität zu Berlin（柏林洪堡大学）；University of Chicago（芝加哥大学）
Uncertainty sampling, a popular active learning algorithm, is used to reduce the amount of data required to learn a classifier, but it has been observed in practice to converge to different parameters depending on the initialization and sometimes to even better parameters than standard training on all the data. In this work, we give a theoretical explanation of this phenomenon, showing that uncertainty sampling on a convex loss can be interpreted as performing a preconditioned stochastic gradient step on a smoothed version of the population zero-one loss that converges to the population zero-o…
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…
Here, we study different update rules in stochastic gradient descent (SGD) for online forecasting problems. The selection of the learning rate parameter is critical in SGD. However, it may not be feasible to tune this parameter in online learning. Therefore, it is necessary to have an update rule that is not sensitive to the selection of the learning parameter. Inspired by the local regret metric that we introduced previously, we propose to use time-smoothed gradients within SGD update. Using the public data set-- GEFCom2014, we validate that our approach yields more stable results than the ot…
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…
A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems
Shenzhen Research Institute of Big Data（深圳大数据研究院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of ISE, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校工业与系统工程系）
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.
Conducting pairwise comparisons is a widely used approach in curating human perceptual preference data. Typically raters are instructed to make their choices according to a specific set of rules that address certain dimensions of image quality and aesthetics. The outcome of this process is a dataset of sampled image pairs with their associated empirical preference probabilities. Training a model on these pairwise preferences is a common deep learning approach. However, optimizing by gradient descent through mini-batch learning means that the "global" ranking of the images is not explicitly tak…
Stochastic gradient descent updates parameters with summation gradient computed from a random data batch. This summation will lead to unbalanced training process if the data we obtained is unbalanced. To address this issue, this paper takes the error variance and error mean both into consideration. The adaptively adjusting approach of two terms trading off is also given in our algorithm. Due to this algorithm can suppress error variance, we named it Variance Suppression Gradient Descent (VSSGD). Experimental results have demonstrated that VSSGD can accelerate the training process, effectively …
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…
University College London（伦敦大学学院）；KAUST（阿卜杜拉国王科技大学）
We study the Stein Variational Gradient Descent (SVGD) algorithm, which optimises a set of particles to approximate a target probability distribution $π\propto e^{-V}$ on $\mathbb{R}^d$. In the population limit, SVGD performs gradient descent in the space of probability distributions on the KL divergence with respect to $π$, where the gradient is smoothed through a kernel integral operator. In this paper, we provide a novel finite time analysis for the SVGD algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration, and rates of convergence …
Nanjing University（南京大学）；the University of Iowa（爱荷华大学）
In this paper, we investigate an online prediction strategy named as Discounted-Normal-Predictor (Kapralov and Panigrahy, 2010) for smoothed online convex optimization (SOCO), in which the learner needs to minimize not only the hitting cost but also the switching cost. In the setting of learning with expert advice, Daniely and Mansour (2019) demonstrate that Discounted-Normal-Predictor can be utilized to yield nearly optimal regret bounds over any interval, even in the presence of switching costs. Inspired by their results, we develop a simple algorithm for SOCO: Combining online gradient desc…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper shows that the implicit bias of gradient descent on linearly separable data is exactly characterized by the optimal solution of a dual optimization problem given by a smoothed margin, even for general losses. This is in contrast to prior results, which are often tailored to exponentially-tailed losses. For the exponential loss specifically, with $n$ training examples and $t$ gradient descent steps, our dual analysis further allows us to prove an $O(\ln(n)/\ln(t))$ convergence rate to the $\ell_2$ maximum margin direction, when a constant step size is used. This rate is tight in both…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Harvard University（哈佛大学）
We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1} \cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are at least the minimum of the input and output dimensions; (ii) weight matrices at initialization are approximately balanced; and (iii) the initial loss is smaller than the loss of any rank-deficient solution. The assumptions on initialization (conditions (ii) and (iii)) are neces…
