Stochastic Gradient Descent (SGD) methods are prominent for training machine learning and deep learning models. The performance of these techniques depends on their hyperparameter tuning over time and varies for different models and problems. Manual adjustment of hyperparameters is very costly and time-consuming, and even if done correctly, it lacks theoretical justification which inevitably leads to "rule of thumb" settings. In this paper, we propose a generic approach that utilizes the statistics of an unbiased gradient estimator to automatically and simultaneously adjust two paramount hyperparameters: the learning rate and momentum. We deploy the proposed general technique for various SGD methods to train Convolutional Neural Networks (CNN's). The results match the performance of the best settings obtained through an exhaustive search and therefore, removes the need for a tedious manual tuning.
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.
Distributed descent-based methods are an essential toolset to solving optimization problems in multi-agent system scenarios. Here the agents seek to optimize a global objective function through mutual cooperation. Oftentimes, cooperation is achieved over a wireless communication network that is prone to delays and errors. There are many scenarios wherein the objective function is either non-differentiable or merely observable. In this paper, we present a cross-entropy based distributed stochastic approximation algorithm (SA) that finds a minimum of the objective, using only samples. We call th…
University of Toronto（多伦多大学）；Princeton University（普林斯顿大学）；Northeastern University（东北大学）
Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice, the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart~(Sim-GDA) in many settings. We prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a …
Nanyang Technological University（南洋理工大学）；Singapore Institute of Manufacturing Technology（新加坡制造技术研究院）
Multi-label learning studies the problem where an instance is associated with a set of labels. By treating single-label learning problem as one task, the multi-label learning problem can be casted as solving multiple related tasks simultaneously. In this paper, we propose a novel Multi-task Gradient Descent (MGD) algorithm to solve a group of related tasks simultaneously. In the proposed algorithm, each task minimizes its individual cost function using reformative gradient descent, where the relations among the tasks are facilitated through effectively transferring model parameter values acros…
Evading adversarial example detection defenses requires finding adversarial examples that must simultaneously (a) be misclassified by the model and (b) be detected as non-adversarial. We find that existing attacks that attempt to satisfy multiple simultaneous constraints often over-optimize against one constraint at the cost of satisfying another. We introduce Orthogonal Projected Gradient Descent, an improved attack technique to generate adversarial examples that avoids this problem by orthogonalizing the gradients when running standard gradient-based attacks. We use our technique to evade fo…
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）
California Institute of Technology（加州理工学院）；Georgia Institute of Technology（佐治亚理工学院）；University of California San Diego（加利福尼亚大学圣迭戈分校）；Nvidia（英伟达）
Many economic games and machine learning approaches can be cast as competitive optimization problems where multiple agents are minimizing their respective objective function, which depends on all agents' actions. While gradient descent is a reliable basic workhorse for single-agent optimization, it often leads to oscillation in competitive optimization. In this work we propose polymatrix competitive gradient descent (PCGD) as a method for solving general sum competitive optimization involving arbitrary numbers of agents. The updates of our method are obtained as the Nash equilibria of a local …
Brigham Young University–Idaho（爱达荷州杨百翰大学）；Purdue University（普渡大学）
Spiking Neural Networks (SNNs) are being explored for their potential energy efficiency benefits due to sparse, event-driven computation. Non-spiking artificial neural networks are typically trained with stochastic gradient descent using backpropagation. The calculation of true gradients for backpropagation in spiking neural networks is impeded by the non-differentiable firing events of spiking neurons. On the other hand, using approximate gradients is effective, but computationally expensive over many time steps. One common technique, then, for training a spiking neural network is to train a …
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
Stochastic gradient descent algorithms for training linear and kernel predictors are gaining more and more importance, thanks to their scalability. While various methods have been proposed to speed up their convergence, the model selection phase is often ignored. In fact, in theoretical works most of the time assumptions are made, for example, on the prior knowledge of the norm of the optimal solution, while in the practical world validation methods remain the only viable approach. In this paper, we propose a new kernel-based stochastic gradient descent algorithm that performs model selection …
