Figure 1. However, it has been recently found that this method, despite its superior empirical performance, may diverge (Reddi et al., 2018). The first thing we do is to... A Common Mistake. neural networks, a.k.a. May 6, 2019 | Written by: IBM Research Editorial Staff. Ghadimi, Saeed, and Guanghui Lan. The Loss Function. The above examples also imply the conditions we derived are essential and “tight”, in the sense that violating them can make an Adam-type algorithm diverge. Intro to optimization in deep learning: Gradient Descent . Optimization is a critical component in deep learning. The bias condition is a new condition introduced by adaptive learning rates. “Stochastic first-and zeroth-order methods for nonconvex stochastic programming.” SIAM Journal on Optimization 23.4 (2013): 2341-2368. We provide a few conditions that guarantee convergence and show these conditions are “tight”, in the sense that we construct specific examples showing that Adam-type algorithms diverge if they violate our conditions. By using our conditions to monitor divergence behavior of Adam, we found the divergence of Adam is exactly due to violation of our bias or variance conditions. This article provides an overview of optimization algorithms and theory for training neural networks. We derived a set of sufficient conditions that can guarantee the convergence of Adam-type algorithms.
Limiting this quantity is a generalization of square-summable stepsizes in SGD. “On the convergence of adam and beyond.” In International Conference on Learning Representations, 2018. We think optimization for neural networks is an interesting topic for theoretical research due to various reasons. Our work builds the theory to understand the behavior for a generic class of adaptive gradient methods for non-convex optimization. In Figures 1 and 2, we show the growth of quantities (a)-(c), which further determine the variance and bias condition for the convergence of Adam-type methods. Zeyu Zhu, Nan Li, Ruoyu Sun, Huijing Zhao, Donghao Xu: Off-road Autonomous Vehicles Traversability Analysis and Trajectory Planning Based on Deep Inverse Reinforcement Learning… While well-known stochastic gradient descent (SGD) takes steps in a given direction based on the gradient alone, Adam-type optimization algorithms look at previous updates to take into account the momentum and use an adaptive learning rate. Figure 2. Let us, for sake of simplicity, let us assume our network has only two parameters. When and why can a neural network be successfully trained? Thus, as a byproduct of our analysis, we established the first convergence result of AdaGrad and AMSGrad in non-convex optimization, which can be of independent interest. In practice, this... Gradient Descent. Ghadimi, Saeed, and Guanghui Lan. Optimization for deep learning: theory and algorithms posted @ 2020-01-09 20:43 微笑sun 阅读( 1336 ) 评论( 0 ) 编辑 收藏 刷新评论 刷新页面 返回顶部 When we use gradient-based algorithms in optimization, we want the decreases in objective predicted by gradient directions to dominate increases brought by other factors. big data optimization), nonconvex optimization in machine learning (e.g. The current focus of speech technology research at IBM Research AI is around Spoken […].
“On the convergence of a class of adam-type algorithms for non-convex optimization.” In International Conference on Learning Representations, 2019. “Adaptive subgradient methods for online learning and stochastic optimization.” Journal of Machine Learning Research 12 Jul (2011): 2121-2159. In Figure 2, AMSGrad converges but Adam diverges because of violating the bias condition.