Oden Institute, UT Austin（德克萨斯大学奥斯汀分校奥登研究所）；Department of Mathematics, UT Austin（德克萨斯大学奥斯汀分校数学系）
We prove that the norm version of the adaptive stochastic gradient method (AdaGrad-Norm) achieves a linear convergence rate for a subset of either strongly convex functions or non-convex functions that satisfy the Polyak Lojasiewicz (PL) inequality. The paper introduces the notion of Restricted Uniform Inequality of Gradients (RUIG)---which is a measure of the balanced-ness of the stochastic gradient norms---to depict the landscape of a function. RUIG plays a key role in proving the robustness of AdaGrad-Norm to its hyper-parameter tuning in the stochastic setting. On top of RUIG, we develop a…
Guanghua School of Management, Peking University（北京大学光华管理学院）；Center for Applied Statistics, Renmin University of China（中国人民大学应用统计中心）；School of Statistics, Renmin University of China（中国人民大学统计学院）
We study a fully decentralized federated learning algorithm, which is a novel gradient descent algorithm executed on a communication-based network. For convenience, we refer to it as a network gradient descent (NGD) method. In the NGD method, only statistics (e.g., parameter estimates) need to be communicated, minimizing the risk of privacy. Meanwhile, different clients communicate with each other directly according to a carefully designed network structure without a central master. This greatly enhances the reliability of the entire algorithm. Those nice properties inspire us to carefully stu…
Gradient descent ascent (GDA), the simplest single-loop algorithm for nonconvex minimax optimization, is widely used in practical applications such as generative adversarial networks (GANs) and adversarial training. Albeit its desirable simplicity, recent work shows inferior convergence rates of GDA in theory even assuming strong concavity of the objective on one side. This paper establishes new convergence results for two alternative single-loop algorithms -- alternating GDA and smoothed GDA -- under the mild assumption that the objective satisfies the Polyak-Lojasiewicz (PL) condition about …
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）
Stochastic gradient descent (SGD) is commonly used for optimization in large-scale machine learning problems. Langford et al. (2009) introduce a sparse online learning method to induce sparsity via truncated gradient. With high-dimensional sparse data, however, the method suffers from slow convergence and high variance due to the heterogeneity in feature sparsity. To mitigate this issue, we introduce a stabilized truncated stochastic gradient descent algorithm. We employ a soft-thresholding scheme on the weight vector where the imposed shrinkage is adaptive to the amount of information availab…
This paper presents competitive algorithms for a novel class of online optimization problems with memory. We consider a setting where the learner seeks to minimize the sum of a hitting cost and a switching cost that depends on the previous $p$ decisions. This setting generalizes Smoothed Online Convex Optimization. The proposed approach, Optimistic Regularized Online Balanced Descent, achieves a constant, dimension-free competitive ratio. Further, we show a connection between online optimization with memory and online control with adversarial disturbances. This connection, in turn, leads to a …
Noise Balance and Stationary Distribution of Stochastic Gradient Descent
Research Laboratory of Electronics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA；Physics \& Informatics Laboratories, NTT Research, 940 Stewart Drive Sunnyvale, CA 94085, USA；Department of Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan；RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan；Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-0033, Japan
A Local Polyak-Lojasiewicz and Descent Lemma of Gradient Descent For Overparametrized Linear Models
Center for Innovation in Data Engineering and Science (IDEAS) Department of Statistics and Data Science, the Wharton School University of Pennsylvania（创新数据工程与科学中心（IDEAS）统计与数据科学系，沃顿商学院，宾夕法尼亚大学）；Center for Innovation in Data Engineering and Science (IDEAS) University of Pennsylvania（创新数据工程与科学中心（IDEAS）宾夕法尼亚大学）；Center for Innovation in Data Engineering and Science (IDEAS) Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania（创新数据工程与科学中心（IDEAS）应用数学与计算科学研究生组，宾夕法尼亚大学）；Mathematical Institute for Data Science (MINDS) Department of Electrical and Computer Engineering Johns Hopkins University（数据科学数学研究所（MINDS）电气与计算机工程系，约翰霍普金斯大学）；Department of Electrical and Systems Engineering（电气与系统工程系；放射学系，宾夕法尼亚大学）；Department of Radiology University of Pennsylvania
We propose a novel Caputo fractional derivative-based optimization algorithm. Upon defining the Caputo fractional gradient with respect to the Cartesian coordinate, we present a generic Caputo fractional gradient descent (CFGD) method. We prove that the CFGD yields the steepest descent direction of a locally smoothed objective function. The generic CFGD requires three parameters to be specified, and a choice of the parameters yields a version of CFGD. We propose three versions -- non-adaptive, adaptive terminal and adaptive order. By focusing on quadratic objective functions, we provide a conv…
