We derive bounds on the path length $ζ$ of gradient descent (GD) and gradient flow (GF) curves for various classes of smooth convex and nonconvex functions. Among other results, we prove that: (a) if the iterates are linearly convergent with factor $(1-c)$, then $ζ$ is at most $\mathcal{O}(1/c)$; (b) under the Polyak-Kurdyka-Lojasiewicz (PKL) condition, $ζ$ is at most $\mathcal{O}(\sqrtκ)$, where $κ$ is the condition number, and at least $\widetildeΩ(\sqrt{d} \wedge κ^{1/4})$; (c) for quadratics, $ζ$ is $Θ(\min\{\sqrt{d},\sqrt{\log κ}\})$ and in some cases can be independent of $κ$; (d) assuming just convexity, $ζ$ can be at most $2^{4d\log d}$; (e) for separable quasiconvex functions, $ζ$ is $Θ(\sqrt{d})$. Thus, we advance current understanding of the properties of GD and GF curves beyond rates of convergence. We expect our techniques to facilitate future studies for other algorithms.
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both gradient descent and stochastic gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under mild assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by (stochastic) gra…
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.
Cracow University of Technology（克拉科夫理工大学）；Jagiellonian University（雅盖隆大学）
In this paper, we propose a simple, fast and easy to implement algorithm LOSSGRAD (locally optimal step-size in gradient descent), which automatically modifies the step-size in gradient descent during neural networks training. Given a function $f$, a point $x$, and the gradient $\nabla_x f$ of $f$, we aim to find the step-size $h$ which is (locally) optimal, i.e. satisfies: $$ h=arg\,min_{t \geq 0} f(x-t \nabla_x f). $$ Making use of quadratic approximation, we show that the algorithm satisfies the above assumption. We experimentally show that our method is insensitive to the choice of initial…
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.
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.
Information Fusion and Mining Laboratory（信息融合与挖掘实验室）
In this paper, we aim at providing an introduction to the gradient descent based optimization algorithms for learning deep neural network models. Deep learning models involving multiple nonlinear projection layers are very challenging to train. Nowadays, most of the deep learning model training still relies on the back propagation algorithm actually. In back propagation, the model variables will be updated iteratively until convergence with gradient descent based optimization algorithms. Besides the conventional vanilla gradient descent algorithm, many gradient descent variants have also been …
We introduce Kalman Gradient Descent, a stochastic optimization algorithm that uses Kalman filtering to adaptively reduce gradient variance in stochastic gradient descent by filtering the gradient estimates. We present both a theoretical analysis of convergence in a non-convex setting and experimental results which demonstrate improved performance on a variety of machine learning areas including neural networks and black box variational inference. We also present a distributed version of our algorithm that enables large-dimensional optimization, and we extend our algorithm to SGD with momentum…
Weizmann Institute of Science（魏茨曼科学研究所）；TTI Chicago（芝加哥丰田工业大学）
We provide tight finite-time convergence bounds for gradient descent and stochastic gradient descent on quadratic functions, when the gradients are delayed and reflect iterates from $τ$ rounds ago. First, we show that without stochastic noise, delays strongly affect the attainable optimization error: In fact, the error can be as bad as non-delayed gradient descent ran on only $1/τ$ of the gradients. In sharp contrast, we quantify how stochastic noise makes the effect of delays negligible, improving on previous work which only showed this phenomenon asymptotically or for much smaller delays. Al…
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.
Gradient descent and its variants are widely used in machine learning. However, oracle access of gradient may not be available in many applications, limiting the direct use of gradient descent. This paper proposes a method of estimating gradient to perform gradient descent, that converges to a stationary point for general non-convex optimization problems. Beyond the first-order stationary properties, the second-order stationary properties are important in machine learning applications to achieve better performance. We show that the proposed model-free non-convex optimization algorithm returns …
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…
University of California, Berkeley（加州大学伯克利分校）；Microsoft Research, India（微软印度研究院）
Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in $\tilde{O}(1/ε^{7/4})$ iterations, faster than the $\tilde{O}(1/ε^{2})$ iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second…
Inspired by dynamic programming, we propose Stochastic Virtual Gradient Descent (SVGD) algorithm where the Virtual Gradient is defined by computational graph and automatic differentiation. The method is computationally efficient and has little memory requirements. We also analyze the theoretical convergence properties and implementation of the algorithm. Experimental results on multiple datasets and network models show that SVGD has advantages over other stochastic optimization methods.
Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and teste…
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Massachusetts Institute of Technology（麻省理工学院）
Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the particular structure of the Gram matrix induced by the neural network architecture. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm. We further extend our …
It has been shown that gradient descent can yield the zero training loss in the over-parametrized regime (the width of the neural networks is much larger than the number of data points). In this work, combining the ideas of some existing works, we investigate the gradient descent method for training two-layer neural networks for approximating some target continuous functions. By making use the generic chaining technique from probability theory, we show that gradient descent can yield an exponential convergence rate, while the width of the neural networks needed is independent of the size of th…
Deep learning models are often successfully trained using gradient descent, despite the worst case hardness of the underlying non-convex optimization problem. The key question is then under what conditions can one prove that optimization will succeed. Here we provide a strong result of this kind. We consider a neural net with one hidden layer and a convolutional structure with no overlap and a ReLU activation function. For this architecture we show that learning is NP-complete in the general case, but that when the input distribution is Gaussian, gradient descent converges to the global optimu…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
Recent years have seen increased interest in performance guarantees of gradient descent algorithms for non-convex optimization. A number of works have uncovered that gradient noise plays a critical role in the ability of gradient descent recursions to efficiently escape saddle-points and reach second-order stationary points. Most available works limit the gradient noise component to be bounded with probability one or sub-Gaussian and leverage concentration inequalities to arrive at high-probability results. We present an alternate approach, relying primarily on mean-square arguments and show t…
The massive size of modern neural networks has motivated substantial recent interest in neural network quantization. We introduce Stochastic Markov Gradient Descent (SMGD), a discrete optimization method applicable to training quantized neural networks. The SMGD algorithm is designed for settings where memory is highly constrained during training. We provide theoretical guarantees of algorithm performance as well as encouraging numerical results.
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…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We analyze the properties of gradient descent on convex surrogates for the zero-one loss for the agnostic learning of linear halfspaces. If $\mathsf{OPT}$ is the best classification error achieved by a halfspace, by appealing to the notion of soft margins we are able to show that gradient descent finds halfspaces with classification error $\tilde O(\mathsf{OPT}^{1/2}) + \varepsilon$ in $\mathrm{poly}(d,1/\varepsilon)$ time and sample complexity for a broad class of distributions that includes log-concave isotropic distributions as a subclass. Along the way we answer a question recently posed b…
We propose a projected semi-stochastic gradient descent method with mini-batch for improving both the theoretical complexity and practical performance of the general stochastic gradient descent method (SGD). We are able to prove linear convergence under weak strong convexity assumption. This requires no strong convexity assumption for minimizing the sum of smooth convex functions subject to a compact polyhedral set, which remains popular across machine learning community. Our PS2GD preserves the low-cost per iteration and high optimization accuracy via stochastic gradient variance-reduced tech…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper establishes risk convergence and asymptotic weight matrix alignment --- a form of implicit regularization --- of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail, for gradient flow applied to strictly decreasing loss functions (with similar results for gradient descent with particular decreasing step sizes): (i) the risk converges to 0; (ii) the normalized i-th weight matrix asymptotically equals its rank-1 approximation $u_iv_i^{\top}$; (iii) these rank-1 matrices are aligned across layers, meaning $|v_{i+1}^{\top}u_…
College of Computer, National University of Defense Technology（国防科技大学计算机学院）；Scientific Computing & Imaging Institute, University of Utah（犹他大学科学计算与成像研究所）
The stability and generalization of stochastic gradient-based methods provide valuable insights into understanding the algorithmic performance of machine learning models. As the main workhorse for deep learning, stochastic gradient descent has received a considerable amount of studies. Nevertheless, the community paid little attention to its decentralized variants. In this paper, we provide a novel formulation of the decentralized stochastic gradient descent. Leveraging this formulation together with (non)convex optimization theory, we establish the first stability and generalization guarantee…
We study convergence properties of Stochastic Gradient Descent (SGD) for convex objectives without assumptions on smoothness or strict convexity. We consider the question of establishing that with high probability the objective evaluated at the candidate minimizer returned by SGD is close to the minimal value of the objective. We compare this result concerning the final candidate minimzer (i.e. the final model parameters learned after all gradient steps) to the online learning techniques of [Zin03] that take a rolling average of the model parameters at the different steps of SGD.
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…
McGill University（麦吉尔大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
In this paper, we propose an implicit gradient descent algorithm for the classic $k$-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent (Entropy-SGD) for improving the training of deep neural networks. Numerical experiments on various synthetic and real datas…
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…
Google Research（谷歌研究院）；Weizmann Institute of Science（魏茨曼科学研究所）
We study the iteration complexity of stochastic gradient descent (SGD) for minimizing the gradient norm of smooth, possibly nonconvex functions. We provide several results, implying that the $\mathcal{O}(ε^{-4})$ upper bound of Ghadimi and Lan~\cite{ghadimi2013stochastic} (for making the average gradient norm less than $ε$) cannot be improved upon, unless a combination of additional assumptions is made. Notably, this holds even if we limit ourselves to convex quadratic functions. We also show that for nonconvex functions, the feasibility of minimizing gradients with SGD is surprisingly sensiti…
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…
Consider the problem of minimizing functions that are Lipschitz and strongly convex, but not necessarily differentiable. We prove that after $T$ steps of stochastic gradient descent, the error of the final iterate is $O(\log(T)/T)$ with high probability. We also construct a function from this class for which the error of the final iterate of deterministic gradient descent is $Ω(\log(T)/T)$. This shows that the upper bound is tight and that, in this setting, the last iterate of stochastic gradient descent has the same general error rate (with high probability) as deterministic gradient descent.…
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.
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…
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…
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…
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…
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…
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）
