This work studies constrained stochastic optimization problems where the objective and constraint functions are convex and expressed as compositions of stochastic functions. The problem arises in the context of fair classification, fair regression, and the design of queuing systems. Of particular interest is the large-scale setting where an oracle provides the stochastic gradients of the constituent functions, and the goal is to solve the problem with a minimal number of calls to the oracle. Owing to the compositional form, the stochastic gradients provided by the oracle do not yield unbiased estimates of the objective or constraint gradients. Instead, we construct approximate gradients by tracking the inner function evaluations, resulting in a quasi-gradient saddle point algorithm. We prove that the proposed algorithm is guaranteed to find the optimal and feasible solution almost surely. We further establish that the proposed algorithm requires $\mathcal{O}(1/ε^4)$ data samples in order to obtain an $ε$-approximate optimal point while also ensuring zero constraint violation. The result matches the sample complexity of the stochastic compositional gradient descent method for unconstrained problems and improves upon the best-known sample complexity results for the constrained settings. The efficacy of the proposed algorithm is tested on both fair classification and fair regression problems. The numerical results show that the proposed algorithm outperforms the state-of-the-art algorithms in terms of the convergence rate.
Changchun University of Science and Technology（长春理工大学）；Northeast Normal University（东北师范大学）
The plain stochastic gradient descent and momentum stochastic gradient descent have extremely wide applications in deep learning due to their simple settings and low computational complexity. The momentum stochastic gradient descent uses the accumulated gradient as the updated direction of the current parameters, which has a faster training speed. Because the direction of the plain stochastic gradient descent has not been corrected by the accumulated gradient. For the parameters that currently need to be updated, it is the optimal direction, and its update is more accurate. We combine the adva…
The Pennsylvania State University（宾夕法尼亚州立大学）；College of Information Science and Technology（信息科学与技术学院）；Department of Computer Science and Engineering（计算机科学与工程系）
Many machine learning, statistical inference, and portfolio optimization problems require minimization of a composition of expected value functions (CEVF). Of particular interest is the finite-sum versions of such compositional optimization problems (FS-CEVF). Compositional stochastic variance reduced gradient (C-SVRG) methods that combine stochastic compositional gradient descent (SCGD) and stochastic variance reduced gradient descent (SVRG) methods are the state-of-the-art methods for FS-CEVF problems. We introduce compositional stochastic average gradient descent (C-SAG) a novel extension o…
Insight Centre for Data Analytics, NUI Galway（爱尔兰国立大学高威数据分析中心）；Aylien Ltd.（Aylien有限公司）
Gradient descent optimization algorithms, while increasingly popular, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. This article aims to provide the reader with intuitions with regard to the behaviour of different algorithms that will allow her to put them to use. In the course of this overview, we look at different variants of gradient descent, summarize challenges, introduce the most common optimization algorithms, review architectures in a parallel and distributed setting, and investigate additional strategies for op…
There introduce Particle Optimized Gradient Descent (POGD), an algorithm based on the gradient descent but integrates the particle swarm optimization (PSO) principle to achieve the iteration. From the experiments, this algorithm has adaptive learning ability. The experiments in this paper mainly focus on the training speed to reach the target value and the ability to prevent the local minimum. The experiments in this paper are achieved by the convolutional neural network (CNN) image classification on the MNIST and cifar-10 datasets.