Stochastic gradient descent (SGD) is widely used in machine learning. Although being commonly viewed as a fast but not accurate version of gradient descent (GD), it always finds better solutions than GD for modern neural networks. In order to understand this phenomenon, we take an alternative view that SGD is working on the convolved (thus smoothed) version of the loss function. We show that, even if the function $f$ has many bad local minima or saddle points, as long as for every point $x$, the weighted average of the gradients of its neighborhoods is one point convex with respect to the desi…
University of Electronic Science and Technology of China（电子科技大学）
Most density-based clustering methods largely rely on how well the underlying density is estimated. However, density estimation itself is also a challenging problem, especially the determination of the kernel bandwidth. A large bandwidth could lead to the over-smoothed density estimation in which the number of density peaks could be less than the true clusters, while a small bandwidth could lead to the under-smoothed density estimation in which spurious density peaks, or called the "ripple noise", would be generated in the estimated density. In this paper, we propose a density-based hierarchic…
Samara University（萨马拉大学）；Institute of Applied Ecology, Oikon Ltd.（应用生态学研究所，Oikon有限公司）
Current expectations from training deep learning models with gradient-based methods include: 1) transparency; 2) high convergence rates; 3) high inductive biases. While the state-of-art methods with adaptive learning rate schedules are fast, they still fail to meet the other two requirements. We suggest reconsidering neural network models in terms of single-species population dynamics where adaptation comes naturally from open-ended processes of "growth" and "harvesting". We show that the stochastic gradient descent (SGD) with two balanced pre-defined values of per capita growth and harvesting…
Boise State University（博伊西州立大学）；University of Delaware（特拉华大学）
End user AI is trained on large server farms with data collected from the users. With ever increasing demand for IOT devices, there is a need for deep learning approaches that can be implemented (at the edge) in an energy efficient manner. In this work we approach this using spiking neural networks. The unsupervised learning technique of spike timing dependent plasticity (STDP) using binary activations are used to extract features from spiking input data. Gradient descent (backpropagation) is used only on the output layer to perform the training for classification. The accuracies obtained for …
City University of Macau（澳门城市大学）；National University of Defense Technology（国防科技大学）；China University of Geosciences（中国地质大学）；Zhejiang Normal University（浙江师范大学）；Research Institute of Ningbo Cixing Co. Ltd（宁波慈星股份有限公司研究院）；Qilu University of Technology (Shandong Academy of Sciences)（齐鲁工业大学（山东省科学院））；Shandong Computer Science Center (National Supercomputer Center in Jinan)（山东省计算中心（国家超级计算济南中心））；Shandong Fundamental Research Center for Computer Science（山东省计算机科学基础研究中心）
Distributionally robust optimization (DRO) is a widely-used approach to learn models that are robust against distribution shift. Compared with the standard optimization setting, the objective function in DRO is more difficult to optimize, and most of the existing theoretical results make strong assumptions on the loss function. In this work we bridge the gap by studying DRO algorithms for general smooth non-convex losses. By carefully exploiting the specific form of the DRO objective, we are able to provide non-asymptotic convergence guarantees even though the objective function is possibly no…
Carnegie Mellon University（卡内基梅隆大学）；Princeton University（普林斯顿大学）；University of Southern California（南加州大学）
We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step size) effectively enforces the differences between squared norms across different layers to remain invariant without any explicit regularization. This result implies that if the weights are initially small, gradient flow automatically balances the magnitudes of all layers. Using …
University of Montreal（蒙特利尔大学）；University of Oxford（牛津大学）；Politecnico di Milano（米兰理工大学）
The optimization of deep neural networks can be more challenging than traditional convex optimization problems due to the highly non-convex nature of the loss function, e.g. it can involve pathological landscapes such as saddle-surfaces that can be difficult to escape for algorithms based on simple gradient descent. In this paper, we attack the problem of optimization of highly non-convex neural networks by starting with a smoothed -- or \textit{mollified} -- objective function that gradually has a more non-convex energy landscape during the training. Our proposition is inspired by the recent …
Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced $k$-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to the original motivation of minimizing the normalized cut. In this paper we propose a new tight continuous relaxation for any balanced $k$-cut problem and show that a related recently proposed relaxation is in most cases loose leading to poor performance in practice. For the optimiz…