Stanford University（斯坦福大学）；Massachusetts Institute of Technology（麻省理工学院）；Boston University（波士顿大学）
Double descent is a surprising phenomenon in machine learning, in which as the number of model parameters grows relative to the number of data, test error drops as models grow ever larger into the highly overparameterized (data undersampled) regime. This drop in test error flies against classical learning theory on overfitting and has arguably underpinned the success of large models in machine learning. This non-monotonic behavior of test loss depends on the number of data, the dimensionality of the data and the number of model parameters. Here, we briefly describe double descent, then provide…
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…
An open question in the Deep Learning community is why neural networks trained with Gradient Descent generalize well on real datasets even though they are capable of fitting random data. We propose an approach to answering this question based on a hypothesis about the dynamics of gradient descent that we call Coherent Gradients: Gradients from similar examples are similar and so the overall gradient is stronger in certain directions where these reinforce each other. Thus changes to the network parameters during training are biased towards those that (locally) simultaneously benefit many exampl…
There are much recent interests in solving noncovnex min-max optimization problems due to its broad applications in many areas including machine learning, networked resource allocations, and distributed optimization. Perhaps, the most popular first-order method in solving min-max optimization is the so-called simultaneous (or single-loop) gradient descent-ascent algorithm due to its simplicity in implementation. However, theoretical guarantees on the convergence of this algorithm is very sparse since it can diverge even in a simple bilinear problem. In this paper, our focus is to characterize …
Despite the growing prominence of generative adversarial networks (GANs), optimization in GANs is still a poorly understood topic. In this paper, we analyze the "gradient descent" form of GAN optimization i.e., the natural setting where we simultaneously take small gradient steps in both generator and discriminator parameters. We show that even though GAN optimization does not correspond to a convex-concave game (even for simple parameterizations), under proper conditions, equilibrium points of this optimization procedure are still \emph{locally asymptotically stable} for the traditional GAN f…
The ability of neural networks to provide `best in class' approximation across a wide range of applications is well-documented. Nevertheless, the powerful expressivity of neural networks comes to naught if one is unable to effectively train (choose) the parameters defining the network. In general, neural networks are trained by gradient descent type optimization methods, or a stochastic variant thereof. In practice, such methods result in the loss function decreases rapidly at the beginning of training but then, after a relatively small number of steps, significantly slow down. The loss may ev…
We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under…
Imperial College London（帝国理工学院）；University of Oxford（牛津大学）
Stein variational gradient descent (SVGD) is a deterministic particle inference algorithm that provides an efficient alternative to Markov chain Monte Carlo. However, SVGD has been found to suffer from variance underestimation when the dimensionality of the target distribution is high. Recent developments have advocated projecting both the score function and the data onto real lines to sidestep this issue, although this can severely overestimate the epistemic (model) uncertainty. In this work, we propose Grassmann Stein variational gradient descent (GSVGD) as an alternative approach, which per…
Neural networks are achieving state of the art and sometimes super-human performance on learning tasks across a variety of domains. Whenever these problems require learning in a continual or sequential manner, however, neural networks suffer from the problem of catastrophic forgetting; they forget how to solve previous tasks after being trained on a new task, despite having the essential capacity to solve both tasks if they were trained on both simultaneously. In this paper, we propose to address this issue from a parameter space perspective and study an approach to restrict the direction of t…
Yale University（耶鲁大学）；Georgia Institute of Technology（佐治亚理工学院）；Singapore University of Technology and Design（新加坡科技设计大学）
In this paper we study two-player bilinear zero-sum games with constrained strategy spaces. An instance of natural occurrences of such constraints is when mixed strategies are used, which correspond to a probability simplex constraint. We propose and analyze the alternating mirror descent algorithm, in which each player takes turns to take action following the mirror descent algorithm for constrained optimization. We interpret alternating mirror descent as an alternating discretization of a skew-gradient flow in the dual space, and use tools from convex optimization and modified energy functio…