This paper addresses the issue of policy evaluation in Markov Decision Processes, using linear function approximation. It provides a unified view of algorithms such as TD(lambda), LSTD(lambda), iLSTD, residual-gradient TD. It is asserted that they all consist in minimizing a gradient function and differ by the form of this function and their means of minimizing it. Two new schemes are introduced in that framework: Full-gradient TD which uses a generalization of the principle introduced in iLSTD, and EGD TD, which reduces the gradient by successive equi-gradient descents. These three algorithms…
The backpropagation algorithm, which had been originally introduced in the 1970s, is the workhorse of learning in neural networks. This backpropagation algorithm makes use of the famous machine learning algorithm known as Gradient Descent, which is a first-order iterative optimization algorithm for finding the minimum of a function. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. In this paper, we develop an alternative to the backpropagation without…
The purpose of this paper is to improve upon existing variants of gradient descent by solving two problems: (1) removing (or reducing) the plateau that occurs while minimizing the cost function, (2) continually adjusting the learning rate to an "ideal" value. The approach taken is to approximately solve for the learning rate as a function of a trust metric. When this technique is hybridized with momentum, it creates an especially effective gradient descent variant, called NeogradM. It is shown to outperform Adam on several test problems, and can easily reach cost function values that are small…
Compositionality is a basic structural feature of both biological and artificial neural networks. Learning compositional functions via gradient descent incurs well known problems like vanishing and exploding gradients, making careful learning rate tuning essential for real-world applications. This paper proves that multiplicative weight updates satisfy a descent lemma tailored to compositional functions. Based on this lemma, we derive Madam -- a multiplicative version of the Adam optimiser -- and show that it can train state of the art neural network architectures without learning rate tuning.…
Facebook Artificial Intelligence Research（脸书人工智能研究院）；Istituto Italiano di Tecnologia（意大利理工学院）
Neural networks are very powerful learning systems, but they do not readily generalize from one task to the other. This is partly due to the fact that they do not learn in a compositional way, that is, by discovering skills that are shared by different tasks, and recombining them to solve new problems. In this paper, we explore the compositional generalization capabilities of recurrent neural networks (RNNs). We first propose the lookup table composition domain as a simple setup to test compositional behaviour and show that it is theoretically possible for a standard RNN to learn to behave com…
We describe an alternative learning method for neural networks, which we call Blind Descent. By design, Blind Descent does not face problems like exploding or vanishing gradients. In Blind Descent, gradients are not used to guide the learning process. In this paper, we present Blind Descent as a more fundamental learning process compared to gradient descent. We also show that gradient descent can be seen as a specific case of the Blind Descent algorithm. We also train two neural network architectures, a multilayer perceptron and a convolutional neural network, using the most general Blind Desc…
In high dimensions, most machine learning method perform fragile even there are a little outliers. To address this, we hope to introduce a new method with the base learner, such as Bayesian regression or stochastic gradient descent to solve the problem of the vulnerability in the model. Because the mini-batch gradient descent allows for a more robust convergence than the batch gradient descent, we work a method with the mini-batch gradient descent, called Mini-Batch Gradient Descent with Trimming (MBGDT). Our method show state-of-art performance and have greater robustness than several baselin…
Convex composition optimization is an emerging topic that covers a wide range of applications arising from stochastic optimal control, reinforcement learning and multi-stage stochastic programming. Existing algorithms suffer from unsatisfactory sample complexity and practical issues since they ignore the convexity structure in the algorithmic design. In this paper, we develop a new stochastic compositional variance-reduced gradient algorithm with the sample complexity of $O((m+n)\log(1/ε)+1/ε^3)$ where $m+n$ is the total number of samples. Our algorithm is near-optimal as the dependence on $m+…
As part of the effort to understand implicit bias of gradient descent in overparametrized models, several results have shown how the training trajectory on the overparametrized model can be understood as mirror descent on a different objective. The main result here is a characterization of this phenomenon under a notion termed commuting parametrization, which encompasses all the previous results in this setting. It is shown that gradient flow with any commuting parametrization is equivalent to continuous mirror descent with a related Legendre function. Conversely, continuous mirror descent wit…
Most of the recent successful applications of neural networks have been based on training with gradient descent updates. However, for some small networks, other mirror descent updates learn provably more efficiently when the target is sparse. We present a general framework for casting a mirror descent update as a gradient descent update on a different set of parameters. In some cases, the mirror descent reparameterization can be described as training a modified network with standard backpropagation. The reparameterization framework is versatile and covers a wide range of mirror descent updates…
We learn recurrent neural network optimizers trained on simple synthetic functions by gradient descent. We show that these learned optimizers exhibit a remarkable degree of transfer in that they can be used to efficiently optimize a broad range of derivative-free black-box functions, including Gaussian process bandits, simple control objectives, global optimization benchmarks and hyper-parameter tuning tasks. Up to the training horizon, the learned optimizers learn to trade-off exploration and exploitation, and compare favourably with heavily engineered Bayesian optimization packages for hyper…
Non-convex optimization problems are challenging to solve; the success and computational expense of a gradient descent algorithm or variant depend heavily on the initialization strategy. Often, either random initialization is used or initialization rules are carefully designed by exploiting the nature of the problem class. As a simple alternative to hand-crafted initialization rules, we propose an approach for learning "good" initialization rules from previous solutions. We provide theoretical guarantees that establish conditions that are sufficient in all cases and also necessary in some unde…
We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on the step sizes gradient descent converges to a critical point of the loss function, i.e., the square loss in this article. Furthermore, we demonstrate that for almost all initializations gradient descent converges to a global minimum in the case of two layers. In the case of three or more layers we show that gradient descent converges to a global minimum on th…
TTI at Chicago（芝加哥丰田理工学院）；USC（南加利福尼亚大学）；Technion（以色列理工学院）
We show that gradient descent on full-width linear convolutional networks of depth $L$ converges to a linear predictor related to the $\ell_{2/L}$ bridge penalty in the frequency domain. This is in contrast to linearly fully connected networks, where gradient descent converges to the hard margin linear support vector machine solution, regardless of depth.
Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
Stochastic compositional optimization generalizes classic (non-compositional) stochastic optimization to the minimization of compositions of functions. Each composition may introduce an additional expectation. The series of expectations may be nested. Stochastic compositional optimization is gaining popularity in applications such as reinforcement learning and meta learning. This paper presents a new Stochastically Corrected Stochastic Compositional gradient method (SCSC). SCSC runs in a single-time scale with a single loop, uses a fixed batch size, and guarantees to converge at the same rate …
In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate $O\left(\frac{1}{\varepsilon^2}\right)$ improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of Hazan and Kale [2012] with convergence rate $O\left(\frac{1}{\varepsilon^4}\right)$.
One of the most important parts of Artificial Neural Networks is minimizing the loss functions which tells us how good or bad our model is. To minimize these losses we need to tune the weights and biases. Also to calculate the minimum value of a function we need gradient. And to update our weights we need gradient descent. But there are some problems with regular gradient descent ie. it is quite slow and not that accurate. This article aims to give an introduction to optimization strategies to gradient descent. In addition, we shall also discuss the architecture of these algorithms and further…
Department of Computer Science, Stanford University（斯坦福大学计算机科学系）；Department of Computer Science, Columbia University（哥伦比亚大学计算机科学系）；Google Research, Mountain View（谷歌研究院（山景城））
We prove that the evolution of weight vectors in online gradient descent can encode arbitrary polynomial-space computations, even in very simple learning settings. Our results imply that, under weak complexity-theoretic assumptions, it is impossible to reason efficiently about the fine-grained behavior of online gradient descent.
Sorbonne Université（索邦大学）；Institut des Systèmes Intelligents et de Robotique（智能与机器人系统研究院）
In this paper, we provide an overview of first-order and second-order variants of the gradient descent method that are commonly used in machine learning. We propose a general framework in which 6 of these variants can be interpreted as different instances of the same approach. They are the vanilla gradient descent, the classical and generalized Gauss-Newton methods, the natural gradient descent method, the gradient covariance matrix approach, and Newton's method. Besides interpreting these methods within a single framework, we explain their specificities and show under which conditions some of…
In this note, we observe the behavior of gradient flow and discrete and noisy gradient descent in some simple settings. It is commonly noted that addition of noise to gradient descent can affect the trajectory of gradient descent. Here, we run some computer experiments for gradient descent on some simple functions, and observe this principle in some concrete examples.
With a weighting scheme proportional to t, a traditional stochastic gradient descent (SGD) algorithm achieves a high probability convergence rate of O(κ/T) for strongly convex functions, instead of O(κ ln(T)/T). We also prove that an accelerated SGD algorithm also achieves a rate of O(κ/T).