说起BN层，基本上都知道是什么，BN层可以看作和卷积层，池化层等类似的神经网络中的一层。上面说到对训练数据归一化是可以加快收敛的，在神经网络中每一层网络都会接受上一层的输入，而上一层的输入大多数时候都会发生分布偏移的现象，也就是在深层隐层中的输入可能已经不服从(0, 1)的正态分布了。BN层在这里的作用就是将每一层的输出都从新归一化然后输入到下一层，这种方式可以加快网络的收敛速度，除此之外还可以缓解梯度消失的问题。, BN虽说好用，但也有他的一些问题，以及应用的局限性，BN是作用在一个batch上的，当batch size较小时容易引入噪声，在求均值和方差时会不准确，另外就是在预测时无法计算均值和方差，只能使用训练时保存的滑动平均值。除了上面的缺点，BN因为是在batch size个样本，单个channel上计算的，对于NLP中的应用就受限了，因为每个句子的长度不一样，而对于长度较长的句子，后面的词向量的归一化就只有几个甚至一个词可以使用。由于这些原因也就衍生出了其他的normalization的方法，总体来说可以分为两种，一种是和BN一样的对输入归一化，例如layer normalization，LN也是在NLP中用的最多的归一化方法。另一种就是直接对权重归一化，如weight normalization。但感觉这种方法用的很少，应该效果不太好。, 加深网络的深度可以提升网络的性能，这一点已经被太多例子给证明，而且从网络的深度和宽度来看，之前周志华老师一次报告中就提到过加深网络比加宽网络的收益更高。这一点在CNN中已经被证明，但在RNN中并没有这样的情况，Transformer应该有这样的迹象。, 像ResNet，Highway Network等的设计都是为了加深网络，而ResNet也实现了很深的网络层，广泛在CNN，以及Transformer等网络中使用。另外最近比较火的NAS也可以去做这个事情，不一定是寻求更深的网络结构，但是可以寻求更合理的网络结构。而从resnet来看，resnet学习到的这种恒等映射行为至少能保证更深的网络不会比更浅的网络差，此外在反向传播过程中因为这样一个非乘积的操作存在，也可以一定程度缓解梯度消失，梯度爆炸的问题，而梯度消失和梯度爆炸是深层网络中最容易出现的问题。, 也有人在训练一些超深的网络（比如大于1000层），此时你需要一个很好的初始化参数，需要用到resnet和BN层。除此之外，你可能还需要从数据预处理，优化算法，激活函数，正则化等方面去下功夫。, 5，General Algorithms for Training Neural Networks, 一个好的优化算法需要保证两点：一是收敛速度够快，二是能收敛到更优的点，也就是模型在我们给定的性能指标上表现更好。这两点之间并没有太多连续，也就是说收敛更快并不一定能导致收敛更好。接下来就讨论下相关优化算法。, 上面的式子中$F(\theta)$就是一个batch上计算得到的损失函数，在第$t$次迭代更新中，利用SGD优化算法，可以将式子写成：, 一般来说我们都是在batch上做梯度下降，batch是你的训练样本中的一小部分（通常取32， 64， 128），尽量保持每次取得batch不一样。, Reasons for SGD: memory constraint and faster convergence, 为什么使用SGD而不是GD，从内存上来看，单机很难讲整个数据集一起处理，只能处理batch size个数据集。而且SGD从收敛速度上也比GD更快。, 学习速率通常是一个较小的值，用来控制每次梯度下降的量，有时候会用一个常量，但由时候又会根据迭代的次数去减小学习速率，因为在靠近收敛阶段，我们希望模型能降低下降的速度，防止跳过了最小值点。, warmup这个东西是在现在这些较深的网络，数据量较大的时候，而又使用较小的batch size的时候有效的。为什么这么说呢？首先来看下warmup是怎么工作的，公式如下：, $lr_{warmup} = lr * (global step / warmup step)$, 在训练步数小于$warmup step$时，采用上面的学习速率，超过这个步数就采用$lr$。从这里可以看到一开始的学习速率是一个非常小的值，假设$warmup step = 1000$，一开始的学习速率就是你设置的0.001倍。这样的做法是因为一开始初始化的参数离真实分布较远，而采用mini-batch的方式训练，每个batch中样本的方差较大，如果一开始的学习速率比较大，参数的分布容易跑偏，而且一旦跑偏了，在现在这种参数比较多的深度神经网络中就很难再调回来。因此才需要在一开始给一个很小的学习速率，让模型慢慢调整参数，当参数调整的类似真实分布时，再给大一点的学习速率加快模型的收敛。从上面分析中也可以看到，当batch size很大时，或者模型很小时，就不太需要warmup。（注意这个batch size是相对训练集大小而言的，没有绝对的大或小）。, 这种学习速率用的比较少，通常要么固定学习速率，要么让学习速率衰减，但这种是让学习速率在一个范围内循环，也就是说学习速率会下降然后突然上升，再接着下降。这样的话就可以在模型陷入鞍点等地方能更容易地跳出来。, Constant v.s.