Ant Group（蚂蚁集团）；Shanghai Jiao Tong University（上海交通大学）
Multi-task learning (MTL) aims at solving multiple related tasks simultaneously and has experienced rapid growth in recent years. However, MTL models often suffer from performance degeneration with negative transfer due to learning several tasks simultaneously. Some related work attributed the source of the problem is the conflicting gradients. In this case, it is needed to select useful gradient updates for all tasks carefully. To this end, we propose a novel optimization approach for MTL, named GDOD, which manipulates gradients of each task using an orthogonal basis decomposed from the span …
Asynchronous parallel implementations for stochastic optimization have received huge successes in theory and practice recently. Asynchronous implementations with lock-free are more efficient than the one with writing or reading lock. In this paper, we focus on a composite objective function consisting of a smooth convex function $f$ and a block separable convex function, which widely exists in machine learning and computer vision. We propose an asynchronous stochastic block coordinate descent algorithm with the accelerated technology of variance reduction (AsySBCDVR), which are with lock-free …
Mirror Descent-Ascent for mean-field min-max problems
Center for Advanced Intelligence Project, RIKEN（日本东京RIKEN先进智能项目中心）；School of Mathematical and Computer Sciences, Heriot-Watt University（赫里奥特-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学研究所）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）；The Alan Turing Institute（艾伦·图灵研究所）
The optimization with orthogonality has been shown useful in training deep neural networks (DNNs). To impose orthogonality on DNNs, both computational efficiency and stability are important. However, existing methods utilizing Riemannian optimization or hard constraints can only ensure stability while those using soft constraints can only improve efficiency. In this paper, we propose a novel method, named Feedback Gradient Descent (FGD), to our knowledge, the first work showing high efficiency and stability simultaneously. FGD induces orthogonality based on the simple yet indispensable Euler d…
Massachusetts Institute of Technology（麻省理工学院）；California Institute of Technology（加州理工学院）
Despite perfectly interpolating the training data, deep neural networks (DNNs) can often generalize fairly well, in part due to the "implicit regularization" induced by the learning algorithm. Nonetheless, various forms of regularization, such as "explicit regularization" (via weight decay), are often used to avoid overfitting, especially when the data is corrupted. There are several challenges with explicit regularization, most notably unclear convergence properties. Inspired by convergence properties of stochastic mirror descent (SMD) algorithms, we propose a new method for training DNNs wit…
Recommender systems need to mirror the complexity of the environment they are applied in. The more we know about what might benefit the user, the more objectives the recommender system has. In addition there may be multiple stakeholders - sellers, buyers, shareholders - in addition to legal and ethical constraints. Simultaneously optimizing for a multitude of objectives, correlated and not correlated, having the same scale or not, has proven difficult so far. We introduce a stochastic multi-gradient descent approach to recommender systems (MGDRec) to solve this problem. We show that this excee…
Institute of Evaluation and Assessment Research, Academy of Military Science（军事科学院评估论证研究中心）；Command and Control Engineering College, Army Engineering University（陆军工程大学指挥控制工程学院）；Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）
Averaging scheme has attracted extensive attention in deep learning as well as traditional machine learning. It achieves theoretically optimal convergence and also improves the empirical model performance. However, there is still a lack of sufficient convergence analysis for strongly convex optimization. Typically, the convergence about the last iterate of gradient descent methods, which is referred to as individual convergence, fails to attain its optimality due to the existence of logarithmic factor. In order to remove this factor, we first develop gradient descent averaging (GDA), which is …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Department of Mathematics（数学系）；Madison, WI 53706（威斯康星州麦迪逊市 53706（注：该地址非机构，此处仅按要求提取原文中明确出现的机构相关内容））
Sampling from a log-concave distribution function is one core problem that has wide applications in Bayesian statistics and machine learning. While most gradient free methods have slow convergence rate, the Langevin Monte Carlo (LMC) that provides fast convergence requires the computation of gradients. In practice one uses finite-differencing approximations as surrogates, and the method is expensive in high-dimensions. A natural strategy to reduce computational cost in each iteration is to utilize random gradient approximations, such as random coordinate descent (RCD) or simultaneous perturbat…
University of Oxford（牛津大学）；DeepMind（深度思考（DeepMind））
Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new tools to understand and control the dynamics in n-player differentiable games. The key result is to decompose the game Jacobian into two com…
Rakuten Institute of Technology（乐天技术研究所）；University of California, Merced（加州大学默塞德分校）；University of Utah（犹他大学）
We propose and analyze a block coordinate descent proximal algorithm (BCD-prox) for simultaneous filtering and parameter estimation of ODE models. As we show on ODE systems with up to d=40 dimensions, as compared to state-of-the-art methods, BCD-prox exhibits increased robustness (to noise, parameter initialization, and hyperparameters), decreased training times, and improved accuracy of both filtered states and estimated parameters. We show how BCD-prox can be used with multistep numerical discretizations, and we establish convergence of BCD-prox under hypotheses that include real systems of …
Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent
Department of Statistics and Data Science and Department of Computer Science, University of Central Florida（统计与数据科学系和计算机科学系，中央佛罗里达大学）；Department of Computational Mathematics, Science and Engineering and Department of Mathematics, Michigan State University（计算数学、科学与工程系和数学系，密歇根州立大学）；School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan University（数学与统计学系，和湖北计算科学重点实验室，武汉大学）；School of Artificial Intelligence, Wuhan University, National Center for Applied Mathematics in Hubei, and Hubei Key Laboratory of Computational Science, Wuhan, China（人工智能学院，武汉大学，湖北应用数学中心，和湖北计算科学重点实验室，中国武汉）；Institute for Advanced Study, Shenzhen University（先进研究学院，深圳大学）
The cornerstone underpinning deep learning is the guarantee that gradient descent on an objective converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, where there are multiple interacting losses. The behavior of gradient-based methods in games is not well understood -- and is becoming increasingly important as adversarial and multi-objective architectures proliferate. In this paper, we develop new techniques to understand and control the dynamics in general games. The key result is to decompose the second-order dynamics into two compo…
Neural networks are usually trained by some form of stochastic gradient descent (SGD)). A number of strategies are in common use intended to improve SGD optimization, such as learning rate schedules, momentum, and batching. These are motivated by ideas about the occurrence of local minima at different scales, valleys, and other phenomena in the objective function. Empirical results presented here suggest that these phenomena are not significant factors in SGD optimization of MLP-related objective functions, and that the behavior of stochastic gradient descent in these problems is better descri…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；MIT（麻省理工学院）
We develop Temporal Quantile Adjustment (TQA), a general method to construct efficient and valid prediction intervals (PIs) for regression on cross-sectional time series data. Such data is common in many domains, including econometrics and healthcare. A canonical example in healthcare is predicting patient outcomes using physiological time-series data, where a population of patients composes a cross-section. Reliable PI estimators in this setting must address two distinct notions of coverage: cross-sectional coverage across a cross-sectional slice, and longitudinal coverage along the temporal …
We present a unified framework based on primal-dual stochastic mirror descent for approximately solving infinite-horizon Markov decision processes (MDPs) given a generative model. When applied to an average-reward MDP with $A_{tot}$ total state-action pairs and mixing time bound $t_{mix}$ our method computes an $ε$-optimal policy with an expected $\widetilde{O}(t_{mix}^2 A_{tot} ε^{-2})$ samples from the state-transition matrix, removing the ergodicity dependence of prior art. When applied to a $γ$-discounted MDP with $A_{tot}$ total state-action pairs our method computes an $ε$-optimal policy…
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）
An increasing number of data science and machine learning problems rely on computation with tensors, which better capture the multi-way relationships and interactions of data than matrices. When tapping into this critical advantage, a key challenge is to develop computationally efficient and provably correct algorithms for extracting useful information from tensor data that are simultaneously robust to corruptions and ill-conditioning. This paper tackles tensor robust principal component analysis (RPCA), which aims to recover a low-rank tensor from its observations contaminated by sparse corru…
We study the problem of recovery of matrices that are simultaneously low rank and row and/or column sparse. Such matrices appear in recent applications in cognitive neuroscience, imaging, computer vision, macroeconomics, and genetics. We propose a GDT (Gradient Descent with hard Thresholding) algorithm to efficiently recover matrices with such structure, by minimizing a bi-convex function over a nonconvex set of constraints. We show linear convergence of the iterates obtained by GDT to a region within statistical error of an optimal solution. As an application of our method, we consider multi-…