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Gradient descent is a popular algorithm in optimization, and its performance in convex settings is mostly well understood. In non-convex settings, it has been shown that gradient descent is able to escape saddle points asymptotically and converge to local minimizers [Lee et. al. 2016]. Recent studies also show a perturbed version of gradient descent is enough to escape saddle points efficiently [Jin et. al. 2015, Ge et. al. 2017]. In this paper we show a negative result: gradient descent may take exponential time to escape saddle points, with non-pathological two dimensional functions. While o…
We consider the maximum mean discrepancy ($\mathrm{MMD}$) GAN problem and propose a parametric kernelized gradient flow that mimics the min-max game in gradient regularized $\mathrm{MMD}$ GAN. We show that this flow provides a descent direction minimizing the $\mathrm{MMD}$ on a statistical manifold of probability distributions. We then derive an explicit condition which ensures that gradient descent on the parameter space of the generator in gradient regularized $\mathrm{MMD}$ GAN is globally convergent to the target distribution. Under this condition, we give non asymptotic convergence resul…
University of North Carolina - Chapel Hill（北卡罗来纳大学教堂山分校）；Binghamton University（宾汉姆顿大学）；Eastern Illinois University（东伊利诺伊大学）
Gradient descent algorithms have been used in countless applications since the inception of Newton's method. The explosion in the number of applications of neural networks has re-energized efforts in recent years to improve the standard gradient descent method in both efficiency and accuracy. These methods modify the effect of the gradient in updating the values of the parameters. These modifications often incorporate hyperparameters: additional variables whose values must be specified at the outset of the program. We provide, below, a novel gradient descent algorithm, called Gravilon, that us…
University of California at Los Angeles（加州大学洛杉矶分校）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
Gradient descent is a simple and widely used optimization method for machine learning. For homogeneous linear classifiers applied to separable data, gradient descent has been shown to converge to the maximal margin (or equivalently, the minimal norm) solution for various smooth loss functions. The previous theory does not, however, apply to non-smooth functions such as the hinge loss which is widely used in practice. Here, we study the convergence of a homotopic variant of gradient descent applied to the hinge loss and provide explicit convergence rates to the max-margin solution for linearly …
We present a unifying framework for adapting the update direction in gradient-based iterative optimization methods. As natural special cases we re-derive classical momentum and Nesterov's accelerated gradient method, lending a new intuitive interpretation to the latter algorithm. We show that a new algorithm, which we term Regularised Gradient Descent, can converge more quickly than either Nesterov's algorithm or the classical momentum algorithm.
Natural gradient descent (NGD) provided deep insights and powerful tools to deep neural networks. However the computation of Fisher information matrix becomes more and more difficult as the network structure turns large and complex. This paper proposes a new optimization method whose main idea is to accurately replace the natural gradient optimization by reconstructing the network. More specifically, we reconstruct the structure of the deep neural network, and optimize the new network using traditional gradient descent (GD). The reconstructed network achieves the effect of the optimization way…
Google DeepMind（谷歌DeepMind）；University of Oxford（牛津大学）；Canadian Institute for Advanced Research（加拿大高等研究院）
The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm can be cast as a learning problem, allowing the algorithm to learn to exploit structure in the problems of interest in an automatic way. Our learned algorithms, implemented by LSTMs, outperform generic, hand-designed competitors on the tasks for which they are trained, and also generalize well to new tasks with similar structure. We demonstrate this on a nu…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Nevada（内华达大学）
Large-scale optimization problems require algorithms both effective and efficient. One such popular and proven algorithm is Stochastic Gradient Descent which uses first-order gradient information to solve these problems. This paper studies almost-sure convergence rates of the Stochastic Gradient Descent method when instead of deterministic, its learning rate becomes stochastic. In particular, its learning rate is equipped with a multiplicative stochasticity, producing a stochastic learning rate scheme. Theoretical results show accelerated almost-sure convergence rates of Stochastic Gradient De…
Convergence of the gradient descent algorithm has been attracting renewed interest due to its utility in deep learning applications. Even as multiple variants of gradient descent were proposed, the assumption that the gradient of the objective is Lipschitz continuous remained an integral part of the analysis until recently. In this work, we look at convergence analysis by focusing on a property that we term as concavifiability, instead of Lipschitz continuity of gradients. We show that concavifiability is a necessary and sufficient condition to satisfy the upper quadratic approximation which i…
This paper studies the performance of a recently proposed preconditioned stochastic gradient descent (PSGD) algorithm on recurrent neural network (RNN) training. PSGD adaptively estimates a preconditioner to accelerate gradient descent, and is designed to be simple, general and easy to use, as stochastic gradient descent (SGD). RNNs, especially the ones requiring extremely long term memories, are difficult to train. We have tested PSGD on a set of synthetic pathological RNN learning problems and the real world MNIST handwritten digit recognition task. Experimental results suggest that PSGD is …
Learning from the Descent Direction: Adaptive Gradient Descent under One-Sided Hölder Regularity
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机研究所）；Physikalisch-Technische Bundesanstalt (PTB)（德国联邦物理技术研究院）；Technical University of Berlin（柏林工业大学）
In this work, we introduce an asynchronous decentralized accelerated stochastic gradient descent type of method for decentralized stochastic optimization, considering communication and synchronization are the major bottlenecks. We establish $\mathcal{O}(1/ε)$ (resp., $\mathcal{O}(1/\sqrtε)$) communication complexity and $\mathcal{O}(1/ε^2)$ (resp., $\mathcal{O}(1/ε)$) sampling complexity for solving general convex (resp., strongly convex) problems.