Both conditions admit simple and intuitive explanations. Let me digress for a moment. Join us at ICLR to hear more about our work.
In Figure 1, Adam diverges due to violation of the variance condition while other algorithms converge. The Loss Function. We also explore the question of whether these algorithms can converge for non-convex problems under the aforementioned conditions and provide an affirmative answer. Non-Convex Optimization in Deep Learning H umans have been enjoying convex optimization (CO) for many years compared to the few contexts where they had to deal with non …, lightspeed a learning company interview questions, qualified medication aide training indiana, district 214 continuing education classes, triathlon training groups in portland oregon, security forces training reference guides. Divergence due to oscillation of effective stepsizes. Formally, an Adam-type algorithm is of the following form: where x is the optimization variable, gt is a stochastic gradient at step t, β1,t is a non-increasing sequence, and ht is an arbitrary function that outputs a vector having the same dimension as x.
In other words, Adam-type methods will converge to a stationary point with error in the order of 1/√T after running T iterations.
Let us, for sake of simplicity, let us assume our network has only two parameters. Our recent work “On the convergence of a class of Adam-type algorithms for non-convex optimization” (authors: Xiangyi Chen (UMN), Sijia Liu (IBM), Ruoyu Sun (UIUC), Mingyi Hong (UMN)) gives a rather comprehensive answer to the above question. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. In Adam-type algorithms, three factors can lead to increases in objective: variance on gradient, higher-order curvature, and “skewed” update direction. We think optimization for neural networks is an interesting topic for theoretical research due to... Toggle navigation. Published: 13 June 2020. by Springer Science and Business Media LLC. Based on quantities (a)-(c), the convergence can then be guaranteed when the following two conditions are satisfied: 1) variance condition: b/a goes to 0 as T goes to infinity; 2) bias condition: c/a goes to 0 as T goes to infinity. Email: ruoyus@illinois.edu. Intuitively, conditions involve three quantities: (a) the accumulation of effective stepsizes; (b) the accumulation of squared effective stepsizes; (c) the accumulation of oscillation of effective stepsizes. Surprisingly, we found the set of conditions are automatically satisfied by SGD, AdaGrad, and AMSGrad. Ruoyu Sun Department of Industrial and Enterprise Systems Engineering (ISE), and affiliated to Coordinated Science Laboratory and Department of ECE, University of Illinois at Urbana-Champaign, Urbana, IL.