We consider the problem of empirical risk minimization given a database, using the gradient descent algorithm. We note that the function to be optimized may be non-convex, consisting of saddle points which impede the convergence of the algorithm. A perturbed gradient descent algorithm is typically employed to escape these saddle points. We show that this algorithm, that perturbs the gradient, inherently preserves the privacy of the data. We then employ the differential privacy framework to quantify the privacy hence achieved. We also analyze the change in privacy with varying parameters such a…
Particle gradient descent, which uses particles to represent a probability measure and performs gradient descent on particles in parallel, is widely used to optimize functions of probability measures. This paper considers particle gradient descent with a finite number of particles and establishes its theoretical guarantees to optimize functions that are \emph{displacement convex} in measures. Concretely, for Lipschitz displacement convex functions defined on probability over $\mathbb{R}^d$, we prove that $O(1/ε^2)$ particles and $O(d/ε^4)$ computations are sufficient to find the $ε$-optimal so…
We consider the behavior of gradient flow and of discrete and noisy gradient descent. It is commonly noted that the addition of noise to the process of discrete gradient descent can affect the trajectory of gradient descent. In previous work, we observed such effects. There, we considered the case where the minima had codimension 1. In this note, we do some computer experiments and observe the behavior of noisy gradient descent in the more complex setting of minima of higher codimension.
University of Cambridge（剑桥大学）；Eötvös Loránd University（罗兰大学）
In gradient descent, changing how we parametrize the model can lead to drastically different optimization trajectories, giving rise to a surprising range of meaningful inductive biases: identifying sparse classifiers or reconstructing low-rank matrices without explicit regularization. This implicit regularization has been hypothesised to be a contributing factor to good generalization in deep learning. However, natural gradient descent is approximately invariant to reparameterization, it always follows the same trajectory and finds the same optimum. The question naturally arises: What happens …
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
We analyze the inductive bias of gradient descent for weight normalized smooth homogeneous neural nets, when trained on exponential or cross-entropy loss. We analyse both standard weight normalization (SWN) and exponential weight normalization (EWN), and show that the gradient flow path with EWN is equivalent to gradient flow on standard networks with an adaptive learning rate. We extend these results to gradient descent, and establish asymptotic relations between weights and gradients for both SWN and EWN. We also show that EWN causes weights to be updated in a way that prefers asymptotic rel…
Institute of Informatics, University of Warsaw（华沙大学信息学研究所）
We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of…
University of British Columbia（不列颠哥伦比亚大学）；Sony Computer Science Laboratories Inc.（索尼计算机科学实验室公司）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Alberta Machine Intelligence Institute（阿尔伯塔机器智能研究所）
In this paper, we propose new structured second-order methods and structured adaptive-gradient methods obtained by performing natural-gradient descent on structured parameter spaces. Natural-gradient descent is an attractive approach to design new algorithms in many settings such as gradient-free, adaptive-gradient, and second-order methods. Our structured methods not only enjoy a structural invariance but also admit a simple expression. Finally, we test the efficiency of our proposed methods on both deterministic non-convex problems and deep learning problems.
Back-propagation is a popular machine learning algorithm that uses gradient descent in training neural networks for supervised learning, but can be very slow. A number of algorithms have been developed to speed up convergence and improve robustness of the learning. However, they are complicated to implement biologically as they require information from previous updates. Inspired by synaptic competition in biology, we have come up with a simple and local gradient descent optimization algorithm that can reduce training time, with no demand on past details. Our algorithm, named dynamic learning r…
We show that gradient descent converges to a local minimizer, almost surely with random initialization. This is proved by applying the Stable Manifold Theorem from dynamical systems theory.
In this work, we propose an optimization algorithm which we call norm-adapted gradient descent. This algorithm is similar to other gradient-based optimization algorithms like Adam or Adagrad in that it adapts the learning rate of stochastic gradient descent at each iteration. However, rather than using statistical properties of observed gradients, norm-adapted gradient descent relies on a first-order estimate of the effect of a standard gradient descent update step, much like the Newton-Raphson method in many dimensions. Our algorithm can also be compared to quasi-Newton methods, but we seek r…