We’ll present on Wednesday, May 8, during the poster session from 4:30 to 6:30 pm in Great Hall BC. 当你想训练好一个神经网络时，你需要做好三件事情：一个合适的网络结构，一个合适的训练算法，一个合适的训练技巧：, 合适的网络结构：包括网络结构和激活函数，你可以选择更深的卷积网络，然后引入残差连接。可以选择relu做为激活函数，也可以选择tanh，swish等。, 合适的训练算法：通常采用SGD，也可以引入动量和自适应学习速率，也许可以取得更好的效果。, 一般的监督式机器学习可以归结为Representation，optimization 和 generalization 。, Generalization：将训练好的模型用来预测测试集，测试集得到的误差称为测试误差，而测试误差又可以分解为表征误差，优化误差和泛化误差。, 一般来说上面三个问题都是单独分开研究的。Optimization是一个复杂的问题，想研究清楚，还需要进一步分解，可以分解为三个方面：能收敛到一个平衡点，收敛的速度足够快，收敛的平衡点是全局最优点。根据这三个方面也就衍生除了一系列的问题：, 按照上面的可以归纳为：收敛问题，包括梯度弥散，梯度爆炸等；收敛速度问题；全局最优问题，包括局部最小值，陷入平坦地区，鞍点等。, 上面的划分并不是完全准确的，因为这三类问题的边界是比较模糊的，下面会就这三个方面去阐述。, 上面式子中的$\phi$是激活函数，通常无法直接映射到$y_i$，而是映射到${\hat{y}}_i$，但我们希望两者之间靠的很近，因此：, 3，Gradient Descent: Implementation and Basic Analysis, ${\eta} _ t$是学习速率（学习步长），$\nabla F({\theta} _ t)$是梯度值，梯度值通常通过BP（反向传播）算法来求得。而且理论上说在迭代时给定一定的约束下（小的学习速率），梯度下降是会收敛的。, 4.1 Possible Slow Convergence Due to Explosion/Vanishing, 训练神经最困难的问题可能是梯度消失和梯度爆炸。常用的梯度下降可以看作是一种反馈校正机制，将网络输出层的误差反向传播到前面的层，调整前面层的权重来减小误差。在这个传递的过程中，梯度可能在每一层被放大或缩小，从而引起梯度爆炸和梯度消失。举一个一维损失函数的例子：, 上面式子中的$e$是误差，当所有的$w_j = 2$，梯度值会爆炸，当所有的$w_j = 0.5$，梯度会消失。, 在定义域$[-1 + c, 1 - c]$之间是一块平坦区域，在这里的梯度几乎接近于零，当神经网络训练时进入到这块区域，就会发生梯度消失，而模型需要很漫长的时间才可能达到最小值点$x=1$。而在两侧的梯度又很大，会引起梯度爆炸，不过梯度爆炸已经不是什么问题，梯度截断能很好的解决这个问题。对于这个问题当你初始化$w$的值在1的附近时，你很容易抵达最小值点，但是当你初始化在其他值，比如$w=-1$，此时你需要迭代很久才能到达最小值。另外在神经网络中像极小值点，鞍点，陷入这些点的概率是非常非常低的，尤其是高维空间下，反倒是这类平坦区域才是你通常容易遇到的地方。, 作者在这里讨论的梯度消失和梯度爆炸问题都只限于前向网络和卷积网络，不涉及到RNN，因为本质上来看RNN的梯度消失和前向网络的梯度消失原理并不一样，而且难度也更大。从上面也看到一个好的初始化是可以有效的解决这个问题，那么什么是好的初始化？, 梯度爆炸和梯度消失区域是确实存在的，而且会占据很大一部分区域，因此一个好的初始化是有可能避开这些区域的。, 我们可以将初始化值全部置为0或者个别非0，其余全为0的稀疏值，又或者是随机无规律的初始值，这种初始化方法通常效果较差，大多数时候都不work。, 上面三种初始化方法可以见神经网络中的权值初始化方法，我做了一些实验来描述三者的特性。, 归一化可以看作是初始化的扩展，只不过归一化不是作用在初始点，而是作用在后续迭代阶段参数的分布。而归一化最常见的就是batch normalization。除此之外还有layer normalization，instance normalization等等。通常对训练数据我们也会做归一化。, 对训练数据的归一化是非常重要的，吴恩达在他的机器学习课程中就讲过这个问题，如果两个特征的值差得非常大的情况下，模型收敛是非常慢的，而归一化可以加快模型收敛的速度。从理论上来讲数据归一化可以减小海森矩阵的条件数，加快梯度下降的收敛速度。. Choosing the right optimization algorithm for deep learning models is critical to achieve good results in a limited time budget.
When we initialize our weights, we are at point A in the loss landscape. Duchi, John, Elad Hazan, and Yoram Singer. Optimization for deep learning: theory and algorithms . Second, classical optimization theory is far from enough to explain many phenomena. INTERSPEECH is the world’s largest conference devoted to speech processing and applications, and is the premiere conference of the International Speech Communication Association.
First, its tractability despite non-convexity is an intriguing question and may greatly expand our understanding of tractable problems. The variance condition can quantify the possible increase in objective brought by the variance on gradients and the higher-order curvature during optimization. In collaboration with researchers from Harvard Medical School, Mt. “On the convergence of a class of adam-type algorithms for non-convex optimization.” In International Conference on Learning Representations, 2019. In particular, we provide mild sufficient conditions that guarantee the convergence for the Adam-type methods. IBM Research’s Project Debater team worked with the producers of “That’s Debatable” to integrate AI into the traditional debate format.
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